Related papers: Weak expansion properties and large deviation prin…

Recent work of Dylan Thurston gives a condition for when a post-critically finite branched self-cover of the sphere is equivalent to a rational map. We apply D. Thurston's positive criterion for rationality to give a new proof of a theorem…

We study the dynamics of smooth interval maps with non-flat critical points. For every such a map that is topologically exact, we establish the full (level-2) Large Deviation Principle for empirical means. In particular, the Large Deviation…

In this paper we apply techniques from nonstandard analysis to study expansive dynamical systems. Among other results, we provide a necessary and sufficient condition for an expansive homeomorphism on a compact metric space to admit…

For a smooth expanding circle map, we show that the empirical distribution of Lyapunov exponents of periodic points of any fixed period is close to normal, with an error that decreases as the period grows. This establishes a version of the…

We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system $(X; T)$ embeds in the $K$-full shift with $h_{top}(T) < \log K $ and $\sharp Per_n(X; T) \leq…

In this paper we study systems of $N$ uniformly expanding coupled maps when $N$ is finite but large. We introduce self-consistent transfer operators that approximate the evolution of measures under the dynamics, and quantify this…

We prove the existence of a fixed point for mappings which satisfy some asymptotic nonexpansive conditions in Banach spaces which are either nearly uniformly convex or they satisfy that asymptotic centers of bounded sequences are compact.…

We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical…

Let $f$ be an expansive Lorenz map and $c$ be the critical point. The survivor set is denoted as $S_{f}(H):=\{x\in[0,1]: f^{n}(x)\notin H, \forall n\geq 0\}$, where $H$ is a open subinterval. Here we study the hole $H=(a,b)$ with $a\leq c…

Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator…

We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its…

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$)…

The goal of this note is to generalize Thurston's Topological Characterization of Rational Functions to the setting when both the covering degree and the set of marked points are infinite. A relevant class of branched coverings are…

For each pseudo-Anosov map $\phi$ on surface $S$, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$, denoted by $A(S,\phi)$. $A(S,\phi)$ is defined by an interaction between the Thurston norm and dilatation of pseudo-Anosov…

Recently, there has been an increasing interest on nonautonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map $F$ results in statistical behaviour close to that of $F$. We show this fact in…

We prove that periodic asymptotic expansiveness introduced in \cite{em} implies the equidistribution of periodic points to measures of maximal entropy. Then following Yomdin's approach \cite{Yom} we show by using semi-algebraic tools that…

Let $\phi:S^2 \to S^2$ be an orientation-preserving branched covering whose post-critical set has finite cardinality $n$. If $\phi$ has a fully ramified periodic point $p_{\infty}$ and satisfies certain additional conditions, then, by work…

While there may be many Thurston metric geodesics between a pair of points in Teichm\"uller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select…

Topological mating is an combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map $F$ on the…

Brehm's extension theorem states that a non-expansive map on a finite subset of a Euclidean space can be extended to a piecewise-linear map on the entire space. In this note, it is verified that the proof of the theorem is constructive…