2.1 Dimension of measures and Hausdorff content

Definition 2.1. Let $\zeta :\mathbb {R}^{+}\mapsto \mathbb {R}^+$ . Suppose that $\zeta $ is increasing in a neighborhood of $0$ and $\zeta (0)=0$ . The Hausdorff outer measure at scale $t\in (0,+\infty ]$ associated with the gauge $\zeta $ of a set E is defined by

(2) $$ \begin{align} \mathcal{H}^{\zeta}_t (E)=\inf \bigg\{\!\sum_{n\in\mathbb{N}}\zeta (\vert B_n\vert) : \, \vert B_n \vert \leq t, \ B_n \text{ closed ball and } E\subset \bigcup_{n\in \mathbb{N}}B_n\bigg\}. \end{align} $$

The Hausdorff measure associated with $\zeta $ of a set E is defined by

(3) $$ \begin{align} \mathcal{H}^{\zeta} (E)=\lim_{t\to 0^+}\mathcal{H}^{\zeta}_t (E). \end{align} $$

For $t\in (0,+\infty ]$ , $s\geq 0$ , and $\zeta :x\mapsto x^s$ , one simply uses the usual notation $\mathcal {H}^{\zeta }_t (E)=\mathcal {H}^{s}_t (E)$ and $\mathcal {H}^{\zeta } (E)=\mathcal {H}^{s} (E)$ , and these measures are called s-dimensional Hausdorff outer measure at scale $t\in (0,+\infty ]$ and s-dimensional Hausdorff measure, respectively. Thus,

(4) $$ \begin{align} \mathcal{H}^{s}_{t}(E)=\inf \bigg\{\!\sum_{n\in\mathbb{N}}\vert B_n\vert^s : \, \vert B_n \vert \leq t, \ B_n \text{ closed ball and } E\subset \bigcup_{n\in \mathbb{N}}B_n\bigg\}. \end{align} $$

The quantity $\mathcal {H}^{s}_{\infty }(E)$ (obtained for $t=+\infty $ ) is called the s-dimensional Hausdorff content of the set E.

Definition 2.2. Let $\mu \in \mathcal {M}(\mathbb {R}^d)$ . For $x\in {\mathrm {supp}}(\mu )$ , the lower and upper local dimensions of $\mu $ at x are defined as

$$ \begin{align*} \underline\dim_{\mathrm{loc}}(\mu,x)=\liminf_{r\rightarrow 0^{+}}\frac{\log(\mu(B(x,r)))}{\log(r)} \quad\text{and} \quad \overline\dim_{\mathrm{loc}}(\mu,x)=\limsup_{r\rightarrow 0^{+}}\frac{\log (\mu(B(x,r)))}{\log(r)}. \end{align*} $$

Then, the lower and upper Hausdorff dimensions of $\mu $ are defined by

(5) $$ \begin{align} \underline{\dim}_H(\mu)={\mathrm{ess\,inf}}_{\mu}(\underline\dim_{\mathrm{loc}}(\mu,x)) \quad \text{and} \quad \overline{\dim}_P (\mu)={\mathrm{ess\,sup}}_{\mu}(\overline\dim_{\mathrm{loc}}(\mu,x)), \end{align} $$

respectively.

It is known (for more details, see [Reference Falconer16]) that

$$ \begin{align*} \begin{split} \underline{\dim}_H(\mu)&=\inf\{\dim_{H}(E): E\in\mathcal{B}(\mathbb{R}^d),\, \mu(E)>0\}, \\ \overline{\dim}_P (\mu)&=\inf\{\dim_P(E): E\in\mathcal{B}(\mathbb{R}^d),\, \mu(E)=1\}. \end{split} \end{align*} $$

When $\underline \dim _H(\mu )=\overline \dim _P(\mu )$ , this common value is simply denoted by $\dim (\mu )$ and $\mu $ is said to be exact dimensional.

2.2 Weakly conformal IFS

2.2.1 Generalities about contracting IFS

Let $m\geq 2$ be an integer. An IFS is a set $S=\{f_1,\ldots ,f_m\}$ of mappings $f_i:X \to X$ , where $X\subset \mathbb {R}^d$ is a closed set. Moreover, one says that f is differentiable on X if there exists an open set $U\supset X$ on which f is differentiable.

Given an open set $U \subset \mathbb {R}^d$ and $f:U\to \mathbb {R}^d$ a differentiable map, for any $x\in U$ :

• • $f'(x)$ is the differential of f at x;

• • let $k\in \mathbb {N}$ be an integer, we write $\mathcal {L}(\mathbb {R}^k,\mathbb {R}^d)$ the set of linear maps from $\mathbb {R}^k$ to $\mathbb {R}^d.$ For $\ell \in \mathcal {L}(\mathbb {R}^k,\mathbb {R}^d)$ , one denotes

  (6) $$ \begin{align} \| \ell\| =\max_{x\in\mathbb{R}^k \neq 0}\frac{\|\ell(x)\|_{\infty}}{ \| x\|_{\infty}}\quad\text{and}\quad \|{\ell}\| \ =\min_{x\in\mathbb{R}^k \neq 0}\frac{\|\ell(x)\|_{\infty}}{ \| x\|_{\infty}}. \end{align} $$

Let us recall the following result.

Proposition 2.3. (Hutchinson [Reference Hutchinson25])

Let $m \geq 2$ be an integer, $X\subset \mathbb {R}^d$ a closed set, and $S=\{f_1,\ldots ,f_m\}$ a system of $C^1$ maps from X to $X.$ Assume that S is uniformly contracting, i.e.,

$$ \begin{align*}\max_{1\leq i\leq m}\sup_{x\in X}\| f^{\prime}_i (x)\|<1. \end{align*} $$

Then, there exists a unique non-empty compact set K satisfying

$$ \begin{align*}K=\bigcup_{1\leq i\leq m}f_i (K).\end{align*} $$

Moreover, for any $(p_1,\ldots ,p_m)\in (0,1)^m$ , there exists a unique measure $\mu \in \mathcal {M}(\mathbb {R}^d)$ supported on K satisfying

(7) $$ \begin{align} \mu=\sum_{1\leq i \leq m}p_i\mu(f_{i}^{-1}(\cdot)). \end{align} $$

From now on, an IFS designates a uniformly contracting system of $C^1$ maps.

The following notation is used throughout the manuscript.

• • $\Lambda (S)=\{1,\ldots ,m\}$ and $\Lambda (S)^{*}=\bigcup _{k\geq 0}\Lambda (S)^k$ . When there is no ambiguity on the system S involved, one simply writes $\Lambda (S)=\Lambda .$

• • $K_{S}$ denotes the attractor of S (or simply K when the context is clear).

• • For $\underline {i}=(i_1,\ldots ,i_k)\in \Lambda ^{k}$ , the cylinder $[\underline {i}]$ is defined by

  $$ \begin{align*}[\underline{i}]=\{(i_1,\ldots,i_k,x_1,x_2,\ldots):(x_1,x_2,\ldots)\in\Lambda^{\mathbb{N}}\}.\end{align*} $$

  Moreover, if $(\alpha _n)_{n\in \mathbb {N}}$ is a sequence of real numbers, one sets

  $$ \begin{align*}\alpha_{\underline{i}}=\alpha_{i_1}\times \cdots \times \alpha_{i_k}\end{align*} $$
  and
  $$ \begin{align*}f_{\underline{i}}=f_{i_1}\circ\cdots \circ f_{i_k}.\end{align*} $$
  For example, given the probability vector $(p_1,\ldots ,p_m), p_{\underline {i}}=p_{i_1}\times \cdots \times p_{i_k}.$

• • The set $\Lambda ^{\mathbb {N}}$ is endowed with the topology generated by the cylinders. The set of probability measures on the Borel sets with respect to this topology is denoted $\mathcal {M}(\Lambda ^{\mathbb {N}})$ .

• • The shift operator $\sigma :\Lambda ^{\mathbb {N}}\to \Lambda ^{\mathbb {N}}$ is defined for any $(i_1,i_2,\ldots )\in \Lambda ^{\mathbb {N}}$ by

  (8) $$ \begin{align} \sigma((i_1,i_2,\ldots))=(i_2,i_3,\ldots). \end{align} $$

• • The canonical projection of $\Lambda ^{\mathbb {N}}$ on K will be denoted $\pi _{ \Lambda }$ (or simply $\pi $ when there is no ambiguity) and, fixing any $x\in K,$ is defined, for any $(i_1,i_2,\ldots .)\in \Lambda ^{\mathbb {N}}$ , by

  (9) $$ \begin{align} K\ni\pi((i_1,\ldots))=\lim_{k\rightarrow+\infty}f_{i_1}\circ\cdots \circ f_{i_k}(x). \end{align} $$
  It is easily verified that $\pi $ is independent of the choice of $x.$

2.2.2 Weakly conformal IFS and pressure function associated with weakly conformal IFSs

Let us recall the definition of a weakly conformal map.

Definition 2.4. [Reference Feng and Hu19]

Let $m \geq 2$ be an integer, $U\subset \mathbb {R}^d$ an open set, $S=\{f_i\}_{i=1}^m$ , where $f_i:U\to U$ is a $C^{1}$ contraction and K the attractor of S.

One says that S is weakly conformal when, recalling equation (6),

(10) $$ \begin{align} \lim_{k\rightarrow+\infty}\frac{\sup_{(x_i)_{i\in\mathbb{N}}\in\{1,\ldots,m\}^{\mathbb{N}}}(\log \| f_{(x_1 ,\ldots,x_k)}^{\prime}(\pi (\sigma^k (x)))\|-\log \| f_{(x_1 ,\ldots,x_k)}^{\prime}(\pi (\sigma^k (x)))\| )}{k}=0. \end{align} $$

In this case, a measure defined by equation (7) is called a weakly conformal measure.

Example 2.5.

• • If the maps $f_1 ,\ldots ,f_m$ are affine similarities or conformal maps (i.e., verify $\| f^{\prime }(x)(y)\|=\| f^{\prime }(x)\| \cdot \| y\|$ for every $x\in U,y\in \mathbb {R}^d$ ), the system $S=\{f_1,\ldots ,f_m\}$ is weakly conformal. In this case, the IFS is called self-similar or self-conformal and the measures satisfying equation (7) are respectively called self-similar and self-conformal measures. Note that this class of IFSs contains, for instance, every system of holomorphic contracting mappings.

• • Assume that for any $1\leq i\leq m, f_i: \mathbb {R}^d \to \mathbb {R}^d$ is defined by $f_i(x)=A_i x +b_i ,$ where for any $1\leq i\leq m$ , $b_i \in \mathbb {R}^d$ and $A_i \in GL_d (\mathbb {R})$ has its eigenvalues equal in modulus to $0<r_i<1$ , and for any $1 \leq i,j \leq m$ , $A_i A_j=A_j A_i$ . Then, $S=\{f_1,\ldots ,f_m\}$ is weakly conformal.

The pressure function associated with a weakly conformal IFS is naturally related to the dimension of the attractor. It is defined by the following proposition.

Proposition 2.6. Let $m\geq 2$ be an integer, $S=\{f_1 ,\ldots ,f_m\}$ be a $C^{1}$ weakly conformal IFS, and let K be its attractor.

Let us fix $s\geq 0$ and $z\in K.$ The following quantity is well defined and independent of the choice of $z:$

(11) $$ \begin{align} P_z(s)=\lim_{k\rightarrow+\infty}\frac{1}{k}\log \sum_{\underline{i}\in \Lambda^k}\| f_{\underline{i}}^{\prime}(z)\|^s. \end{align} $$

Remark 2.7. Let us mention that the proof of Proposition 2.6 is very standard and does not diverge much from the standard proofs made in [Reference Bowen8, Reference Ruelle34], but strictly speaking, due to the weakly conformal settings, unfortunately, one cannot recover the result from these references. So, for the sake of completeness, a proof is given in §4.

Since $P_z(s) $ does not depend on z, one writes

$$ \begin{align*}P_z (s)=P(s)=\lim_{k\rightarrow+\infty}\frac{1}{k}\log\bigg(\!\sum_{\underline{i}\in\Lambda^k}\vert f_{\underline{i}}(K)\vert ^s\bigg).\end{align*} $$

As said above, the pressure function is naturally connected to the dimension of the attractor K associated with the underlying IFS. More precisely, the following quantity is a natural candidate to be the Hausdorff dimension of K.

Definition 2.8. Let $m\geq 2$ be an integer. Let $S=\{f_1 ,\ldots ,f_m\}$ be a $C^{1}$ weakly conformal IFS and K its attractor.

The unique real number $\dim (S)$ satisfying $P(\dim (S))=0$ is called the conformality dimension of S.

Remark 2.9. If the mappings $f_1 ,\ldots ,f_m$ are affine similarities, then the conformality dimension is called the similarity dimension. It is the real number solution to

(12) $$ \begin{align} \sum_{i=1}^{m}c_i ^s=1, \end{align} $$

where $c_i$ is the contraction ratio of $f_i$ (that is, $c_i=\| f^{\prime }_i \|$ ).

3.1 Dimension of weakly conformal dynamical Diophantine set

Our main result is the following.

Theorem 3.1. Let $m\geq 2$ be an integer. Let U be an open set and let $S=\{f_1:U\to U, \ldots ,f_m:U\to U\}$ be a $C^{1}$ weakly conformal IFS with attractor K. For every $\delta>0,$ set

(13) $$ \begin{align} W(x_0,\delta)=\limsup_{\underline{i} \in\Lambda^*}B(f_{\underline{i}}(x_0),\vert f_{\underline{i}}(K)\vert^{\delta}). \end{align} $$

Then, we have the following.

1. (1) For any $x_0\in U,$ for any $\delta <1$ ,

   $$ \begin{align*}W(x_0,\delta)=K.\end{align*} $$

2. (2) Assume in addition that $\dim _H (K)=\dim (S)$ , then for any $x_0\in K$ , for any $\delta \geq 1$ ,

   $$ \begin{align*} \dim_H W(x_0,\delta)=\frac{\dim_H (K)}{\delta}. \end{align*} $$

Let us provide concrete examples of IFSs for which Theorem 3.1 applies and examples of IFS to which the conclusion of Theorem 3.1 does not hold.

• • Theorem 3.1 applies to any self-similar IFS S on $\mathbb {R}$ satisfying the exponential separation condition [Reference Hochman24, Theorem 1.4] or having algebraic contraction ratio and no exact overlaps (i.e., for every $k\in \mathbb {N}$ , every $\underline {i}\neq \underline {j}\in \Lambda ^k, f_{\underline {i}}\neq f_{\underline {j}}$ ) with $\dim (S)\leq 1$ (due to a result of Rapaport [Reference Rapaport31]). For instance, define $g_1,g_2$ , and $g_3$ as three mappings $\mathbb {R}\to \mathbb {R}$ by setting for every $x\in \mathbb {R}$ , $g_1(x)=\tfrac 14x$ , $g_2(x)=\tfrac 14(x+1)$ , and $g_3 (x)=\tfrac 14(x+t)$ , where $t\in \mathbb {R}\setminus \mathbb {Q}.$ Then, Theorem 3.1 applies to $S=\{g_1,g_2,g_3\}$ .

• • Let $m\geq 2$ be an integer and $0<c_1,\ldots ,c_m<1$ be m real numbers satisfying

  $$ \begin{align*}c_1+\cdots +c_m \leq 1.\end{align*} $$
  Then (due to a result of Hochman, [Reference Hochman24]), for Lebesgue-almost every choice of $a_1,\ldots ,a_m \in \mathbb {R}$ , Theorem 3.1 applies to the IFS $S=\{f_1,\ldots ,f_m\}$ where, for every $1\leq i\leq m, f_i (x)=c_i x+a_i$ .

• • It is relatively easy to see that Theorem 3.1 cannot hold when $\dim (S)>d.$ One could take, for instance, so many similarities so that for every $1\leq \delta \leq 2$ ,

  $$ \begin{align*}\dim_H \Big(\!\limsup_{\underline{i}\in\Lambda^{*}}B(f_{\underline{i}}(x_0),\vert f_{\underline{i}}(K)\vert^{\delta})\Big)=d .\end{align*} $$
  To be more explicit, fix t a badly approximable (by rationals) number and define $\phi _1 (x)={x}/{2}$ , $\phi _2(x)=({x+1})/{2}$ , $\phi _3 (x)=({x+t})/{2}$ , $\phi _4(x)=({x+1+t})/{2}$ , and $\Lambda =\{1,\ldots ,4\}$ . Then, it is proved in [Reference Baker2, Theorem 2.10] that for every $x_0\in [0,1+t],$
  $$ \begin{align*}\dim_H\bigg(\limsup_{\underline{i}\in \Lambda^*}B\bigg(\phi_{\underline{i}}(x_0),\frac{1}{4^{\vert \underline{i} \vert}}\bigg)\bigg)=1.\end{align*} $$

In §3.1, we establish that the result applies to a large class of weakly conformal IFSs, namely the weakly conformal IFS satisfying the asymptotically weak separation condition (AWSC).

3.2 An application in the case of homogeneous self-similar IFS

In Theorem 3.1, the choice of the radii of the balls plays an important role on the hypothesis that one needs to assume to be able to estimate $\dim _H W(x,\delta ).$ In the case of homogeneous IFSs, we will be able to treat the general case where the radii are simply given by a mapping $\phi :\mathbb {N}\to \mathbb {R}^+.$ In our settings, we will call a system of affine similarities $S=\{f_i\}_{1\leq i\leq m}$ homogeneous if for every $1\leq i,j\leq m,$

$$ \begin{align*}\| f^{\prime}_i \|=\| f^{\prime}_j \|.\end{align*} $$

Theorem 3.3. Let S be an homogeneous self-similar IFS of common contraction ratio $0<c<1$ . Let $\phi :\mathbb {N}\to \mathbb {R}^+$ be such that $\lim _{n\rightarrow +\infty }\phi (n)=0$ and set

(14) $$ \begin{align} s_{\phi}=\inf\bigg\{s\geq 0 :\sum_{k\geq 0}\sum_{\underline{i}\in\Lambda^k}\phi(k)^{s\dim(S)}<+\infty\bigg\}. \end{align} $$

Assume that $\dim _H K=\dim (S)$ . Then,

$$ \begin{align*}\dim_H W(x_0,\phi):= \dim_H\limsup_{\underline{i}\in\Lambda^*}B(f_{\underline{i}}(x_0),\phi(\vert\underline{i}\vert))=\min\{1,s_{\phi}\}\dim(S).\end{align*} $$

Let us provide a short application to Theorem 3.3. Let $t\in (0,1)$ be a transcendental number and $\mathcal {A}=\{q_1=0,\ldots ,q_N\}\subset \mathbb {Q},$ where $N\geq 2$ satisfies $\log N \leq -\log t$ . Define $S=\{f_1(x)=tx+q_1,\ldots ,f_N(x)=tx+q_N\}.$ It is established in [Reference Rapaport and Varju32] that, denoting K as the attractor of S, one has

$$ \begin{align*}\dim_H K = \dim S =\frac{\log N}{-\log t}.\end{align*} $$

In addition, one has

$$ \begin{align*}\mathcal{P}_{\mathcal{A},t,n}:=\bigg\{P(t)=\sum_{k=0}^n a_i t^i:a_0,\ldots,a_n\in\mathcal{A}\bigg\}=\{f_{\underline{i}}(0),\underline{i}\in\{1,\ldots,m\}^{n+1}\}.\end{align*} $$

So, given $\phi :\mathbb {N}\to \mathbb {R}^+,$ writing

$$ \begin{align*}W_{\mathcal{A},t}(\phi)=\bigg\{x\in\mathbb{R}^d : \vert x-P(t)\vert\leq \phi(\deg P)\text{ for infinitely many }P\in\bigcup_{n\geq 0}\mathcal{P}_{\mathcal{A},n,t} \bigg\}, \end{align*} $$

a direct application of Theorem 3.3 yields that

$$ \begin{align*}\dim_H W_{\mathcal{A},t}(\phi)= \min\{1,s_{\phi}\}\frac{\log N}{-\log t},\end{align*} $$

where $s_{\phi }$ is defined as in equation (14). Finally, we mention that, for the sake of clarity, the present article deals with the case of IFSs without exact overlaps, but one easily could extend the results to the case where the IFS has exact overlaps under suitable assumption (related to the Garcia entropy of the IFS). In particular, $\dim _H W_{\mathcal {A},t}(\phi ) $ can also be estimated for t algebraic. The short note [Reference Daviaud11] has been made available to explain how one should proceed in this case.

3.3 The classical shrinking target problem

As mentioned in §1, thanks to the techniques developed in this article, we are able to extend the dimension results established by Hill and Velani in [Reference Hill and Velani23] with a great deal of generality. Let us explain first why one needs a different hypothesis to extend the result of [Reference Hill and Velani23] than in Theorem 3.1. Reading the rest of the article, the reader will find that the proof of Theorem 3.1 is almost completely geometrical (that is, the proof is essentially non-symbolic). The combination of this fact and the fact that establishing dimension results in the context of self-similar IFSs is, in general, difficult and relies on deep ideas (see [Reference Feng and Hu19, Reference Hochman24, Reference Rapaport31, Reference Rapaport and Varju32, Reference Shmerkin36]), which makes it somewhat surprising that Theorem 3.1 holds under the rather weak hypothesis that the similarity dimension and the dimension of the attractor agree. This is completely due to the specific choice of the radii. More precisely, the choice of the necessary hypotheses in Theorem 3.1 are made with regard to the analyzing measures one will need to consider to solve this specific Diophantine problem. In the case of the classical shrinking target problem, these analyzing measures will be Gibbs measures, so the hypothesis one will require to deal with it will be made to ensure that the projections of Gibbs measure have the expected dimension on K. For the sake of simplicity, we will place ourselves in the case where the IFS is conformal and satisfies the so-called bounded distortion property, which we recall.

Definition 3.5. Let $S=\{f_1,\ldots ,f_m\}$ be a conformal IFS. We say that S satisfies the bounded distortion property if there exists $\kappa>0$ such that for every $n\in \mathbb {N},$ for every $\underline {i}\in \{1,\ldots ,m\}^n$ and every $x,y \in K,$ one has

(15) $$ \begin{align} \kappa^{-1}\leq\frac{\| f_{\underline{i}}'(x) \|}{\| f_{\underline{i}}'(y) \|} \leq \kappa. \end{align} $$

Before stating our main theorem, we recall the following result (see for instance [Reference Allen and Bárány1, equation (2.6), p. 7]).

Theorem 3.6. Let $s> 0$ be a real number. Assume that $S=\{f_1,\ldots ,f_m\}$ is conformal and satisfies the bounded distortion property, then there exists a unique ergodic measure $\nu _s \in \mathcal {M}(\Lambda ^{\mathbb {N}})$ satisfying that for every $n\in \mathbb {N}$ and for every $\underline {i}\in \Lambda ^n,$ one has

(16) $$ \begin{align} C^{-1}\leq\frac{\nu_s ([\underline{i}])}{\vert f_{\underline{i}}(K)\vert^s e^{-nP(s)}}\leq C, \end{align} $$

where $C\geq 1$ is a constant independent of $\underline {i}.$ We will write $\mu _s=\nu _s \circ \pi ^{-1}$ as the projection of $\nu _s$ on the attractor of S.

Remark 3.7. By ergodicity, one easily proves that there exists $h_s\geq 0$ such that for $\nu _s$ -almost every $\underline {i}=(i_n)_{n\in \mathbb {N}}\in \{1,\ldots ,m\}^{\mathbb {N}},$ one has

$$ \begin{align*}\lim_{n\to +\infty}\frac{\log \nu_s([i_1,\ldots,i_n])}{\log \vert f_{(i_1,\ldots,i_n)}(K)\vert}=h_s.\end{align*} $$

Our main result regarding the Hausdorff dimension of shrinking targets set for overlapping conformal IFSs is the following.

Theorem 3.8. Let S be a conformal IFS satisfying the bounded distortion property, given by equation (15), and let $\phi :\mathbb {N}\to \mathbb {R}^+$ be a mapping satisfying $\phi (n)\to 0$ . Set

$$ \begin{align*}\alpha=\liminf_{n\to 0}\frac{-\log \phi(n)}{n}\end{align*} $$

and denote s the solution to $P(s)={s\alpha }/{\delta }.$ Assume that $\dim _{H}\!y{\mu }_s =h_s$ and $0<\alpha <+\infty $ , then for every $\delta \geq 1,$ one has

$$ \begin{align*}\dim_H \limsup_{\underline{i}\in\Lambda^*}B(f_{\underline{i}}(x_0),\vert f_{\underline{i}}(K) \vert^\delta\phi(\vert \underline{i}\vert))=\frac{s}{\delta}.\end{align*} $$

To provide a large class of IFSs to which Theorem 3.8 applies, we start by recalling the definition of the so-called multifractal scaling function of a measure. Let $\mu \in \mathcal {M}(\mathbb {R}^d)$ and define

$$ \begin{align*}\Theta_{\mu}(q,r)= \inf\bigg\{\!\sum_{i\in I} \mu(B(x_i,r))^q\bigg\},\end{align*} $$

where the infimum is taken over all countable collections of balls $\{B(x_i,r) \}_{i \in I}$ satisfying the following two properties:

1. (1) $x_i \in {\mathrm {supp}}(\mu )$ for every $i \in I$ ; and

2. (2) $B(x_i,r) \cap B(x_j,r) =\emptyset $ for $i\neq j.$

Then, the multifractal scaling function of $\mu $ is defined as

(17) $$ \begin{align} \tau_{\mu}(q)=\liminf_{r\to 0}\frac{\log (\Theta_{\mu}(q,r))}{ \log r}. \end{align} $$

Such mappings were first considered in the context of turbulence by Frisch and Parisi in [Reference Frisch, Parisi and Ghil21]. The mapping $\tau _\mu $ is often connected to the so-called multifractal spectrum of $\mu $ . More precisely, for $h\geq 0,$ denote by $E_{\mu ,h}$ the set of points of lower $\mu $ -local dimension h, that is,

$$ \begin{align*}E_{\mu,h}=\{x: \underline{\dim}(\mu,x)=h\}.\end{align*} $$

Many natural measures satisfy for every $h\geq 0$ that

$$ \begin{align*}\dim_H E_{\mu,h}=\tau_\mu^*(h),\end{align*} $$

where $\tau _\mu ^*(h)=\inf _{q\in \mathbb {R}}\{qh-\tau _{\mu }(q)\}.$ We refer, for instance, to [Reference Brown, Michon and Peyrière9] for a complete mathematical foundation and to [Reference Feng18, Reference Olsen29, Reference Shmerkin36] for results regarding the multifractal analysis of self-similar measures.

When S is conformal and satisfies the bounded distortion property, it is classical (see [Reference Allen and Bárány1] for instance) that there exists a unique ergodic measure $\nu _0$ on $\{1,\ldots ,m\}^{\mathbb {N}}$ such that for every $k\in \mathbb {N},$ for every $\underline {i}\in \{1,\ldots ,m\}^k$ , one has

(18) $$ \begin{align} C^{-1}\leq\frac{\nu_0([\underline{i}])}{\vert f_{\underline{i}}(K)\vert^{\dim(S)}}\leq C, \end{align} $$

where $C>0$ is independent of $\underline {i}.$ In addition, when the IFS is self-similar, $\nu _0$ is simply the self-similar measure corresponding to the similarity dimension (see Remark 2.9). We now introduce the following condition.

Definition 3.10. Let S be a conformal IFS satisfying equation (15), let K be its attractor, let $\nu _{0}$ be as in equation (18), and set $\mu _0 =\nu _0 \circ \pi ^{-1},$ where $\pi $ is the canonical projection from $\{1,\ldots ,m\}^{\mathbb {N}}$ to K. We say that S satisfies condition $(A)$ if for every $q> 1,$

$$ \begin{align*}\tau_{\mu_0}(q)=(q-1)\dim(S).\end{align*} $$

We mention that condition $(A)$ has been proved by Barral and Feng in [Reference Barral and Feng4] to be equivalent to the so-called asymptotically weak separation condition without exact-overlaps mentioned at the end of §3.1 (see Proposition 4.11 thereafter).

Thus (due to Remark 4.22 below), the following result holds true.

In §4, we recall and establish some geometric properties of a weakly conformal IFS. We also establish Proposition 4.10, which justifies that one can apply Theorem 3.1 to weakly conformal IFS satisfying the AWSC.

The mass transference principle for weakly conformal measures, which is key to prove Theorem 3.1, is established in §5.

In §6, Theorem 3.1 is established.

Section 7 is dedicated to the proof of Theorem 3.3.

4.1 Some general estimates

Let $m \geq 2$ be an integer. One collects some useful geometric results on a $C^{1}$ weakly conformal IFS.

Consider a $C^{1}$ weakly conformal IFS $S=\{f_1 ,\ldots ,f_m\}$ with attractor K and for every $x\in K$ , $k\in \mathbb {N}$ , and $\underline {i}=(i_1,\ldots ,i_k)\in \Lambda ^k$ , write

$$ \begin{align*}c_{\underline{i}}(x)=\| f_{\underline{i}}'(x)\|.\end{align*} $$

Let us recall the following result established as [Reference Feng and Hu19, Lemma 5.4].

Lemma 4.1. [Reference Feng and Hu19]

For any $c>1$ , there exists a constant $D(c)>0$ such that, for every $k\in \mathbb {N}$ , for every $\underline {i}\in \Lambda ^k$ and every $x,y\in K$ ,

(19) $$ \begin{align} D(c)^{-1}c^{-k} \| f_{\underline{i}}^{\prime}(x)\|\cdot\| x-y\|\leq \| f_{\underline{i}}(x)-f_{\underline{i}}(y)\|\leq D(c)c^k \| f_{\underline{i}}^{\prime}(x)\|\cdot\| x-y\| \end{align} $$

(20) $$ \begin{align} D(c)^{-1}c^{-k} \| f_{\underline{i}}^{\prime}(x)\|\leq \vert f_{\underline{i}}(K)\vert\leq D(c)c^k \| f_{\underline{i}}^{\prime}(x)\|. \end{align} $$

Note that for every $k\in \mathbb {N}$ and every $x\in K,$ one has

$$ \begin{align*} c^{ \pm k}\| f_{\underline{i}}^{\prime}(x)\| =\| f_{\underline{i}}^{\prime}(x)\|^{1+({\pm k\log c}/{\log \| f_{ \underline{i}} ^{\prime}(x)\|})}. \end{align*} $$

Moreover, since there exists two constants $C_1, C_2>0$ such that for every $1\leq i\leq m$ and every $x\in K$ ,

$$ \begin{align*} C_1 \leq\| f_{i}'(x)\|\leq\| f_i ' (x)\|\leq C_2,\end{align*} $$

there also exists two constants $0< t_1 \leq t_2$ such that

$$ \begin{align*}t_1\leq \frac{k}{\log \| f_{ \underline{i}} ^{\prime}(x)\|}\leq t_2 .\end{align*} $$

Combining this fact with Lemma 4.1, for any $\theta>0$ , there exists $\widetilde {C}_{\theta }>0$ such that for every $k\in \mathbb {N}$ , every $\underline {i}\in \Lambda ^k$ , and every $x,y\in K$ ,

(21) $$ \begin{align} \widetilde{C}^{-1}_{\theta}c_{\underline{i}}(x)^{1+\theta}\| x-y\|\leq \| f_{\underline{i}}(x)-f_{\underline{i}}(y)\| \leq \widetilde{C}_{\theta}c_{\underline{i}}(x)^{1-\theta}\| x-y\|. \end{align} $$

In particular, there also exists $\widehat {C}_{\theta }$ such that for every $\underline {i}\in \Lambda ^{*}$ and every $x\in K,$ one has

(22) $$ \begin{align} \widehat{C}_{\theta}^{-1}c_{\underline{i}}^{1+\theta}(x)\vert K\vert\leq \vert f_{ \underline{i}}(K) \vert\leq \widehat{C}_{\theta}c_{\underline{i}}^{1-\theta}(x)\vert K\vert. \end{align} $$

Let us remark that equation (22) also implies that there exist $0<\alpha \leq \beta <1$ as well as $C_{\alpha },C_{\beta }>0$ such that, for any $k\in \mathbb {N},$

(23) $$ \begin{align} C_{\alpha}\alpha^k \leq \vert f_{\underline{i}}(K)\vert\leq C_{\beta}\beta^{k}. \end{align} $$

4.2 Proof of Proposition 2.6

As mentioned in §2, the proof of Proposition 2.6 is very standard and does not diverge much from the proof in the conformal case. Unfortunately, one cannot derive directly the result from classical cases (see [Reference Bowen8, Reference Ruelle34], for instance) and those computations in the weakly conformal case do not seem to be written explicitly in the literature so for the seek of completeness, it is done below.

Proof. Assume first that the limit exists in $\mathbb {R}\cup \{-\infty \}$ , and let us show that it is independent of the choice of z and that the limit is $>-\infty $ . Let $ c>1$ be a real number. Recalling Lemma 4.1, for any $k\in \mathbb {N},$ one has

(24) $$ \begin{align} \log\bigg(\!\sum_{\underline{i}\in\Lambda^k}D(c)^{-s}c^{-sk}\vert f_{\underline{i}}(K)\vert ^s\bigg)&\leq \log\bigg(\!\sum_{\underline{i}\in \Lambda^k}\| f_{\underline{i}}' (z)\|^s\bigg)\nonumber\\ &\leq \log\bigg(\!\sum_{\underline{i}\in\Lambda^k}D(c)^s c^{sk}\vert f_{\underline{i}}(K)\vert^s \bigg). \end{align} $$

Since equation (24) holds for any $c>1$ , one gets that

(25) $$ \begin{align} \lim_{k\rightarrow+\infty}\frac{1}{k}\bigg(\log\bigg( \sum_{\underline{i}\in \Lambda^k}\| f_{\underline{i}}^{\prime}(z)\|^s \bigg)-\log\bigg(\!\sum_{\underline{i}\in\Lambda^k}\vert f_{\underline{i}}(K)\vert ^s\bigg)\bigg)=0, \end{align} $$

which proves that this quantity does not depend on z. Moreover, there exists $b>0$ so that for any $k\in \mathbb {N}$ , any $\underline {i}\in \Lambda ^k$ , and any $x\in K$ ,

$$ \begin{align*}\| f_{\underline{i}}'(x)\|\geq b^{k}.\end{align*} $$

This implies that if $P_z(s)$ is well defined, then $P_z(s)>-\infty .$

Let us now prove that the limit exists. For $k\in \mathbb {N},$ write

(26) $$ \begin{align} g_k =\log\bigg(\!\sum_{\underline{i}\in \Lambda^k}\vert f_{\underline{i}}(K)\vert^s \bigg). \end{align} $$

As in the conformal case, the existence of the pressure relies on a sub-additivity argument.

Lemma 4.3. For any $\varepsilon>0,$ there exists a constant $M_{\varepsilon }>0$ such that for any $n,m\in \mathbb {N},$ one has

(27) $$ \begin{align} g_{n+m}\leq M_{ \varepsilon}+m\varepsilon+g_n +g_m. \end{align} $$

Furthermore, any sequence $(g_n)_{n\in \mathbb {N}}$ verifying equation (27) is such that $({g_n}/{n})_{n\in \mathbb {N}}$ converges in $\mathbb {R}\cup \{-\infty \}$ .

Proof. Let us start by proving the second statement. Let $(g_n)_{n\in \mathbb {N}}$ be a sequence satisfying equation (27). Fix $\varepsilon>0$ and $M_{\varepsilon }$ satisfying equation (27). For any $q\in \mathbb {N}^*, b\in \mathbb {N}^*, 0\leq r< q$ , one has

$$ \begin{align*} &g_{bq+r}\leq bg_{q}+g_r +(bq+r) \varepsilon+(b+1)M_{\varepsilon} \\&\quad \Rightarrow \frac{g_{bq+r}}{bq+r}\leq \frac{bq}{bq+r}\cdot \frac{g_q}{q}+\frac{(b+1)M_{\varepsilon}+g_r}{bq+r}+\varepsilon. \end{align*} $$

Fixing q large enough independently of b so that $({(b+1)M_{\varepsilon }}/{bq})\leq \varepsilon $ , for any large $b\in \mathbb {N}^*$ , one has

$$ \begin{align*}\frac{g_{bq+r}}{bq+r}\leq (1+\varepsilon)\frac{g_q}{q}+2\varepsilon.\end{align*} $$

This implies that

$$ \begin{align*}\limsup_{n\rightarrow+\infty}\frac{g_n}{n}\leq (1+\varepsilon)\liminf_{n\rightarrow+\infty}\frac{g_n}{n}+2\varepsilon.\end{align*} $$

Letting $\varepsilon \to 0$ proves the statement.

One now shows that $g_n$ satisfies equation (27).

Let $k\in \mathbb {N}$ and $\underline {i}\in \Lambda ^k.$ Let us begin by the following lemma.

Lemma 4.4. Let $D(c)$ be again defined as in Lemma 4.1, then one has, for any $ \underline {j}\in \Lambda ^{*},$

(28) $$ \begin{align} \tfrac{1}{2}D(c)^{-2}c^{-2k} \vert f_{\underline{i}}(K)\vert\cdot \vert f_{\underline{j}}(K) \vert\leq \vert f_{\underline{i}\underline{j}}(K))\vert\leq 2D(c)^{2}c^{2k} \vert f_{\underline{i}}(K)\vert\cdot \vert f_{\underline{j}}(K) \vert. \end{align} $$

Proof. Let us start by establishing the lower bound. Let $x,y\in K$ be such that

(29) $$ \begin{align} \| f_{ \underline{j}}(x)-f_{\underline{j}}(y)\|\leq\vert f_{\underline{j}}(K)\vert\leq 2\| f_{\underline{j}}(x)-f_{\underline{j}}(y)\|. \end{align} $$

By Lemma 4.1, one has

(30) $$ \begin{align} D(c)^{-1}c ^{-k}\| f_{\underline{i}}^{\prime}(f_{\underline{j}}(x))\|\cdot \| f_{\underline{j}}(x)-f_{\underline{j}}(y)\|\leq \| f_{\underline{i}\underline{j}}(x)-f_{\underline{i}\underline{j}}(y)\|\leq \vert f_{\underline{i}\underline{j}}(K)\vert \end{align} $$

and

(31) $$ \begin{align} \| f_{\underline{i}}^{\prime}(f_{\underline{j}}(x))\|\geq D(c)^{-1}c^{-k}\vert f_{\underline{i}}(K)\vert. \end{align} $$

Combining equations (29), (30), and (31), one obtains

$$ \begin{align*} \tfrac{1}{2}D(c)^{-2}c^{-2k}\vert f_{\underline{i}}(K)\vert \cdot \vert f_{\underline{j}}(K)\vert\leq \vert f_{\underline{i}\underline{j}}(K)\vert. \end{align*} $$

Let us focus now on the upper bound. Let $x,y\in K$ be such that

(32) $$ \begin{align} \| f_{\underline{i}\underline{j}}(x)-f_{\underline{i}\underline{j}}(y)\|\geq \tfrac{1}{2}\vert f_{\underline{i}\underline{j}}(K)\vert. \end{align} $$

Using again Lemma 4.1, one has

(33) $$ \begin{align} \| f_{\underline{i}\underline{j}}(x)-f_{\underline{i}\underline{j}}(y)\|&\leq D(c)c^k \| f_{\underline{i}}'(f_{\underline{j}}(x))\| \cdot \| f_{ \underline{j}}(x)-f_{\underline{j}}(y) \|\nonumber\\&\leq D(c)^2 c^{2k}\vert f_{\underline{i}}(K)\vert \cdot \vert f_{\underline{j}}(K)\vert. \end{align} $$

The upper bound is obtained by combining equations (32) and (33).

By Lemma 4.4, for any $c>1$ and any $n,n^{\prime }\in \mathbb {N},$ one has

$$ \begin{align*} g_{n+n'}= &\,\, \log\bigg(\!\sum_{\underline{i}\in\Lambda^{n+n'}}\vert f_{\underline{i}}(K) \vert^s\bigg)=\log\bigg(\!\sum_{\underline{i}\in \Lambda^n,\underline{j}\in\Lambda^{n'}}\vert f_{\underline{i}\underline{j}}(K)\vert^s\bigg)\\ \leq &\,\, \log\bigg(\!\sum_{\underline{i}\in \Lambda^n,\underline{j}\in\Lambda^{n'}}2^s D(c)^{2s}c^{2sn}\vert f_{\underline{i}}(K)\vert^s \vert f_{\underline{j}}(K)\vert^s\bigg)\\ = &\,\, n \cdot 2s\log(c)+\log(2^s D(c)^{2s})+\log\bigg(\bigg(\!\sum_{\underline{i}\in\Lambda^n}\vert f_{\underline{i}}(K)\vert^s\bigg)\times(\sum_{\underline{j}\in\Lambda^{n^{\prime}}}\vert f_{\underline{j}}(K)\vert^s)\bigg)\\ \leq &\,\, 2sn\log(c)+\log(2^s D(c)^{2s})+g_{n}+g_{n'}. \end{align*} $$

Fixing $c=e^{{\varepsilon }/{2s}}$ , one has $2s\log (c)=\varepsilon $ and setting $M_{\varepsilon }=\log (2^s D(c)^{2s})$ shows that $(g_n)_{n\in \mathbb {N}}$ satisfies the condition of Lemma 4.3.

Lemma 4.3 together with equation (25) concludes the proof of Proposition 2.6.

4.3 Lyapunov exponent and dimension of weakly conformal measures

Let $m\geq 2$ and let us fix a $C^1$ weakly conformal IFS $S=\{f_1,\ldots ,f_m\}$ with attractor K.

Given $x=(x_n)_{n\in \mathbb {N}}\in \Lambda ^{\mathbb {N}},$ the following quantity, called Lyapunov exponent of S at x, defines a logarithmic shrinking rate associated with S at x. See [Reference Feng and Hu19, Proposition 5.6].

Proposition 4.5. [Reference Feng and Hu19]

For $x=(x_n)_{n\in \mathbb {N}}\in \Lambda ^{\mathbb {N}},$ the Lyapunov exponent of S at x is defined, when the following limit exists, as

(34) $$ \begin{align} \unicode{x3bb}(x)=-\lim_{n\rightarrow+\infty}\frac{\log \vert f_{x_1}\circ \cdots \circ f_{x_n}(K) \vert}{n}. \end{align} $$

Moreover, for every ergodic measure $\nu \in \mathcal {M}(\Lambda ^{\mathbb {N}})$ (with respect to the shift $\sigma $ ), there exists $\unicode{x3bb} _{\nu }\geq 0$ such that for $\nu $ -almost any $x=(x_n)_{n \in \mathbb {N}}$ ,

(35) $$ \begin{align} \unicode{x3bb}(x)=\int \unicode{x3bb}(y)\ d\nu(y):=\unicode{x3bb}_{\nu}. \end{align} $$

The following consequence of Proposition 4.5 will be useful later (see Proposition 6.7).

Corollary 4.7. Let $((p_1 ^{(k)},\ldots ,p_m ^{(k)}))_{k\in \mathbb {N}}\in ([0,1]^m)^{\mathbb {N}}$ be a sequence of probability vectors such that $(p_1^{(k)},\ldots ,p_m^{(k)})\to (p_1,\ldots ,p_m).$ Denote for $k\in \mathbb {N} \nu ,\nu _{k}\in \mathcal {M}(\Lambda ^{\mathbb {N}})$ the measures defined for any cylinder $[(i_1,\ldots i_n)]$ by

$$ \begin{align*}\nu_{k}([(i_1,\ldots,i_k)])=p_{i_1}^{(k)} \cdot \cdots \cdot p_{i_n}^{(k)}\text{ and } \nu([(i_1,\ldots,i_n)])=p_{i_1} \cdot \cdots \cdot p_{i_n}.\end{align*} $$

Then, $\nu _k \underset {k\to +\infty }{\rightarrow } \nu $ weakly, so that

$$ \begin{align*}\lim_{k\rightarrow+\infty}\unicode{x3bb}_{\nu_k}= \unicode{x3bb}_{\nu}. \end{align*} $$

Let us also recall the following fundamental result established by Feng and Hu [Reference Feng and Hu19, Theorem 2.2].

4.4 The AWSC condition

4.4.1 Definition and known results

To provide a larger class of weakly conformal IFSs to which Theorem 3.1 applies, we recall the definition of the AWSC [Reference Feng18]. First, given an IFS $S=\{f_i\}_{i\in \Lambda },$ let us define

(38) $$ \begin{align} \Lambda^{(k)}=\{\underline{i}=(i_1,\ldots,i_n) \in\Lambda^{*}:2^{-k-1}<\vert f_{\underline{i}}(K)\vert \leq 2^{-k}\}. \end{align} $$

Definition 4.9. One says that $S=\{f_i\}_{i\in \Lambda }$ satisfies the AWSC [Reference Feng18] when, writing for $k\in \mathbb {N}$ ,

(39) $$ \begin{align} t_k(S)=\max_{x\in\mathbb{R}^d}\#\{f_{\underline{i}}:\underline{i}\in\Lambda^{(k)}\text{ and } f_{\underline{i}}(K)\cap B(x,2^{-k})\neq \emptyset\}, \end{align} $$

one has

$$ \begin{align*}\frac{\log t_k (S)}{k}\to 0 .\end{align*} $$

Let us also note here that when the IFS $S $ has no exact overlaps (i.e., for any $\underline {i}\neq \underline {j}\in \Lambda ^*$ , $f_{\underline {i}}\neq f_{\underline {j}}$ ), one also has

(40) $$ \begin{align} t_k(S)=\max_{x\in\mathbb{R}^d}\#\{\underline{i}:\underline{i}\in\Lambda^{(k)}\text{ and } f_{\underline{i}}(K)\cap B(x,2^{-k})\neq \emptyset\}. \end{align} $$

The introduction of the AWSC condition is motivated by the following result.

Proposition 4.10. Let S be a weakly conformal IFS satisfying the AWSC with no exact overlaps. Then, its attractor K satisfies

$$ \begin{align*}\dim_H K =\dim(S).\end{align*} $$

As mentioned in §3.3, the advantage of the ASWC is that it can be reformulated in terms of a condition regarding the multifractal spectrum of some natural measures (given by Lemma 4.21) associated with a weakly conformal IFS.

The following proposition was established by Barral and Feng in [Reference Barral and Feng4] in the case of self-similar measures but readily adapts in the case of weakly conformal measures.

Proposition 4.11. [Reference Barral and Feng4]

Let $S=\{f_1,\ldots ,f_m\}$ be a conformal IFS and let K be its attractor. Assume that there exists a measure $\nu _0 \in \mathcal {M}(\{1,\ldots ,m\}^{\mathbb {N}})$ such that for every $\underline {i}\in \{1,\ldots ,m\}^k$ , one has

(41) $$ \begin{align} C^{-1}\leq\frac{\nu_0([\underline{i}])}{\vert f_{\underline{i}}(K)\vert^{\dim(S)}}\leq C, \end{align} $$

where $C>0$ is independent of $\underline {i}.$ Set $\mu _0 =\nu _0 \circ \pi ^{-1}$ . If for every $q\geq 1,$ one has

$$ \begin{align*}\tau_{\mu_0}(q)=(q-1)\dim(S),\end{align*} $$

where $\tau _{\mu _0}$ is the scaling function defined by equation (17), then $\dim _{H}\!y{\mu }_0 =\dim (S), S$ satisfies the AWSC, and S has no exact-overlaps.

Proposition 4.11 combined with a well-known result established by Shmerkin in [Reference Shmerkin36] regarding the multifractal spectrum of self-similar measures on $\mathbb {R}$ satisfying the exponential separation yields the following result established as [Reference Barral and Feng4, Theorem 1.3].

Theorem 4.12. [Reference Barral and Feng4]

Let S be a self-similar IFS on $\mathbb {R}$ satisfying the exponential separation condition [Reference Hochman24, Theorem 1.4]. Then, S satisfies the AWSC if and only if $\dim (S)\leq 1$ .

4.4.2 Some technical results regarding the AWSC

For $k\in \mathbb {N}$ and $x\in \mathbb {R}^d,$ recall equation (38) and define

(42) $$ \begin{align} \widetilde{\Lambda}^{(k)}=\{\underline{i}=(i_1,\ldots,i_n) \in\Lambda^{*}:\vert f_{\underline{i}}(K)\vert \leq 2^{-k}<\vert f_{(i_1,\ldots i_{k-1})}(K)\vert\}. \end{align} $$

Moreover, for $x\in K,$ we set

$$ \begin{align*} T_k(x)&=\{f_{\underline{i}}: \ f_{\underline{i}}(K)\cap B(x,2^{-k})\neq \emptyset , \ \underline{i}\in\widetilde{\Lambda}^{(k)}\}, \\ T^{\prime}_k(x)&=\{f_{\underline{i}}: \ f_{\underline{i}}(K)\cap B(x,2^{-k})\neq \emptyset , \ \underline{i}\in\Lambda^{(k)}\}. \end{align*} $$

Note that S satisfies AWSC if and only if $ \lim _{k\rightarrow +\infty }\max _{x\in \mathbb {R}^d}({\log \#T_k(x)}/{k})=0.$

Proposition 4.13. One has

$$ \begin{align*}\lim_{k\rightarrow+\infty}\max_{x\in\mathbb{R}^d}\frac{\log\# T_k(x)}{k}=0 \ \Longleftrightarrow \ \lim_{k\rightarrow+\infty}\max_{x\in\mathbb{R}^d}\frac{\log \# T^{\prime}_k(x)}{k}=0.\end{align*} $$

Proof. By equation (23), there exists $0<\alpha <\tfrac 12< \beta \leq 1$ such that for every $k\in \mathbb {N}$ ,

$$ \begin{align*}\alpha^k\leq\vert f_{\underline{i}}(K)\vert\leq \beta^k .\end{align*} $$

Remark 4.14.

1. (1) For every $k\in \mathbb {N}$ and every $\underline {i}=(i_1,\ldots ,i_n)\in \Lambda ^{(k)},$ one has

   $$ \begin{align*} C(\alpha,\beta)^{-1}k \leq k \frac{-\log 2} {\log \beta}+1\leq n \leq 2 k \frac{-\log 2} {\log \alpha}\leq C(\alpha,\beta) k.\end{align*} $$

2. (2) For every $c>1$ , by Lemma 4.4, for every $\underline {i}=(i_1,\ldots ,i_n)\in \widetilde {\Lambda }^{(k)}$ ,

   $$ \begin{align*}D(c)^{-2} \min_{1\leq j\leq m}\vert f_{j}(K)\vert c^{-2 C(\alpha,\beta)k}2^{-k}\leq \vert f_{\underline{i}}(K)\vert \leq 2^{-k}.\end{align*} $$

   In particular, for any $k\in \mathbb {N}$ large enough, one has

   (43) $$ \begin{align} c^{-{1}/{2} C(\alpha,\beta)k}2^{-k}\leq \vert f_{\underline{i}}(K)\vert \leq 2^{-k}. \end{align} $$

Lemma 4.15. For every $\varepsilon _0>0$ , there exists $k_{\varepsilon }\in \mathbb {N}$ such that for every $k\geq k_{ \varepsilon _0}$ , for every $\underline {i}=(i_1,\ldots ,i_{n})\in \Lambda ^{(k)},$ there exists $0\leq p \leq \varepsilon _0 k$ such that $(i_1,\ldots ,i_{n-p})\in \widetilde {\Lambda }^{(k)}.$

Proof. Fix $\varepsilon ={\varepsilon _0}/{2C(\alpha ,\beta )}$ and $c>1$ such that $c^{1-\varepsilon }\beta ^{\varepsilon }<1$ . By Lemma 4.4, for any $(i_1,\ldots ,i_n)\in \Lambda ^*$ and $0\leq p\leq n,$

$$ \begin{align*} \vert f_{(i_1,\ldots,i_{n})}(K)\vert\leq D(c)^2 c^{n-p}\vert f_{(i_1,\ldots,i_{n-p})}(K)\vert \times \vert f_{(i_{n-p+1},\ldots,i_n)}(K)\vert.\end{align*} $$

In particular, for $p\geq n\varepsilon ,$

$$ \begin{align*} \vert f_{(i_1,\ldots,i_{n-p})}(K)\vert \geq \frac{D(c)^{-2}c^{-(n-p)}}{\vert f_{(i_{n-p+1},\ldots,i_n)}(K)\vert}2^{-k}\geq 2^{-k}\frac{D(c)^{-2}}{(c^{1-\varepsilon}\beta^{\varepsilon})^n}. \end{align*} $$

This yields, for k large enough and $p_{}=\lfloor \varepsilon C(\alpha ,\beta )k\rfloor +1 \leq 2C( \alpha ,\beta )\varepsilon k$ , that

$$ \begin{align*}\vert f_{(i_1,\ldots,i_{n-p})}(K)\vert>2^{-k}.\end{align*} $$

As a consequence, there must exist $p\leq 2C( \alpha ,\beta )\varepsilon k$ such that $(i_1,\ldots ,i_{n-p})\in \widetilde {\Lambda }^{(k)}.$

Lemma 4.16. For every $c>1$ , for every $\varepsilon>0,$ for every k large enough (depending on c) and every $x\in \mathbb {R}^d,$ one has

(44) $$ \begin{align} &\# T_k(x)\nonumber\\&\ \ \leq k\bigg\lfloor \frac{ ({C(\alpha,\beta)}/{2})\log c}{\log 2} \bigg\rfloor C_d c^{kd{(C(\alpha,\beta)}/{2})}\max_{k\leq k" \leq k(1+\lfloor { (({C(\alpha,\beta)}/{2})\log c})/{\log 2} \rfloor)}\max_{y\in\mathbb{R}^d}\# T^{\prime}_{k'}(y) \\&\qquad\qquad\qquad\qquad\qquad\qquad\text{and } \quad \# T^{\prime}_k(x) \leq m^{k\varepsilon}\# T_k(x).\nonumber \end{align} $$

Proof. Remark that, for each $ k' $ such that

$$ \begin{align*}c^{-({1}/{2}) C(\alpha,\beta)k}2^{-k}\leq 2^{-k'}\leq 2^{-k},\end{align*} $$

there exists a constant $C_d$ (which depends on $d,\alpha $ , and $\beta $ ) so that each ball $B(x,2^{-k})$ can be covered by less than

$$ \begin{align*} C_d c^{kd({C(\alpha,\beta)}/{2})}\end{align*} $$

balls of radius $2^{-k'}.$ This implies that

$$ \begin{align*}\#\{f_{\underline{i}}: \ \underline{i}\in T_k(x)\cap \Lambda^{(k')} \}\leq C_d c^{kd({C(\alpha,\beta)}/{2})}\max_{y\in\mathbb{R}^d}\# T^{\prime}_{k'}(y). \end{align*} $$

Since one has

$$ \begin{align*}\widetilde{\Lambda}^{(k)}\subset \bigcup_{k'=k}^{k(1+\lfloor { (({C(\alpha,\beta)}/{2})\log c})/{\log Z} \rfloor)}\Lambda^{(k')},\end{align*} $$

it holds that

(45) $$ \begin{align} &\# T_k(x)\nonumber\\ &\ \ \leq k\bigg\lfloor \frac{ ({C(\alpha,\beta)}/{2})\log c}{\log 2} \bigg\rfloor C_d c^{kd({C(\alpha,\beta)}/{2})}\max_{k\leq k" \leq k(1+\lfloor { (({C(\alpha,\beta)}/{2})\log c})/{\log 2} \rfloor)}\max_{y\in\mathbb{R}^d}\# T^{\prime}_{k'}(y). \end{align} $$

Moreover, by Lemma 4.15, there exists $\phi _k:T^{\prime }_k (x)\to T_k(x)$ defined by

$$ \begin{align*}\phi_k((i_1,\ldots,i_n))=(i_1,\ldots i_{n-p}) \end{align*} $$

with $0\leq p\leq k\varepsilon $ . The mapping $\phi _k$ verifies that, for every $(i_1,\ldots i_{n'})\in T_k(x) $ ,

$$ \begin{align*}\# \phi_k^{-1}(\{(i_1,\ldots i_{n'})\})\leq m^{k\varepsilon}.\end{align*} $$

This implies that

$$ \begin{align*}\# T^{\prime}_k(x) \leq m^{k\varepsilon}\# T_k(x).\\[-33pt] \end{align*} $$

Taking the log of the estimates of Lemma 4.16 and letting k tend to infinity concludes the proof.

4.4.3 AWSC and dimension regularity

The following notion was introduced by Barral and Feng in [Reference Barral and Feng4].

Definition 4.17. [Reference Barral and Feng4]

Let $S=\{f_1,\ldots ,f_m\}$ be a weakly conformal IFS. For $P=(p_1,\ldots ,p_m)\in [0,1]^m$ a probability vector, denote again by $\nu _P\in \mathcal {M}(\Lambda ^*)$ the measure satisfying for every $(i_1,\ldots ,i_n)\in \Lambda ^*$ , $\nu _P([i_1,\ldots ,i_n])=p_{i_1}\times \cdots \times p_{i_n}$ and $\mu _P=\nu _P\circ \pi ^{-1}$ . The IFS S is said to be dimension regular if, for every probability vector P,

(46) $$ \begin{align} \dim(\mu_P)=\min\bigg\{\frac{-\sum_{1\leq i\leq m}p_i \log(p_i)}{\unicode{x3bb}_{\nu_P}}, d\bigg\}, \end{align} $$

where $\unicode{x3bb} _{\nu _P}$ is defined by equation (35).

Remark 4.18.

• • When S is self-similar, calling $0<c_1,\ldots ,c_m<1$ the contraction ratios of the similarities $f_1,\ldots ,f_m$ , for any probability vector $(p_1,\ldots ,p_m), \mu $ , and $\nu $ as in Definition 4.17, one has

  (47) $$ \begin{align} \dim(\mu)=\min\bigg\{\frac{-\!\sum_{1\leq i\leq m}p_i \log(p_i)}{\unicode{x3bb}_{\nu}}, d\bigg\}= \min\bigg\{\frac{\sum_{1\leq i\leq m}p_i \log(p_i)}{\sum_{i=1}^m p_i \log (c_i )}, d\bigg\}. \end{align} $$

• • As proved in [Reference Hochman24], any self-similar IFS on $\mathbb {R}$ satisfying the exponential separation condition [Reference Hochman24, Theorem 1.4] is dimension regular.

We will prove the following result which implies Proposition 4.10.

Proposition 4.19. Assume that $S=\{f_1,\ldots ,f_m\}$ satisfies the AWSC without exact overlaps. Then, S is dimension regular and $\dim (S)=\dim _H(K).$

Had the IFS been conformal and satisfying some bounded distortion properties, the proof of Proposition 4.19 would follow directly from the existence of appropriated Gibbs measures. Unfortunately, such measures do not always exist in the weakly conformal case, but some measures that are close enough from satisfying the desired properties still exist as established by the following lemma.

Lemma 4.20. Let $\varepsilon>0$ and $s\geq 0$ be real numbers. There exists $k\in \mathbb {N}$ and a probability vector $(p_{\underline {i}})_{\underline {i}\in \Lambda ^k}$ such that the weakly conformal measure $\nu $ associated with $S'=\{f_{\underline {i}}\}_{\underline {i}\in \Lambda ^k}$ and $(p_{\underline {i}})_{\underline {i}\in \Lambda ^k}$ verifies, for any $p\in \mathbb {N}$ and $\underline {i}_1 ,\ldots ,\underline {i}_p \in \Lambda ^{k},$

(48) $$ \begin{align} e^{-kp\varepsilon}\frac{\vert f_{\underline{i}_1 \cdots \underline{i}_p}(K)\vert^s}{e^{pkP(s)}}\leq \nu([\underline{i}_1 \cdots \underline{i}_p])\leq e^{kp\varepsilon}\frac{\vert f_{\underline{i}_1 \cdots \underline{i}_p}(K)\vert^s}{e^{pkP(s)}}. \end{align} $$

Proof. Fix $\varepsilon>0$ and $c>1$ small enough so that $8s\log c\leq \varepsilon .$

By Lemma 4.3, there exists $k\in \mathbb {N}$ so large that the constant named $D(c)$ in Lemma 4.1 verifies $ ({\log D(c)}/{k})\leq \log c $ and

(49) $$ \begin{align} \bigg\vert \frac{1}{k}\log \sum_{\underline{i}\in\Lambda^k}\vert f_{\underline{i}}(K)\vert^s -P(s)\bigg\vert\leq \frac{\varepsilon}{2}. \end{align} $$

Writing again $g_k =\log \sum _{\underline {i}\in \Lambda ^k}\vert f_{\underline {i}}(K)\vert ^s,$ let us define the probability vector $(p_{\underline {i}})_{\underline {i}\in \Lambda ^k}$ by setting

$$ \begin{align*}p_{\underline{i}}=\frac{\vert f_{\underline{i}}(K)\vert^s}{e^{g_k}}.\end{align*} $$

Let $\nu $ be the weakly conformal measure associated with $S'=\{f_{\underline {i}}\}_{\underline {i}\in \Lambda ^k}$ and $(p_{\underline {i}})_{\underline {i}\in \Lambda ^k}.$ Applying Lemma 4.4, for any $p\in \mathbb {N}, \underline {i}_1,\ldots ,\underline {i}_p\in \Lambda ^k,$ ,

(50) $$ \begin{align} D(c)^{-2p}c^{-2kp}\leq \frac{\vert f_{\underline{i}_1}\circ \cdots \circ f_{\underline{i}_p}(K)\vert}{\prod_{j=1}^p \vert f_{\underline{i}_j}(K)\vert}\leq D(c)^{2p}c^{2kp}. \end{align} $$

Also,

(51) $$ \begin{align} D(c)^{2sp} c^{2skp}=e^{pk 2s\cdot (({\log D(c)}/{k})+\log c)}\leq e^{({\varepsilon}/{2}) pk}. \end{align} $$

As a consequence, for any $p\in \mathbb {N}$ and any $\underline {i}_1,\ldots ,\underline {i}_p \in \Lambda ^k,$ one has

$$ \begin{align*} \nu([\underline{i}_1 \cdots \underline{i}_p])=p_{\underline{i}_1}\cdot \cdots \cdot p_{\underline{i}_p}=\frac{\prod_{j=1}^p \vert f_{\underline{i}_j}(K)\vert^s}{e^{pg_k}}=\frac{\prod_{j=1}^p \vert f_{\underline{i}_j}(K)\vert^s}{e^{kp(({g_k}/{k})-P(s))}e^{pkP(s)}}. \end{align*} $$

Using equations (49), (50), and (51) concludes the proof.

Remark 4.21. The measure $\nu $ can be extended over $\Lambda ^{\mathbb {N}}$ by the usual arguments. Moreover, for any $\underline {i}=(i_1,\ldots ,i_n)\in \Lambda ^{*},$ write $n_1=k\lfloor {n}/{k} \rfloor $ and $n_2=k(\lfloor {n}/{k} \rfloor +1).$ Consider ${\underline {j}\in \Lambda ^{n_1}}$ such that $[\underline {i}]\subset [\underline {j}]$ and $\underline {\ell }=(\ell _1,\ldots ,\ell _{n_2-n})\in \Lambda ^{n_2-n},$ one has

(52) $$ \begin{align} e^{-n_2\varepsilon}\frac{\vert f_{(i_1, \ldots , i_{n},\ell_1,\ldots,\ell_{n_2-n})}(K)\vert^s}{e^{n_2 P(s)}}\leq\nu([\underline{i}\underline{\ell}]) \leq \nu([\underline{i}])\leq \nu([\underline{j}])\leq e^{n_1\varepsilon}\frac{\vert f_{(i_1,\ldots,i_{n_1})}(K)\vert^s}{e^{n_1 P(s)}}. \end{align} $$

By Lemma 4.4, there exists a constant $C>0$ such that, uniformly on $\underline {i},\underline {j},\underline {i}\underline {\ell },$ one has

$$ \begin{align*} C^{-1}\leq\min\bigg\{\frac{\vert f_{\underline{j}}(K)\vert}{\vert f_{\underline{i}}(K)\vert},\frac{\vert f_{\underline{i}}(K)\vert}{\vert f_{\underline{i}\underline{\ell}}(K)\vert}\bigg\} \leq\max\bigg\{\frac{\vert f_{\underline{j}}(K)\vert}{\vert f_{\underline{i}}(K)\vert},\frac{\vert f_{\underline{i}}(K)\vert}{\vert f_{\underline{i}\underline{\ell}}(K)\vert}\bigg\}\leq C. \end{align*} $$

Hence, there exists a constant $\gamma _{s,\varepsilon }$ such that for any $\underline {i}=(i_1,\ldots ,i_n)\in \Lambda ^{*}$ , one has

(53) $$ \begin{align} \gamma_{s,\varepsilon}^{-1}e^{-n\varepsilon}\frac{\vert f_{\underline{i}}(K)\vert^s}{e^{nP(s)}}\leq \nu([\underline{i}])\leq \gamma_{s,\varepsilon}e^{n\varepsilon}\frac{\vert f_{\underline{i}}(K)\vert^s}{e^{nP(s)}}. \end{align} $$

Let us now prove Proposition 4.19.

Proof. Call K the attractor of S. Let us show first that if any system S satisfying the AWSC also verifies that, for any weakly conformal measure $\mu \in \mathcal {M}(\mathbb {R}^d)$ associated with a probability vector $(p_1,\ldots ,p_m)$ and S,

(54) $$ \begin{align} \dim(\mu)=\frac{-\!\sum_{1\leq i\leq m}p_i \log p_i}{\unicode{x3bb}_{\nu}}, \end{align} $$

where $\nu $ is the measure associated on $\Lambda ^{\mathbb {N}},$ then $\dim (S)=\dim _H (K).$

Fix $\varepsilon>0$ and consider $k\in \mathbb {N}$ , $S'=\{f_{\underline {i}}\}_{\underline {i}\in \Lambda ^k}$ , and $\nu $ as in Lemma 4.20 applied with $s=\dim (S).$ Note that, since S satisfies the AWSC, so does $S'$ . Then, considering the measure $\mu =\nu \circ \pi ^{-1}$ , where $\pi $ is the canonical projection, one has

$$ \begin{align*} \dim(S)-\varepsilon\leq \dim (\mu)=\frac{-\!\sum_{\underline{i}\in\Lambda^k}p_{\underline{i}}\log p_{\underline{i}}}{\unicode{x3bb}_{\nu}}\leq \dim(S)+\varepsilon. \end{align*} $$

This proves that $\dim _H (K)\geq \dim (S)-\varepsilon .$ Since it always holds that $\dim _H(K)\leq \dim (S)$ (see [Reference Falconer15]) and $\varepsilon $ is arbitrary,

$$ \begin{align*}\dim_H(K)=\dim(S).\end{align*} $$

Let us show that, for any system satisfying the AWSC, equation (54) holds for every weakly conformal measure $\mu $ .

Let $\mu \in \mathcal {M}(\mathbb {R}^d)$ be a weakly conformal measure associated with S and a probability vector $(p_1,\ldots ,p_m)$ and $\nu \in \mathcal {M}(\Lambda ^{\mathbb {N}})$ such that $\mu =\nu \circ \pi ^{-1}.$

It comes from the proof of Theorem 4.8 [Reference Feng and Hu19] (applied to $\mu $ ) that for any $\varepsilon>0,$ for $\mu $ -almost any $x\in K$ such that $\mu ^{\pi ^{-1}(\{x\})}$ exists and satisfies the two first items of Theorem 4.8, there exists $n_0$ large enough so that, for any $n\geq n_0$ , there exists $\underline {i}_1,\ldots ,\underline {i}_{N_n}$ such that:

• • for any $1\leq j\leq N_n,$

  (55) $$ \begin{align} e^{-n(\unicode{x3bb}+\varepsilon)}\leq\vert f_{\underline{i}_j}(K)\vert\leq e^{-n(\unicode{x3bb}-\varepsilon)}; \end{align} $$

• • one has

  (56) $$ \begin{align} \mu^{\pi^{-1}(\{x\})}\bigg(\bigcup_{1\leq j\leq N_n}[\underline{i}_j]\bigg)\geq \frac{1}{2}; \end{align} $$

• • for any $1\leq j\leq N_n,$

  (57) $$ \begin{align} e^{-n(h+\varepsilon)}\leq\mu^{\pi^{-1}(\{x\})}([\underline{i}_j])\leq e^{-n(h-\varepsilon)}. \end{align} $$

Assume that $h>0$ and take $0<\varepsilon <\min \{ {h}/{2},{ \unicode{x3bb} }/{2}\}.$

Combining equations (56) and (57), one gets

(58) $$ \begin{align} N_n \geq \tfrac{1}{2}e^{n(h-\varepsilon)}. \end{align} $$

Note that $\#\{k:e^{-n(\unicode{x3bb} +\varepsilon )}\leq 2^{-k}\leq e^{-n(\unicode{x3bb} -\varepsilon )}\}\leq ({2n\varepsilon }/{\log 2}).$ As a consequence, there exists $k\in [{n(\unicode{x3bb} -\varepsilon )}/{\log 2},{n(\unicode{x3bb} +\varepsilon )}/{\log 2}]$ such that

(59) $$ \begin{align} \# \Lambda^{(k)}\cap \{[\underline{i}_j]\}_{1\leq j\leq N_n}\geq \frac{N_n}{{2n\varepsilon}/{\log 2}}\geq \frac{({1}/{2})e^{{nh}/{2}}}{{2n\varepsilon}/{\log 2}}. \end{align} $$

Since for any $\leq j \leq N_n, [ \underline {i}_j]\cap \pi ^{-1}(\{x\})\neq \emptyset ,$ one also has $f_{\underline {i}_j}(K)\subset B(x,e^{-n(\unicode{x3bb} -\varepsilon )}),$ so that, writing $n'=\lfloor {n(\unicode{x3bb} -\varepsilon )}/{\log 2} \rfloor $ , one has

(60) $$ \begin{align} \#\{\underline{i}\in \Lambda^{(n')}:f_{\underline{i}}(K)\cap B(x,2^{-n'})\}\geq \frac{({1}/{2})e^{{nh}/{2}}}{{2n\varepsilon}/{\log 2}}. \end{align} $$

In particular, recalling equation (40) and Proposition 4.13,

$$ \begin{align*}\frac{\log t_k}{k}\nrightarrow 0 \end{align*} $$

and S does not satisfy the AWSC. As a consequence, S satisfies the AWSC implying $h=0$ , which, recalling the last item of Theorem 4.8, concludes the proof.

Remark 4.22. For simplicity, we established the result for weakly conformal measures, but Theorem 4.8, that is, [Reference Feng and Hu19, Theorem 2.1], is actually stated for ergodic measures and a careful reader will notice that the same proof combined with [Reference Feng and Hu19, Theorem 2.1] actually shows that if the IFS S satisfies AWSC, for every ergodic measure $\nu \in \mathcal {M}(\Lambda ^{\mathbb {N}})$ (with respect to the shift $\sigma $ ), one has

$$ \begin{align*}\dim_H \nu\circ \pi^{-1} =\frac{h(\nu)}{\unicode{x3bb}_{\nu}},\end{align*} $$

where $\unicode{x3bb} _{\nu }$ is the Lyapunov exponent associated with $\nu $ and $h(\nu )$ is defined as the positive number for which, for $\nu $ -almost $(x_n)_{n\in \mathbb {N}},$ one has

$$ \begin{align*}h(\nu)=\lim_{n \to +\infty}\frac{-\log \nu([(x_1,\ldots,x_n])}{n}.\end{align*} $$

We also state the following corollary, which will be useful later in the manuscript.

Corollary 4.23. If S is weakly conformal, then any weakly conformal measure $\mu $ satisfies equation (54). So by Corollary 4.7, $\dim _{H}{\mu }$ depends continuously on the choice of the probability vector.

5.1 Essential content for weakly conformal measures

Estimates on essential contents for weakly conformal measures are now established. These estimates are similar to that established in [Reference Daviaud10, Theorem 2.6] for self-similar measures, but the introduction of the weakly conformal settings brings a number of geometrical difficulties, so for the sake of completeness, we provide all the details regarding these estimates.

Theorem 5.5. Let S be a $C^{1}$ weakly conformal IFS of $\mathbb {R}^d$ .

Let K be the attractor of S and $\mu $ be a measure on K which satisfies that, for every $\underline {i}\in \bigcup _{k\geq 1}\{1,\ldots ,m\}^k$ , $\mu \circ f_{\underline {i}}^{-1}$ is absolutely continuous with respect to $\mu $ . Then, we have the following.

For any $0\leq s<\underline {\dim }_H(\mu )$ , for any $0<\varepsilon \leq \tfrac 12$ , there exists a constant $c=c(d,\mu ,s,\varepsilon )>0 $ depending on the dimensions d, $\mu $ , s, and $\varepsilon $ only, such that for any ball $B=B(x,r)$ centered on K and $r\leq 1$ , for any open set $\Omega $ , one has

(64)

For any $s>\overline {\dim }_H(\mu )$ , $ \mathcal {H}^{\mu ,s}_{\infty }(\Omega )=0.$

• • The system S is not assumed to verify any separation condition.

• • When the maps are similarities, one still has, for any $s>\overline {\dim }_H(\mu )$ , $ \mathcal {H}^{\mu ,s}_{\infty }(\Omega )=0$ , but for $s<\underline {\dim } (\mu )$ , there exists a constant $c(d,\mu ,s)$ such that the following more precise estimates hold true [Reference Daviaud10, Theorem 2.6]:

  (65)

  

• • When the measure $\mu $ is weakly conformal (Definition 2.3) or the projection of a Gibbs measure, $\mu $ is exact dimensional (see [Reference Feng and Hu19, Theorem 2.1] for instance), which implies that $\underline {\dim }_{H}{\mu } =\overline {\dim }_{H}{\mu }$ . In this case, Theorem 5.5 provides a complete description of the essential Hausdorff content, except at $s=\dim _{H}{\mu }$ (this case must depend in general on the separation property of the IFS).

Before proving Theorem 5.5, we prove that projections of quasi-Bernoulli measures satisfy the assumptions of Theorem 5.5.

Let $\nu \in \mathcal {M}(\{1,\ldots ,m\}^{\mathbb {N}})$ be a measure. We call $\nu $ a quasi-Bernoulli measure if there exists $C>0$ such that for every $\underline {i},\underline {j}\in \bigcup _{k\geq 1}\{1,\ldots ,m\}^k,$ one has

$$ \begin{align*}C^{-1}\leq\frac{\nu([\underline{i}\underline{j}])}{\nu([\underline{i}])\times \nu([\underline{j}])}\leq C.\end{align*} $$

Proof. Note that since for every $1\leq i\leq m, \nu ([i])\neq 0,$ one has for every $n\in \mathbb {N}$ and every $\underline {i}=(i_1,\ldots ,i_n)\in \{1,\ldots ,m\}^n,$

$$ \begin{align*}\nu([\underline{i}])\geq \frac{1}{C^n}\prod_{k=1}^n \nu([i_k])>0.\end{align*} $$

In addition, for every Borel set $E\subset \mathbb {R}^d$ , one has

$$ \begin{align*} \nu(\pi^{-1}(E)\cap [\underline{i}])&=\nu\Big\{\underline{i}(x_n)_{n\in\mathbb{N}}:f_{\underline{i}}\Big(\lim_{n\to+\infty}f_{x_1}\circ \cdots \circ f_{x_n}(0)\Big)\in E\Big\}\\ &=\nu\Big\{\underline{i}(x_n)_{n\in\mathbb{N}}:\lim_{n\to+\infty}f_{x_1}\circ \cdots \circ f_{x_n}(0)\in f_{\underline{i}}^{-1}( E)\Big\}\\ &=\nu\{\underline{i}(x_n)_{n\in\mathbb{N}}:(x_n)_{n\in\mathbb{N}}\in \pi^{-1}\circ f_{\underline{i}}^{-1}( E)\}. \end{align*} $$

This implies that

$$ \begin{align*} C^{-1}\nu(\underline{i})\nu(\pi^{-1}\circ f_{\underline{i}}^{-1}( E))\leq\nu(\pi^{-1}(E)\cap [\underline{i}])\leq C\nu(\underline{i})\nu(\pi^{-1}\circ f_{\underline{i}}^{-1}( E)),\end{align*} $$

that is,

$$ \begin{align*}C^{-1}\nu(\underline{i})\mu(f_{\underline{i}}^{-1}(E))\leq\nu(\pi^{-1}(E)\cap [\underline{i}])\leq C\nu(\underline{i})\mu(f_{\underline{i}}^{-1}(E)).\end{align*} $$

Since, for every $n\in \mathbb {N}$ ,

$$ \begin{align*} \mu(E)=\nu(\pi^{-1}(E))=\sum_{\underline{j}\in\{1,\ldots,m\}^n}\nu(\pi^{-1}(E)\cap [\underline{j}]),\end{align*} $$

we have

$$ \begin{align*} C^{-1} \sum_{\underline{j}\in\{1,\ldots,m\}^n}\nu(\underline{j})\mu(f_{\underline{j}}^{-1}(E))\leq \mu(E)\leq C\sum_{\underline{j}\in\{1,\ldots,m\}^n}\nu(\underline{j})\mu(f_{\underline{j}}^{-1}(E)). \end{align*} $$

So,

$$ \begin{align*}\mu(E)=0\Longrightarrow C^{-1}\nu(\underline{i})\mu(f_{\underline{i}}^{-1}(E))=0 \Longrightarrow \mu(f_{\underline{i}}^{-1}(E))=0,\end{align*} $$

which proves that $\mu \circ f_{\underline {i}}^{-1}$ is absolutely continuous with respect to $\mu .$

Let us now prove Theorem 5.5.

Proof. As mentioned above, the proof of Theorem 5.5 is similar to the proof of (65) from [Reference Daviaud10], only it is made significantly more technical by the assumption that the IFS is weakly conformal.

Let us recall the well-known Besicovitch covering theorem.

Theorem 5.7. [Reference Mattila28]

There exists $Q_{d} \in \mathbb {N}^{\star }$ , a constant depending only on the dimension d, such that for every $E\subset [0,1]^{d}$ , for every set $\mathcal {F}=\{B(x, r_{(x)} ): x\in E, r_{(x)}>0 \}$ , there exists $\mathcal {F}_1,\ldots ,\mathcal {F}_{Q_{d}}$ finite or countable sub-families of $\mathcal {F}$ such that:

• • $\text { for all } 1\leq i\leq Q_{d}$ , $\text { for all } L\neq L'\in \mathcal {F}_i$ , one has $L \cap L'=\emptyset .$

• • E is covered by the families $\mathcal {F}_i$ , that is,

  (66) $$ \begin{align} E\subset \bigcup_{1\leq i\leq Q_{d}}\bigcup_{L\in \mathcal{F}_i}L. \end{align} $$

The proof of Theorem 5.7 relies on the following geometric lemma, which will also be used.

Lemma 5.8. [Reference Mattila28]

Let $\mathcal {B} =(B_n )_{n\in \mathbb {N}}$ be a family of balls and B a ball such that:

1. (i) $\text { for all } \ n\geq 1$ , $\vert B_n\vert \geq \tfrac 12\vert B\vert ;$

2. (ii) $\text { for all } \ n_1 \neq n_2 \geq 1 $ , the center of $B_{n_1}$ does not belong to $B_{n_2}$ .

Then, B intersects less than $Q_{d}$ elements of $\mathcal {B}$ , where $Q_{d}$ is the same constant as in Theorem 5.7.

Let us first prove the above estimates for balls. We fix $S=\{f_1,\ldots ,f_m\}$ a weakly conformal IFS and we denote by K its attractor.

Proposition 5.9. Let $\mu $ be a measure supported on K satisfying that for every $\underline {i}\in \bigcup _{n\geq 1}\{1,\ldots ,m\}^n$ , $\mu (f_{\underline {i}}^{-1})$ is absolutely continuous with respect to $\mu $ . Then, for any $0<\varepsilon \leq \underline {\dim }_H(\mu )$ , any $0\leq \varepsilon ^{\prime }\leq \tfrac 12$ such that $\underline {\dim }_H (\mu )-\varepsilon +\varepsilon ^{\prime }>0$ , there exists a constant $\chi ( d,\mu ,\varepsilon ,\varepsilon ^{\prime })>0 $ such that for any ball $B=B(x,r)$ with $x\in K$ and $r\leq 1$ , one has

In addition, for any $s>\overline {\dim _H}( \mu )$ , $\mathcal {H}^{\mu ,s}_{\infty }(B)=0.$

Proof. Note first that, by definition of $\mathcal {H}^{\mu ,s}_{\infty }$ (Definition 5.1), for any $s>\overline {\dim }_H(\mu )$ , $\mathcal {H}^{\mu ,s}_{\infty }(B)=0.$

Let us consider $0\leq s<\underline {\dim }_H (\mu )$ and start by a few remarks.

Set $\alpha =\underline {\dim }_H(\mu )$ , and let $\varepsilon>0$ and $\rho>0$ be two real numbers. One defines

$$ \begin{align*}E_{\mu} ^{\alpha, \rho,\varepsilon}=\{x\in\mathbb{R}^d :\text{ for all } r\leq \rho, \ \mu(B(x,r))\leq r^{\alpha-\varepsilon}\}.\end{align*} $$

By definition of $\underline {\dim }_{H}{\mu }$ (Definition 2.2), for every $\varepsilon>0,$

$$ \begin{align*}\mu\bigg(\bigcup_{\rho>0}E_{\mu} ^{\alpha, \rho,\varepsilon}\bigg)=1.\end{align*} $$

Let $\varepsilon>0$ and $0<\rho _{\varepsilon }\leq 1$ be two real numbers such that $\mu (E_{\mu } ^{\alpha , \rho _{\varepsilon },\varepsilon })\geq \tfrac 12$ and write $E=E_{\mu }^{\alpha ,\rho _{\varepsilon }, \varepsilon }.$

Write $c_{\underline {i}}=\vert f_{\underline {i}}(K)\vert .$ Let us fix $\underline {i}=(i_1,\ldots ,i_k)\in \Lambda ^{*}.$ For any $x\in K$ and $r>0$ , by equations (21) and (22) applied with $\theta =\varepsilon ^{\prime }$ , one has

$$ \begin{align*} f_{\underline{i}}(B(x,r))\supset B(f_{\underline{i}}(x_0),\widehat{C}_{\varepsilon'}c_{\underline{i}}(x_0)^{1-\varepsilon'}r)\supset B\bigg(f_{\underline{i}}(x_0),\frac{\widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}}{\vert K\vert^{-({1+\varepsilon'})/({1-\varepsilon'})}} c_{\underline{i}} ^{({1+\varepsilon'})/({1-\varepsilon'})}r\bigg). \end{align*} $$

Remember that $\varepsilon '\leq \tfrac 12.$ Since $({1+\varepsilon '})/({1-\varepsilon '})\leq 1+4\varepsilon '$ ,

(67) $$ \begin{align} f_{\underline{i}}(B(x,r))\supset B(f_{\underline{i}}(x_0),\widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}\cdot\vert K\vert^{({1+\varepsilon'})/({1-\varepsilon'})} c_{\underline{i}} ^{1+4\varepsilon'}r). \end{align} $$

Writing $\mu _{\underline {i}}=\mu (f_{\underline {i}}^{-1})$ , equation (67) yields

(68) $$ \begin{align} \nonumber E_{\underline{i}}&:=f_{\underline{i}}(E)\\\nonumber &=\{f_{\underline{i}}(x)\in K :\text{ for all } \ r\leq \rho_{\varepsilon}, \ \mu(B(x,r))\leq r^{\alpha -\varepsilon}\}\\\nonumber &\subset\bigg\{f_{\underline{i}}(x), x\in K :\text{ for all } \ r \leq \rho_{\varepsilon},\\\nonumber &\quad \mu (f_{\underline{i}} ^{-1}(B(f_{\underline{i}}(x_0),\widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}\cdot\vert K\vert^{({1+\varepsilon'})/({1-\varepsilon'})} c_{\underline{i}} ^{1+4\varepsilon'}r) ))\\\nonumber &\leq \bigg(\frac{\widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}\cdot\vert K\vert^{({1+\varepsilon'})/({1-\varepsilon'})} c_{\underline{i}} ^{1+4\varepsilon'}r}{\widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}\cdot\vert K\vert^{({1+\varepsilon'})/({1-\varepsilon'})} c_{\underline{i}} ^{1+4\varepsilon'}}\bigg)^{\alpha -\varepsilon}\bigg\}\\\nonumber &=\bigg\{y\in f_{\underline{i}}(K) : \text{ for all } \ r'\leq \widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}\cdot\vert K\vert^{({1+\varepsilon'})/({1-\varepsilon'})} c_{\underline{i}} ^{1+4\varepsilon'}\rho_{\varepsilon}, \\& \quad \mu_{\underline{i}}(B(y,r'))\leq \bigg(\frac{r'}{\widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}\cdot\vert K\vert^{({1+\varepsilon'})/({1-\varepsilon'})} c_{\underline{i}} ^{1+4\varepsilon'}}\bigg)^{\alpha -\varepsilon}\bigg\}. \end{align} $$

Notice also that

$$ \begin{align*}\mu_{\underline{i}}(E_{\underline{i}})=\mu(E)\geq \tfrac{1}{2}.\end{align*} $$

We are now ready to estimate the $\mu $ -essential content of a ball B centered in K.

Let us write

(69) $$ \begin{align} \gamma(S,\varepsilon')=\widehat{C}_{\varepsilon'}^{-{2}/({1-\varepsilon'})}\cdot\vert K\vert^{({1+\varepsilon'})/({1-\varepsilon'})}. \end{align} $$

Let $B=B(x,r)$ with $x\in K$ and $r\leq c_0:=\min _{z\in K}\min _{1\leq i\leq m}\| f_i '(z) \|$ .

Since $x\in K$ , there exists $\underline {i }=(i_1,\ldots ,i_k)\in \Lambda ^{*}$ such that:

• • $x\in f_{\underline {i}}(K)$ ;

• • $\vert f_{\underline {i}}(K)\vert \leq \tfrac 13\vert B\vert $ ;

• • $\vert f_{(i_1,\ldots ,i_{k-1})}(K)\vert \geq \tfrac 13\vert B\vert .$

By equation (22), for any $y\in K$ , one has

(70) $$ \begin{align} \vert f_{ \underline{i}}(K)\vert \geq \widehat{C}_{\varepsilon'}^{-1}\| f_{ \underline{i}}'(y)\|^{1+\varepsilon'}\vert K\vert \end{align} $$

and

(71) $$ \begin{align} \| f_{\underline{i}}(y)\|&=\| f^{\prime}_{(i_1,\ldots,i_{n-1})}(f_n (x))\circ f^{\prime}_{i_n}(x) \|\geq \| f^{\prime}_{(i_1,\ldots,i_{n-1})}(f_n (x))\| c_0 \nonumber\\ &\geq \vert f_{(i_1,\ldots,i_{n-1}}(K) \vert^{{1}/({1-\varepsilon'})}\widehat{C}_{\varepsilon'}^{{-1}/({1-\varepsilon'})}\cdot \vert K\vert^{{-1}/({1-\varepsilon'})}c_0. \end{align} $$

Combining equations (70) and (71), one obtains

(72) $$ \begin{align} c_{\underline{i}}=\vert f_{\underline{i}}(K)\vert &\geq \widehat{C}_{\varepsilon'}^{-1-({1+\varepsilon'})/({1-\varepsilon'})}\vert K\vert^{{-2\varepsilon'}/({1-\varepsilon'})}c_0^{1+\varepsilon'}\vert f_{(i_1,\ldots,i_{n-1})}(K)\vert^{({1+\varepsilon'})/({1-\varepsilon'})}\nonumber \\ & \geq \widehat{C}_{\varepsilon'}^{-1-({1+\varepsilon'})/({1-\varepsilon'})}\vert K\vert^{{-2\varepsilon'}/({1-\varepsilon'})}c_0^{1+\varepsilon'}r^{1+4\varepsilon'}. \end{align} $$

Note that .

Consider a set $A\subset B$ verifying $\mu (A)=\mu (B).$ One aims to give a lower bound for the Hausdorff content of A which depends only on B, d, $\varepsilon $ , $\varepsilon ^{\prime }$ , and the measure $\mu $ .

Consider a sequence of balls $(L_n=B( x_n,\ell _n))_{n\geq 1}$ covering $A\cap E_{\underline {i}} $ such that $\ell _{n}<\gamma (S,\varepsilon ')\rho _{\varepsilon }c_{ \underline {i}}^{1+4\varepsilon '}$ and $ x_n\in A\cap E_{\underline {i}} $ .

Since $\mu _{\underline {i}}$ is absolutely continuous with respect to $\mu $ , it holds that $\mu _{\underline {i}}(A)=1.$

By equation (68) applied to each ball $L_n$ , $n\in \mathbb {N}$ , one has $({\vert L_n \vert }/{\gamma (S,\varepsilon ' )c_{\underline {i}}^{1+4\varepsilon ^{\prime }}})^{\alpha -\varepsilon }\geq \mu _{\underline {i}}(L_n)$ , so that, recalling equation (72),

(73) $$ \begin{align} \sum_{n\in\mathbb{N}}| L_n| ^{\alpha-\varepsilon}\ &\geq \sum_{n\in\mathbb{N}}(\gamma(S,\varepsilon')c_{\underline{i}}^{1+4\varepsilon^{\prime}}) ^{\alpha -\varepsilon}\mu_{\underline{i}}(L _n )\geq (\gamma(S,\varepsilon')c_{\underline{i}}^{1+4\varepsilon^{\prime}}) ^{\alpha -\varepsilon}\mu_{\underline{i}}\bigg(\bigcup_{n\in \mathbb{N} }L_n \bigg)\nonumber\\ &\geq (\gamma(S,\varepsilon')c_{\underline{i}}^{1+4\varepsilon^{\prime}}) ^{\alpha -\varepsilon}\mu_{\underline{i}}(E_{\underline{i}})\geq \tfrac{1}{2}(\gamma(S,\varepsilon')c_{\underline{i}}^{1+4\varepsilon^{\prime}}) ^{\alpha -\varepsilon}\nonumber \\ &\geq \kappa(\mu,\varepsilon^{\prime},\varepsilon)r^{(1+4\varepsilon')^2(\alpha-\varepsilon)}\geq \kappa(\mu,\varepsilon^{\prime},\varepsilon)r^{(1+16\varepsilon')(\alpha-\varepsilon)} , \end{align} $$

where $\kappa (\mu ,\varepsilon ^{\prime },\varepsilon )=\tfrac 12\gamma (S,\varepsilon ')^{\alpha -\varepsilon }\cdot (\widehat {C}_{\varepsilon '}^{-1-({1+\varepsilon '})/({1-\varepsilon '})}\vert K\vert ^{{-2\varepsilon '}/({1-\varepsilon '})}c_0^{1+\varepsilon '})^{(1+4\varepsilon ')(\alpha -\varepsilon )}.$

This series of inequalities holds for any sequence of balls $(L_n)_{n\in \mathbb {N}}$ with radius less than $\gamma (S,\varepsilon ')\rho _{\varepsilon }c_{\underline {i}}^{1+4\varepsilon ^{\prime }} $ centered in $A\cap E_{\underline {i}}$ . One now proves that one can freely remove those constraints on the center and the radius of the balls used to cover $A\cap E_{\underline {i}}$ , up to a multiplicative constant.

Consider balls $( L_n=B(x_n,\ell _n))_{n\geq 1}$ covering $A\cap E_{\underline {i}} $ such that $\ell _{n}<\gamma (S,\varepsilon ')\rho _{\varepsilon }c_{\underline {i}}^{1+4\varepsilon ^{\prime }} $ but $x_n$ does not necessarily belongs to $A\cap E_{\underline {i}} $ .

Let $n\in \mathbb {N}$ . One constructs recursively a sequence of balls $(L_{n,j})_{1\leq j\leq J_n}$ such that the following properties hold for any $1\leq j\leq J_n$ :

• • $L_{n,j}$ is centered on $A\cap E_{\underline {i}}\cap L_n$ ;

• • $A\cap E_{\underline {i}}\cap L_n \subset \bigcup _{1\leq j \leq J_n}L_{n,j}$ ;

• • for all $1\leq j\leq J_n, \vert L_{n,j}\vert = \vert L_n \vert $ ;

• • the center of $L_{n,j}$ does not belong to any $L_{n,j^{\prime }}$ for $1\leq j^{\prime }\neq j\leq J_n$ .

To achieve this, simply consider $y_1\in A\cap E_{\underline {i}}\cap L_n$ and set $L_{1,n}=B(y_1,\ell _n)$ . If $A\cap E_{\underline {i}}\cap L_n \nsubseteq L_{1,n}$ , consider $y_2 \in A\cap E_{\underline {i}}\cap L_n \setminus L_{1,n}$ and set $L_{2,n}=B(y_2 ,\ell _n)$ . If $A\cap E_{\underline {i}}\cap L_n \nsubseteq L_{1,n}\cup L_{2,n}$ , consider $y_3 \in A\cap E_{\underline {i}}\cap L_n \setminus L_{1,n}\cup L_{2,n}$ and set $L_{3,n}=B(y_3 ,\ell _n)$ , and so on.

Note that for any $1\leq j\leq J_n$ , any ball $L_{j,n}$ has radius $\ell _n$ , intersects $L_n$ (which also has radius $\ell _n$ ), and, because $y_j \notin \bigcup _{1 \leq j^{\prime }\neq j\leq J_n}L_{j^{\prime },n}$ , it holds that for any $j\neq j^{\prime }$ , $\tfrac 13L_{n,j}\cap \tfrac 13L_{n,j^{\prime }}=\emptyset $ . A volume argument yields that $J_n \leq Q_{d, 1/3},$ where $Q_{d, 1/3}$ is constant which only depends on the dimension d and the contraction factor $\tfrac 13.$

Hence, denoting by $(\widetilde L_n)_{n\in \mathbb {N}} $ the collection of the corresponding balls centered on $A\cap E_{\underline {i}} $ associated with all the balls $L_n$ , one has by equation (73) applied to $(\widetilde L_n)_{n\in \mathbb N}$ ,

$$ \begin{align*}\sum_{n\in\mathbb{N}}|L_n| ^{\alpha-\varepsilon}\geq \frac{1}{Q_{d,{1}/{3}}} \sum_{n\in\mathbb{N}}|\widetilde L_n| ^{\alpha-\varepsilon} \geq \frac{\kappa (\mu,\varepsilon^{\prime},\varepsilon)}{Q_{d,{1}/{3}}} r ^{(1+4\varepsilon^{\prime})(\alpha -\varepsilon)}.\end{align*} $$

Remark also that any ball of radius smaller than $c_{\underline {i}}$ can be covered by at most $({2c_{\underline {i}}^{-4\varepsilon '}}/{\gamma (S,\varepsilon ')\rho _{\varepsilon }})^d$ balls of radius $\gamma (S,\varepsilon ')\rho _{\varepsilon }c_{\underline {i}}^{1+4\varepsilon ^{\prime }}$ . Moreover, by equation (72),

$$ \begin{align*} c_{\underline{i}}^{-4\varepsilon^{\prime}}\leq (\widehat{C}_{\varepsilon'}^{-1-({1+\varepsilon'})/({1-\varepsilon'})}\vert K\vert^{{-2\varepsilon'}/({1-\varepsilon'})}c_0^{1+\varepsilon'})^{-4\varepsilon'}r^{-4\varepsilon' \cdot(1+4\varepsilon')}.\end{align*} $$

Setting

$$ \begin{align*}\widehat{\kappa}(\mu,\varepsilon,\varepsilon^{\prime},d)=\bigg(\frac{2(\widehat{C}_{\varepsilon'}^{-1-({1+\varepsilon'})/({1-\varepsilon'})}\vert K\vert^{{-2\varepsilon'}/({1-\varepsilon'})}c_0^{1+\varepsilon'})^{-4\varepsilon'}}{\gamma(S,\varepsilon')\rho_{\varepsilon}}\bigg)^d ,\end{align*} $$

any ball of radius less than $c_{\underline {i}}$ can be covered by less than $\widehat {\kappa }(\mu ,\varepsilon ,\varepsilon ^{\prime },d)r^{-4d\varepsilon ' \cdot (1+4\varepsilon ')}$ balls of radius less than $\gamma (S,\varepsilon ')\rho _{\varepsilon }c_{\underline {i}}^{1+4\varepsilon ^{\prime }}$ .

This proves that for any sequence of balls $\widehat {L}_n$ with $\vert \widehat {L}_n \vert \leq c_{\underline {i}}$ covering $A\cap E_{\underline {i}}$ , recalling equation (73), it holds that

(74) $$ \begin{align} \sum_{n\in\mathbb{N}}|\widehat{L}_n| ^{\alpha-\varepsilon} &\geq Q_{d,{1}/{3}}^{-1}\widehat{\kappa}(\mu,\varepsilon,\varepsilon^{\prime},d)^{-1}r^{4d\varepsilon' \cdot(1+4\varepsilon')}\kappa(\mu,\varepsilon^{\prime},\varepsilon)r^{(1+16\varepsilon')(\alpha-\varepsilon)} \end{align} $$

(75) $$ \begin{align} &\geq Q_{d,{1}/{3}}^{-1}\widehat{\kappa}(\mu,\varepsilon,\varepsilon^{\prime},d)^{-1}\kappa(\mu,\varepsilon^{\prime},\varepsilon)r^{(1+16\varepsilon')(\alpha-\varepsilon)+4d\varepsilon' \cdot(1+4\varepsilon')}. \end{align} $$

Recalling that $\vert E_{\underline {i}}\vert \leq c_{\underline {i}}$ and Definition 4, since equation (74) is valid for any covering $(\widehat {L}_n )_{n\in \mathbb {N}}$ of $A\cap E_{\underline {i}}$ with $\vert L_n \vert \leq c_{\underline {i}}$ , one has

(76) $$ \begin{align} \vert B\vert^{\alpha-\varepsilon}&\geq \mathcal{H}^{\alpha-\varepsilon}_{\infty}(A)\geq \mathcal{H}^{\alpha-\varepsilon}_{\infty}(A \cap E_{\underline{i}})\nonumber\\ &\geq Q_{d,{1}/{3}}^{-1}\widehat{\kappa}(\mu,\varepsilon,\varepsilon^{\prime},d)^{-1}\kappa(\mu,\varepsilon^{\prime},\varepsilon)r^{(1+16\varepsilon')(\alpha-\varepsilon)+4d\varepsilon' \cdot(1+4\varepsilon')}. \end{align} $$

Taking the inf over all the set $A\subset B$ satisfying $\mu (A)=\mu (B)$ , one obtains

$$ \begin{align*}\vert B\vert^{\alpha-\varepsilon}\geq \mathcal{H}^{\mu,s}_{\infty}(B)\geq Q_{d,{1}/{3}}^{-1}\widehat{\kappa}(\mu,\varepsilon,\varepsilon^{\prime},d)^{-1}\kappa(\mu,\varepsilon^{\prime},\varepsilon)r^{(1+16\varepsilon')(\alpha-\varepsilon)+4d\varepsilon' \cdot(1+4\varepsilon')}. \end{align*} $$

The results stands for balls of diameter less than $c_0 .$

Set

$$ \begin{align*}\varepsilon^{\prime}_0=16\varepsilon'(\alpha-\varepsilon)+4d\varepsilon' \cdot(1+4\varepsilon')\end{align*} $$

and write

$$ \begin{align*}\gamma(d, \mu, \varepsilon,\varepsilon^{\prime}_0)= c_0 ^{\alpha-\varepsilon+\varepsilon^{\prime}_0} Q_{d,{1}/{3}}^{-1}\widehat{\kappa}(\mu,\varepsilon,\varepsilon^{\prime}_0,d)^{-1}\kappa(\mu,\varepsilon^{\prime}_0,\varepsilon).\end{align*} $$

For any ball of radius less than $1$ centered on K, one has

$$ \begin{align*}\vert B\vert^{\alpha-\varepsilon}\geq \mathcal{H}^{\mu,\alpha-\varepsilon}_{\infty}(B)\geq \gamma(d, \mu, \varepsilon,\varepsilon^{\prime}_0) r^{\alpha-\varepsilon+\varepsilon^{\prime}_0}.\\[-37pt] \end{align*} $$

The estimates of Theorem 5.5 are now established in the case of general open sets.

Recall that by item $(5)$ of Proposition 5.14, for any $s>\dim (\mu )$ and any set E, $ \mathcal {H}^{\mu ,s}_{\infty }(E)=0.$

Let us fix $s<\dim (\mu )$ , $\varepsilon ^{\prime }>0$ and set $\varepsilon '=\min \{({\dim (\mu )-s})/{2},\tfrac 12 \}>0.$

Since $K\cap \Omega \subset \Omega $ and $\mu (K\cap \Omega )=\mu (\Omega )$ , it holds that

$$ \begin{align*}\mathcal{H}^{\mu,s}_{\infty}(\Omega)\leq \mathcal{H}^{s}_{\infty}(\Omega \cap K).\end{align*} $$

It remains to show that there exists a constant $c( d,\mu ,s,\varepsilon ^{\prime })$ such that for any open set $\Omega $ , the converse inequality

$$ \begin{align*}c(d,\mu,s,\varepsilon^{\prime})\mathcal{H}^{s+\varepsilon^{\prime}}_{\infty}(\Omega \cap K)\leq \mathcal{H}^{\mu,s}_{\infty}(\Omega)\end{align*} $$

holds.

Let $E \subset \Omega $ be a Borel set such that $\mu (E)=\mu (\Omega )$ and

(77) $$ \begin{align} \mathcal{H}^{s}_{\infty}(E)\leq 2\mathcal{H}^{\mu,s}_{\infty}(\Omega). \end{align} $$

Let $\{L_n \}_{n\in \mathbb {N}}$ be a covering of E by balls verifying

(78) $$ \begin{align} \mathcal{H}^{s}_{\infty}(L)\leq\sum_{n\geq 0}\vert L_n \vert^s \leq 2\mathcal{H}^{s}_{\infty}(E). \end{align} $$

The covering $(L_n)_{n\in \mathbb {N}}$ will be modified into a covering $(\widetilde {L}_n)_{n\in \mathbb {N}}$ verifying the following properties:

• • $K\cap \Omega \subset \bigcup _{n\in \mathbb {N}}\widetilde {L}_n$ ;

• • $\bigcup _{n\in \mathbb {N}}L_n \subset \bigcup _{n\in \mathbb {N}}\widetilde {L}_n$ ;

• •

  $$ \begin{align*}\sum_{n\geq 0}\vert \widetilde{L}_n \vert^{s+\varepsilon^{\prime}} \leq 8. 2^{s+\varepsilon^{\prime}}\frac{Q_{d}^2}{\gamma( d,\mu,\varepsilon,\varepsilon^{\prime})}\sum_{n\geq 0}\vert L_n \vert ^s,\end{align*} $$

where $Q_{d}$ and $\gamma ( d,\mu ,\varepsilon ,\varepsilon ^{\prime })$ are the constants arising from Theorem 5.7 and Proposition 5.9.

The last item together with equations (77) and (78) then immediately imply that

$$ \begin{align*}\frac{\gamma( d,\mu,\varepsilon,\varepsilon^{\prime})}{8. 2^{s+\varepsilon^{\prime}} Q_{d}^2}\mathcal{H}^{s+\varepsilon^{\prime}}_{\infty}(K\cap \Omega)\leq \mathcal{H}^{\mu,s}_{\infty}(\Omega).\end{align*} $$

Setting $c(d,\mu ,\varepsilon ,\varepsilon ^{\prime })={\gamma ( d,\mu ,\varepsilon ,\varepsilon ^{\prime })}/{8. 2^{s+\varepsilon ^{\prime }} Q_{d}^2} $ will then conclude the proof.

Let us start the construction of the sequence $(\widetilde {L}_n )_{n \in \mathbb {N}}.$

Let $ \Delta =( K \setminus \bigcup _{n\in \mathbb {N}}B_n )\cap \Omega $ . For every $x\in \Delta $ , fix $0< r_x \leq 1$ such that $B(x,r_x)\subset \Omega .$ One of the following alternatives must occur:

1. (1) for any ball $L_n$ such that $L_n \cap B(x,r_x)\neq \emptyset $ , $\vert L_n \vert \leq r_x$ ; or

2. (2) there exists $n_x \in \mathbb {N}$ such that $L_{n_x}\cap B(x,r_x)\neq \emptyset $ and $\vert L_{n_x}\vert \geq r_x$ .

Consider the set $S_1$ of points of X for which the first alternative holds.

By Theorem 5.7, it is possible to extract from the covering of $S_1$ , $\{B(x,r_x),x\in S_1\}$ , $Q_{d}$ families of pairwise disjoint balls, $\mathcal {F}_1 ,\ldots ,\mathcal {F}_{Q_{d}}$ such that

$$ \begin{align*}S_1 \subset \bigcup_{1\leq i\leq Q_{d}}\bigcup_{L\in\mathcal{F}_i}L.\end{align*} $$

Now, any ball $L_n$ intersecting a ball $L\in \bigcup _{1\leq i\leq Q_{d}}\mathcal {F}_i$ must satisfy $\vert L_n \vert \leq L.$ In particular, since for any $1\leq i\leq Q_{d}$ , the balls of $\mathcal {F}_i$ are pairwise disjoint, applying Lemma 5.8 to the ball of $\mathcal {F}_i$ intersecting L, we get that the ball $L_n$ intersects at most $Q_{d}$ balls of $\mathcal {F}_i$ and hence at most $Q_{d}^2$ balls of $\bigcup _{1\leq i\leq Q_{d}}\mathcal {F}_i .$

Let $L\in \bigcup _{1\leq i\leq Q_{d}}\mathcal {F}_i .$ One aims at replacing all the balls $L_n$ intersecting L by the ball $2L$ .

For any $1\leq i \leq Q_{d}$ and any ball $L\in \mathcal {F}_i$ , denote by $\mathcal {G}_L$ the set of balls $L_n$ intersecting L. Since $E\subset \bigcup _{n\in \mathbb {N}}L_n$ and $\mu (E)=\mu (\Omega )$ , one has $E\cap L\subset \bigcup _{B\in \mathcal {G}_L}B$ and $\mu (E\cap L)=\mu (L)$ . By Definition 5.1 and Proposition 5.9, this implies that

(79) $$ \begin{align} \gamma( d,\mu,\varepsilon,\varepsilon^{\prime})\vert L\vert^{s+\varepsilon^ {\prime}} \leq\mathcal{H}^{\mu,s}_{\infty}(L)\leq \sum_{B\in\mathcal{G}_L}\mathcal{H}^{\mu,s}_{\infty}(B)\leq \sum_{B\in\mathcal{G}_L}\vert B\vert^s. \end{align} $$

Replace the balls of $\mathcal {G}_L$ by the ball $\widehat {L}=2L$ (recall that $\bigcup _{B\in \mathcal {G}_{L}}B \subset 2L$ ). The new sequence of balls so obtained by the previous construction applied to all the balls $L\in \bigcup _{\leq i\leq Q_{d}}\mathcal {F}_i$ is denoted by $(\widehat {L}_k)_{1\le k \le K}$ , where $0\le K\le +\infty $ .

It follows from the construction and equation (79) that $S_1 \subset \bigcup _{1\le k\le K}\widehat {L}_k$ and

(80) $$ \begin{align} \sum_{1\le k\le K}\bigg (\frac{\vert \widehat{L}_k \vert}{2}\bigg)^{s+\varepsilon^{\prime}} \leq \frac{Q_{d}^2}{\gamma(d,\mu,\varepsilon,\varepsilon^{\prime})}\sum_{n\geq 0}\vert L_n \vert^s. \end{align} $$

However, since for any $x\in S_2 = \Delta \setminus S_1$ , there exists $n_x\in \mathbb {N}$ such that ${L}_{n_x}\cap B(x,r_x)\neq \emptyset $ and $r_x \leq \vert {L}_{n_x}\vert $ , one has $S_2\subset \bigcup _{n\in \mathbb {N}}2{L}_n$ , so that

$$ \begin{align*} \bigg (\bigcup_{n\in\mathbb N}L_n\bigg )\cup \bigg(K\cap \Omega \setminus \bigcup_{n\in\mathbb N}L_n\bigg ) \subset \bigg (\bigcup_{1\le k\le K}\widehat L_k\bigg )\cup \bigg ( \bigcup_{n\in\mathbb N}2{L}_n\bigg ). \end{align*} $$

Putting the elements of $(\widehat {L}_k)_{1\le k \le K}$ and $(2L_n)_{n\ge 0}$ in a single sequence $(\widehat { L}_n)_{n\ge 0}$ , writing $(\widetilde {L}_n :=2\widehat {L}_n)_{n\in \mathbb {N}}$ , by construction, $K\cap \Omega \subset \bigcup _{n\in \mathbb N}\widetilde {L}_n$ and due to equation (80),

$$ \begin{align*} \mathcal{H}^ {s+\varepsilon^{\prime}}_{\infty}(K\cap \Omega)&\leq\sum_{n\geq 0}\vert \widetilde{L}_n \vert^{s+\varepsilon^{\prime}} \leq 2^{s+\varepsilon^{\prime}}\bigg(\frac{Q_{d}^2}{\gamma(d,\mu,\varepsilon,\varepsilon^{\prime})}+1\bigg)\sum_{n\geq 0}\vert L_n \vert^s \\ &\leq 8.2^{s+\varepsilon^{\prime}} \frac{Q_{d}^2}{\gamma( d,\mu,\varepsilon,\varepsilon^{\prime})}\mathcal{H}^{\mu,s}_{\infty}(\Omega). \end{align*} $$

The proof is concluded now by setting

$$ \begin{align*}c( d,\mu,s,\varepsilon^{\prime})=\frac{\gamma( d,\mu,\dim(\mu)-s,\varepsilon^{\prime})}{Q_{d}^2 8 .2^{s+\varepsilon^{\prime}}}.\\[-47pt] \end{align*} $$

5.2 Mass transference principle in the weakly conformal case

Combining Theorem 5.4 with Theorem 5.5 and Proposition 5.3 yields the following result.

Proof. One proves the first item of Theorem 5.11.

Fix $\mu \in \mathcal {M}(\mathbb {R}^d)$ supported on K satisfying that for every $\underline {i}\in \bigcup _{k\geq 1}\{1,\ldots ,m\}^k, \mu \circ f_{\underline {i}}^{-1}$ is absolutely continuous with respect to $\mu .$ Let $(B_n)_{n\in \mathbb {N}}$ be a $\mu $ -a.c. sequence of balls centered on K satisfying $\vert B_n \vert \to 0$ . Let us fix $\varepsilon>0$ .

Let us start with a lemma whose proof can be found in [Reference Daviaud12, Lemma 4.9].

Lemma 5.13. Let $\mu \in \mathcal {M}(\mathbb {R}^d)$ . Let $\mathcal {B} =(B_n :=B(x_n ,r_n))_{n\in \mathbb {N}}$ be a $\mu $ -a.c. sequence of balls of $ \mathbb {R}^d$ . Then, for every $\varepsilon>0$ , there exists a $\mu $ -a.c. sub-sequence $(B_{\phi (n)})_{n\in \mathbb {N}}$ of $\mathcal {B} $ such that for every $n\in \mathbb {N}$ , $\mu (B_{\phi (n)})\leq (r_{\phi (n)})^{\underline {\dim }_H (\mu )-\varepsilon }.$

By Lemma 5.13, up to an extraction, one can assume that $\mu (B_n)\leq \vert B_n\vert ^{\dim (\mu )-{\varepsilon }/{4}}.$

The following proposition is proved in [Reference Daviaud10, Proposition 3.12].

Proposition 5.14. Let $\mu \in \mathcal {M}(\mathbb {R}^d)$ , $s\geq 0$ , and $A\subset \mathbb {R}^d $ be a Borel set. The s-dimensional $\mathcal {H}^{\mu ,s}_{\infty }(\cdot )$ outer measure satisfies the following properties.

1. (1) If $\vert A\vert \leq 1$ , the mapping $s\geq 0\mapsto \mathcal {H}^{\mu ,s}_{\infty }(A)$ is decreasing from $\mathcal {H}^{\mu ,0}_{\infty }(A)=1$ to $\lim _{t\to +\infty }\mathcal {H}^{\mu ,t}_{\infty }(A)=0$ .

2. (2) $0\leq \mathcal {H}^{\mu ,s}_{\infty }(A)\leq \min \{\vert A\vert ^s, \mathcal {H}^{s}_{\infty }(A)\}$ .

3. (3) For every subset $ B\subset A$ with $\mu (A)=\mu (B)$ , $\mathcal {H}^{\mu ,s}_{\infty }(A)=\mathcal {H}^{\mu ,s}_{\infty }(B).$

4. (4) For every $\delta \geq 1$ , $\mathcal {H}^{\mu ,{s}/{\delta }}_{\infty }(A)\geq (\mathcal {H}^{\mu ,s}_{\infty }(A))^{{1}/{\delta }}.$

5. (5) For every $ s>\overline {\dim }_H (\mu )$ , $\mathcal {H}^{\mu ,s}_{\infty }(A)=0.$

Also, by Theorem 5.5 and item $(5)$ of Proposition 5.14, there exists a constant $c(d,\mu ,\dim (\mu )-{\varepsilon }/{2},{\varepsilon }/{4})$ such that, for any $n\in \mathbb {N}$ , for any $\delta>1$ ,

Taking n large enough so that $\vert B_n \vert ^{-{\varepsilon }/{4}}\geq c(d,\mu ,\underline {\dim }_H(\mu ))-{\varepsilon }/{2},{\varepsilon }/{4})$ , one gets

(84)

Defining , using equation (84) and Theorem 5.4 with $s_{ \varepsilon }{\kern-1pt}={\kern-1pt} ({\underline {\dim }_H(\mu ){\kern-1pt}-{\kern-1pt}\varepsilon }/{\delta })$ and letting $\varepsilon \to 0$ finishes the proof of the first item.

Assume now that the sequence $(B_n)_{n\in \mathbb {N}}$ satisfies only $\mu (\limsup _{n\rightarrow +\infty }B_n).$ Then, by Proposition 5.3, $(2B_n)_{n\in \mathbb {N}}$ is $\mu $ -a.c.

Since, for any $\varepsilon>0$ ,

$$ \begin{align*}\limsup_{n\rightarrow+\infty}(2B_n )^{\delta+\varepsilon}\subset \limsup_{n\rightarrow+\infty}B_n^{\delta},\end{align*} $$

applying the first item of Theorem 5.11 to $(2B_n )_{n\in \mathbb {N}}$ , one gets

$$ \begin{align*}\dim_H \Big(\!\limsup_{n\rightarrow+\infty}B_n ^{\delta}\Big)\geq \frac{\underline{\dim}_H(\mu)}{\delta +\varepsilon}.\end{align*} $$

Since $\varepsilon $ was arbitrary, the second item is proved.

6.1 Proof of item (1) of Theorem 3.1

Write $s=\dim _H(K)$ . The notation of the proof of Theorem 5.5 is adopted in this section.

Lemma 6.4. For any $x_0\in U$ and any $\delta <1$ ,

$$ \begin{align*}\limsup_{\underline{i}\in\Lambda^*}B(f_{\underline{i}}(x_0),\vert f_{\underline{i}}(K)\vert^{\delta})=K.\end{align*} $$

Proof. Note first that, since K is the (compact) attractor of S,

$$ \begin{align*}\limsup_{\underline{i}\in\Lambda^*}B(f_{\underline{i}}(x_0),\vert f_{\underline{i}}(K)\vert^{\delta})\subset K.\end{align*} $$

We now prove the converse inclusion.

Let $c>1$ . By Lemma 4.1 and Remark 4.2 applied with

$$ \begin{align*}X= \bigcup_{\underline{i}\in\Lambda ^*}f_{\underline{i}}(x_0)\cup K,\end{align*} $$

there exists $D(c)>0$ such that for any $y\in K$ and any $\underline {i}=(i_1,\ldots ,i_n)\in \Lambda ^*,$

(85) $$ \begin{align} \| f_{\underline{i}}(x_0)-f_{\underline{i}}(y)\|\leq D(c)c^n \| f^{\prime}_{\underline{i}}(y)\|\cdot \| x_0-y\|. \end{align} $$

By Lemma 4.1 and equation (20), one has

(86) $$ \begin{align} \| f_{\underline{i}}'(y)\|\leq D(c)c^n \vert f_{\underline{i}}(K)\vert. \end{align} $$

Combining equations (85) and (86), one gets

(87) $$ \begin{align} \| f_{\underline{i}}(x_0)-f_{\underline{i}}(y)\|\leq \max_{z\in K}d(x_0,z)D(c)^2 c^{2n}\vert f_{\underline{i}}(K)\vert. \end{align} $$

Recall that there exist $0<t_1<t_2$ so that, uniformly on n and $\underline {i}\in \Lambda ^n,$

$$ \begin{align*}t_1\leq \frac{\log \| f_{\underline{i}}\|}{n}\leq t_2.\end{align*} $$

Set $\varepsilon =\delta -1>0,$ taking $c=e^{{t_1 \varepsilon }/{4}}$ and writing $\kappa (S,\varepsilon ,x_0)=\max _{z\in K}d(x,z)D(c)^2,$ one gets

(88) $$ \begin{align} \| f_{\underline{i}}(x_0)-f_{\underline{i}}(y)\|\leq \kappa(S,\varepsilon,x_0)\vert f_{\underline{i}}(K)\vert^{1-{\varepsilon}/{2}}. \end{align} $$

In particular, since $\vert f_{\underline {i}}(K)\vert \to 0,$ for n large enough, for any $\underline {i}\in \Lambda ^{n}$ ,

$$ \begin{align*}f_{\underline{i}}(K)\subset B(f_{\underline{i}}(x_0),\vert f_{\underline{i}}(K)\vert^{1-\varepsilon}).\end{align*} $$

Recalling that

$$ \begin{align*}K= \bigcup_{\underline{i}\in\Lambda^n}f_{ \underline{i}}(K), \end{align*} $$

one concludes that $K\subset \limsup _{\underline {i}\in \Lambda ^*}B(f_{\underline {i}}(x_0),\vert f_{\underline {i}}(K)\vert ^{1-\varepsilon }).$