On the rotation sets of generic homeomorphisms on the torus

Abstract

We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

Sensitivity and historic behavior for continuous maps on Baire metric spaces

“…In the last decades, though, there has been an intense study concerning the set of points for which Cesàro averages do not converge. Contrary to the previous measure-theoretical description, the set of the irregular points may be Baire generic and, moreover, have full topological pressure, full metric mean dimension or full Hausdorff dimension (see [2,5,3,4,22,24,31]). In [8], the first and the fourth named authors obtained a simple and unifying criterion, using first integrals, to guarantee that I(T, ϕ) is Baire generic whenever T : X → X is a continuous dynamics acting on a compact metric space X.…”

Genericity of historic behavior for maps and flows

“…This result supports Boltzman ergodic hypothesis but fails to describe the behavior and the complexity of the set of points at which the sequence of Birkhoff averages has no limit. Nowadays there is a well established theory to assess how big is the irregular set (also called the set of points with historic behavior): contrary to the previous measure theoretical description, the set of these non-typical points may be Baire generic and, moreover, have full topological pressure, full Hausdorff dimension or full metric mean dimension, as attested in [2,6,7,20,23,33] and references therein.…”

Entropy of irregular points for some dynamical systems

“…If X(f, ϕ) is nonempty, for a topologically mixing subshift of finite type (or any topologically conjugate system), by [1] this set carries full topological entropy. This result was generalized to maps satisfying the almost specification property ( [22], see also [16] for the context of flows) or the orbit gluing property (see [14] and further references therein). Moreover, by [6], assuming the asymptotic average shadowing property (AASP for short) and a certain condition on the measure center, the set X(f, ϕ) is either residual or empty.…”