Related papers: A Note on Measure-Expansive Diffeomorphisms

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal…

It is known that Iterated Function Systems generated by orientation preserving homeomorphisms of the unit interval admit a unique invariant measure on $(0,1)$. The setup for this result is the positivity of Lyapunov exponents at both fixed…

Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and…

For a certain parametrized family of maps on the circle, with critical points and logarithmic singularities where derivatives blow up to infinity, a positive measure set of parameters was constructed in [19], corresponding to maps which…

For any Tychonoff space $X$ let $C_p(X)$ (resp., $C^*_p(X)$) be the set of all continuous (resp., and bounded) functions on $X$ with the pointwise convergence topology. Given Tychonoff spaces $X$ and $Y$, Uspenskij \cite{us} proved that if…

We study H-structures associated to SU-rank 1 measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of…

It is known that every ${\rm C}^r$-orbifold, $1\leq r\leq\infty$, has a compatible ${\rm C}^s$-differential structure, for every $s$, where $r< s\leq\omega$. We prove that if two reduced ${\rm C}^r$-orbifolds, $2\leq r\leq\omega$, are ${\rm…

Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic…

On a Peano continuum, all local stable and unstable components of a continuum-wise expansive homeomorphism are non trivial. In particular, there is sensitive dependence on initial conditions. This generalizes results in \cite{h,l} about…

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between arbitrary locally finite separable metric measure spaces, assuming no metric hypotheses on either…

In this paper, we introduce the concept of S-expansiveness for local homeomorphisms and demonstrate that a class of extended symbolic dynamics, known as zip shift maps, are S-expansive and possess the shadowing property. Furthermore, we…

In this paper we consider expansive homeomorphisms of compact spaces with a hyperbolic metric presenting a self-similar behavior on stable and unstable sets. Several application are given related to Hausdorff dimension, entropy,…

Two structures $M, N$ in the same language are called probably isomorphic if they (or, in case of metric structures, their completions) are isomorphic after forcing with the Lebesgue measure algebra. We show that, if $M$ and $N$ are…

Let $\Lambda$ be an isolated non-trival transitive set of a $C^1$ generic diffeomorphism $f\in\Diff(M)$. We show that the space of invariant measures supported on $\Lambda$ coincides with the space of accumulation measures of time averages…

For a certain parametrized family of maps on the circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit…

Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most…

An inner-distal homeomorphism is one such that each of its proximal cells has empty interior. In locally connected spaces, we prove these homeomorphisms have the following properties: Every $cw$-distal homeomorphism is inner-distal but not…

Consider a definable complete d-minimal expansion $(F, <, +, \cdot, 0, 1, \dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and $2 \le r <\infty$. We prove that the set of definable…

We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily…

This article consists in applications of [arXiv:2511.14232] in the case of homemomorphisms of higher genus surfaces whose homological rotation set is big enough -- a class of dynamics that is open. We first prove a structure theorem for the…