Related papers: Piecewise contracting maps on the interval: Hausdo…

We use the concept of Baire Ergodicity and Ergodic Formalism introduced to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the {\em wandering intervals attractors}. As a…

In this note a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space the generalized topological…

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the…

A $1$-Lipschitz map $f$ from a convex compact set to itself has fixed points. This consequence of Brouwer's or Schauder's fixed point theorem has more elementary proofs by approximating $f$ by $\lambda$-contractions, $f_\lambda$. We study…

Piecewise Translations is a class of dynamical systems which arises from some applications in computer science, machine learning, and electrical engineering. In dimension 1 it can also be viewed as a non-invertible generalization of…

Let $\{S_i\}_{i=1}^\ell$ be an iterated function system (IFS) on $\R^d$ with attractor $K$. Let $(\Sigma,\sigma)$ denote the one-sided full shift over the alphabet $\{1,..., \ell\}$. We define the projection entropy function $h_\pi$ on the…

Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We prove a formula relating the topological entropy of f_c to the Hausdorff dimension of the set of rays landing on the real Julia set, and to the Hausdorff dimension of…

We consider piecewise $C^2$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points.…

We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr\rho_A^n for any integer n is calculated as the four-point function of a particular type of twist fields…

Let $-1<\lambda<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $\lambda$-affine map, that is, there exist points $0=c_0<c_1<\cdots <c_{n-1}<c_n=1$ and real numbers $b_1,\ldots,b_n$ such that $f(x)=\lambda x+b_i$ for every $x\in…

In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study…

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of…

We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In…

We study the long-term behavior of the iteration of a random map consisting of Lipschitz transformations on a compact metric space, independently and randomly selected according to a fixed probability measure. Such a random map is said to…

We show that if $G$ is a topological graph, and $f$ is continuous map, then the induced map $\tilde{f}$ acting on the hyperspace $C(G)$ of all connected subsets of $G$ by natural formula $\tilde{f}(C)=f(C)$ carries the same entropy as $f$.…

We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit,…

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1…

We evaluate the entanglement entropy of exactly solvable Hamiltonians corresponding to general families of three-dimensional topological models. We show that the modification to the entropic area law due to three-dimensional topological…

We study the jump of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^r([0; 1])$ with $h_{top}(f) > \frac{\log^+ \|f'\|_\infty}{r}$. To this end we…

We deal with the finite family $\mathcal{F}$ of continuous maps on the Hausdorff space. A nonempty compact subset $A$ of such space is called a strict attractor if it has an open neighborhood $U$ such that…