Related papers: Thomae's function and the space of ergodic measure…

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is…

We give an explicit description of all finite Borel measures on Noetherian topological spaces X, and characterize them as objects dual to a space of functions on X. We use these results to study the asymptotic behavior of continuous…

In this paper we develop a rigorous foundation for the study of integration and measures on the space $\mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study several interrelated $\sigma$-algebras and a…

We describe the behaviour of the homotopy similarity relations and finite-order invariants under the function $[X,Y]\to[X,Z]$ induced by a map $Y\to Z$ strongly $r$-similar to the constant map.

Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant…

If a function $f$, acting on a Euclidean space $\mathbb{R}^n$, is "almost" orthogonally additive in the sense that $f(x+y)=f(x)+f(y)$ for all $(x,y)\in\bot\setminus Z$, where $Z$ is a "negligible" subset of the $(2n-1)$-dimensional manifold…

Let S_1 and S_2 be ergodic extensions of finite measure preserving transformations T_1 and T_2, where the extensions are by rotations of a compact group G. Then there is an N-valued function k, measurable with respect to the factor T_1, so…

Previous work introduced two measure-conjugacy invariants: the $f$-invariant (for actions of free groups) and $\Sigma$-entropy (for actions of sofic groups). The purpose of this paper is to show that the $f$-invariant is a special case of…

By using a suitable Banach space on which we let the transfer operator act, we make a detailed study of the ergodic theory of a unimodal map $f$ of the interval in the Misiurewicz case. We show in particular that the absolutely continuous…

The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the $T_{3 \frac{1}{2}}$ space. Motivated by the generalized Arzel\`a-Ascoli theorem for Tichonoff space $T$ via Wallman…

We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic $u$-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special…

We study invariant ergodic measures for quasiperiodically forced circle homeomorphisms and derive that either the system is uniquely ergodic or any such measure is associated to some invariant multigraph.

Let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with either $0 < \lambda(A) < 1 < \lambda(X)$ and $0 < \mu(B) < \mu(Y)$, or the other way…

This paper presents some numerical experiments in relation with the theoretical study of the ergodic short-term behaviour of discretizations of expanding maps done in arXiv:2206.07991 [math.DS]. Our aim is to identify the phenomena driving…

In this paper, we study how the cohomology of nilpotent groups is affected by Lipschitz maps. We show that, given a smooth Lipschitz map $f$ between two simply-connected nilpotent Lie groups $G$ and $H$, there is a map $\psi$ that induces…

Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. The estimates are uniform with respect to all graphs of a given finite length, and the constant factors in the…

We effect thermodynamical formalism for the non-uniformly hyperbolic $C^\infty$ map of the two dimensional torus known as the Katok map. It is a slowdown of a linear Anosov map near the origin and it is a local (but not small) perturbation.…

For a non-elementary subgroup of the mapping class group of a surface, we study its invariant Radon measures on the space of measured laminations, by classifying them on the recurrent measured laminations. In particular, given a…

Let $f: \mathbb{T}^2 \to \mathbb{T}^2$ be a homeomorphism homotopic to the identity and $F: \mathbb{R}^2 \to \mathbb{R}^2$ a lift of $f$ such that the rotation set $\rho(F)$ is a line segment of rational slope containing a point in…

Invariant ergodic measures for generalized Boole type transformations are studied using an invariant quasi-measure generating function approach based on special solutions to the Frobenius--Perron operator. New two-dimensional Boole type…