Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-02-06T06:53:03.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Deviations in the Franks–Misiurewicz conjecture

Part of: Low-dimensional dynamical systems Topological dynamics

Published online by Cambridge University Press:  26 February 2019

ALEJANDRO PASSEGGI  and
MARTÍN SAMBARINO
ALEJANDRO PASSEGGI
Affiliation:
UdelaR, Facultad de Ciencias, Uruguay email alepasseggi@gmail.com, samba@cmat.edu.uy
MARTÍN SAMBARINO
Affiliation:
UdelaR, Facultad de Ciencias, Uruguay email alepasseggi@gmail.com, samba@cmat.edu.uy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if there exists a counter example for the rational case of the Franks–Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set.

Keywords

Franks–Misiurewicz conjecturerotation theorytopological dynamics

MSC classification

Primary: 37E45: Rotation numbers and vectors
Secondary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37B45: Continua theory in dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2533 - 2540
Check for updates
Copyright
© Cambridge University Press, 2019

References

Boyland, P., de Carvalho, A. and Hall, T.. New rotation sets in a family of torus homeomorphisms. Invent. Math. 204(3) (2016), 895–937.Google Scholar
Calvez, P. L. and Tal, F. A.. Forcing theory for transverse trajectories of surface homeomorphisms. Invent. Math. 212 (2018), 619–729.Google Scholar
Dávalos, P.. On annular maps of the torus and sublinear diffusion. J. Inst. Math. Jussieu 17(4) (2018), 913–978.Google Scholar
Franks, J. and Misiurewicz, M.. Rotation sets of toral flows. Proc. Amer. Math. Soc. 109(1) (1990), 243–249.Google Scholar
Franks, J.. Recurrence and fixed points of surface homeomorphisms. Ergod. Th. & Dynam. Sys. 8* (1988), 99–107 Charles Conley Memorial Issue.Google Scholar
Franks, J.. Realizing rotation vectors for torus homeomorphisms. Trans. Amer. Math. Soc. 311(1) (1989), 107–115.Google Scholar
Jäger, T.. Linearization of conservative toral homeomorphisms. Invent. Math. 176(3) (2009), 601–616.Google Scholar
Jäger, T. and Passeggi, A.. On torus homeomorphisms semiconjugate to irrational rotations. Ergod. Th. & Dynam. Sys. 35 (2015), 2114–2137.Google Scholar
Jäger, T. and Tal, F. A.. Irrational rotation factors for conservative torus homeomorphisms. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1537–1546.Google Scholar
Kocsard, A.. On the dynamics of minimal homeomorphisms of $\mathbb{T}^{2}$ which are not pseudo-rotations. Preprint, 2016, arXiv:1611.03784.Google Scholar
Passeggi, K. and Sambarino. The Franks–Misiurewicz conjecture for extensions of irrational rotations. Preprint, 2016, arXiv:1611.05498.Google Scholar
Kwapisz, J.. A toral diffeomorphism with a nonpolygonal rotation set. Nonlinearity 8(4) (1995), 461–476.Google Scholar
Llibre, J. and MacKay, R. S.. Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Th. & Dynam. Sys. 11(1) (1991), 115–128.Google Scholar
Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. Lond. Math. Soc. (2) 40(3) (1989), 490–506.Google Scholar
Misiurewicz, M. and Ziemian, K.. Rotation sets and ergodic measures for torus homeomorphisms. Fund. Math. 137(1) (1991), 45–52.Google Scholar