Growth and mixing

Krzysztof Fraczek; Leonid Polterovich

Home/Archive/arXiv

arXiv · 0706.0954

Growth and mixing

Krzysztof Fraczek·Leonid Polterovich

View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin-Shapiro sequence.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Krzysztof Fraczek, Leonid Polterovich. 2007-11-08. Growth and mixing. https://arxiv.org/abs/0706.0954

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Conformal hyperbolic distances and holonomies of Anosov diffeomorphisms

We use conformal hyperbolic distances to analyze the stable/unstable holonomies of Anosov diffeomorphisms. The main result is that the lengths of the stable/unstable curves in a conformal hyperbolic distance are preserved by the stable/unstable holonomies, whenever they are well-defined. In particular, the existence of a conformal hyperbolic distance with well-defined length of stable/unstable curves ensures that the holonomies are globally defined. We exhibit a conformal hyperbolic distance with well-defined length of stable/unstable curves for each real Anosov diffeomorphism, which are Anosov diffeomorphisms with a one-dimensional finest dominated splitting where contraction/expansion in each space is constant along orbits.

math.DS↗

arXiv2026

Some consequences of shadowing and entropy-expansiveness

Given any continuous self-map of a compact metric space, we prove that a non-empty uniformly rigid subset must be zero-dimensional, under assumptions of shadowing and h-expansiveness.

math.DS↗

arXiv2026

On ball expanding maps

We study the dynamical properties of ball expanding maps, a class of continuous self-maps defined on compact metric spaces. For a ball expanding map, we show that: (1) the set of periodic points is dense in the chain recurrent set; (2) if the topological entropy of the map is zero, then the chain recurrent set is finite; (3) the map has only finitely many chain components; (4) if the space is perfect, then the topological entropy of the map is positive; and (5) if the space is connected, then the map is locally eventually onto and hence mixing. Several examples are also provided.

math.DS↗