We give examples of non-integrable measurable functions for which there are ‘many’ rotations such that the arithmetic (ergodic) averages exist for almost every x. We also show that if the above ergodic averages exist for almost every x for a set of rotations of positive measure, then the function should be integrable on [0, 1].

We determine the Hausdorff and Minkowski dimensions of some self-affine Sierpinski sponges, extending results of McMullen and Bedford. This result is used to show that every compact set invariant under an expanding toral endomorphism which is a direct sum of conformal endomorphisms supports an invariant measure of full dimension.

We combine recurrence properties of polynomials and IP-sets and show that polynomials evaluated along IP-sequences also give rise to Poincaré sets for measure-preserving systems, that is, sets of integers along which the analogue of the Poincaré recurrence theorem holds. This is done by applying to measure-preserving transformations a limit theorem for products of appropriate powers of a commuting family of unitary operators.

We construct linked twist maps of the two-dimensional torus which are Bernoulli and possess infinite entropy. Inparticular, we construct a Bernoulli toral linked twist map B* of infinite entropy which has smooth, absolutely continuous local (un)stable manifolds and positive Lyapunov exponents defined almost everywhere. This map is continuous at each point save those on two line segments.

B* is shown to be stochastically stable under the following random perturbation: apply the map to a point p and then jump (all points move the same distance and in the same direction) according to a B-process (not necessarily an independent process) such that the expected distance moved is equal to r. Stochastic stability means that given α > 0 if r > 0 is sufficiently small then the perturbed and unperturbed systems are α-congruent. We prove a similar stability result for B* under a perturbation in which the random jump described above is distributed according to a general stochastic process. These stability results are also shown to hold (in a slightly modified form) for a general class of finite-entropy toral linked twist maps under the same perturbations.

We give a condition for automorphisms α and β on finite von Neumann algebras which induces the tensor product formula for entropies: H(α ⊗ β) = H(α) + H(β). As an application, the Bernoulli shift (1/n, 1/n, …, 1/n) has extensions to ergodic outer automorphisms {αk; k = 1,2, …} on the hyperfinite II1 factor R with the entropies H(αk) = (1/2)kn log n.

We find necessary conditions for a generalized Sinai billiard to be ergodic, independent of the geometry of the disks; i.e. if the conditions are not satisfied, then we can always construct special geometries for which the system is not ergodic. We prove non-ergodicity by constructing an elliptic periodic orbit and showing that this orbit satisfies the non-degeneracy condition of KAM theory. By a combination of theory and numerical calculation involving a computer algebra system (Mathematica), we show that the first Birkhoff invariant is non-zero. We apply this result to produce non-ergodic Hamiltonian flows that are generated by smooth potentials of finite range (repelling potentials and Lennard–Jones potentials) and by geodesic flows on the torus.

Given a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

In this paper we obtain regularity results for the Livsic cohomology equation for cocycles with values in diffeomorphism groups and Lie groups.

We consider a class of skew-products with base the one-sided two-shift equipped with the (½, ½)Bernoulli measure and fibre either or the circle. We give conditions for the first kind to be isomorphic to the base itself and use this result to establish an isomorphism with the base for the second kind when a certain irrationalis very well approximated by rationals. A consequence is that there is a circle action of automorphisms commuting with theone-sided two-shift.

We provide an algorithm to compute the first non-zero derivative of the return map r ↦ L(r, ε) of a planar vector field which is a polynomial perturbation of . It yields a new method for finding the centre conditions in the case of a homogeneous perturbative part.

An IP set in ℕ is a subset of ℕ which coincides with the set of finite sums taken from an infinite sequence in ℕ with not necessarily distinct terms. It has been established by H. Furstenberg and others that there is a rich connection between IP sets and idempotents in the enveloping semigroup E(X) of a compact metric dynamical system. Our aim in this paper is to further develop this program by considering special elements of E(X) with various IP properties.

By Fourier methods, it is possible to prove central limit theorems for Riesz-Raikov sums for any real θ > 1 and for a quite large class of functions f.

The same problem is solved here for Hölder-continuous functions f and Pisot–Vijayaragavan numbers θ in a totally different way: it is shown that the question is equivalent to working with ergodic sums for suitable functions F on [0;l], and T denoting the θ-transformation x ↦ θx mod 1. In addition, limit theorems are proved on the θ-shifts, for any real θ 1.

An α-rotation orbit for a self-map of the circle is an orbit whose cyclic order on the circle is the same as for any orbit of rotation by a. In this paper, we classify sets which are minimal sets comprised of a-rotation orbits for degree d αself-coverings of the circle.

Let T be an ergodic automorphism defined on a standard Borel probability space for which T and T−1 are isomorphic. We investigate the form of the conjugating automorphism. It is well known that if T is ergodic having a discrete spectrum and S is the conjugation between T and T−1, i.e. S satisfies TS = ST−1 then S2 = I the identity automorphism. We show that this result remains true under the weaker assumption that T has a simple spectrum. If T has the weak closure property and is isomorphic to its inverse, it is shown that the conjugation S satisfies S4 = I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.

In this paper, structural stability of discrete-time linear random dynamical systems is studied. A random dynamical system is called structurally stable with respect to a random norm if it is topologically conjugate to any random dynamical system which is sufficiently close to it in this norm. We prove that a discrete-time linear random dynamical system is structurally stable with respect to its Lyapunov norms if and only if it is hyperbolic.

We introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

We show that near the geodesic flow of a compact Riemannian manifold with no conjugate points which is expansive, every expansive geodesic flow has no conjugate points. We also prove that in the above hypotheses the geodesic flow istopologically stable.

This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

Let G be a countable group of automorphisms of a Lebesgue space (X, m) and let [G] be the full group of G. For a pair of countable ergodic subgroups H1 and H2 of [G], the following problem is considered: when are the full subgroups [H1] and [H2] conjugate in the normalizer N[G] = {g ∈ Aut X: g[G]g-1 = [G]} of [G]. A complete solution of the problem is given in the case when [G] is an approximately finite group of type II and [H] is admissible, in the sense that there exists an ergodic subgroup [H0] of [G] and a countable subgroup Γ ⊂ N[H0] consisting of automorphisms which are outer for [H0], such that [H0] ⊂ [G] and the full subgroup [Ho, Γ] generated by [H0] and Γ coincides with [G].

We study the pressure function for critically finite polynomials and analyze the case when this function fails to be real analytic.