Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ergodic Formalism for topological Attractors and historic behavior

Vilton Pinheiro

Abstract

We introduce the concepts of Baire Ergodicity and Ergodic Formalism, employing them to study topological and statistical attractors. Specifically, we establish the existence and finiteness of such attractors and provide applications for maps of the interval, Viana maps, non-uniformly expanding maps, partially hyperbolic systems, strongly transitive dynamics, and skew-products. In a dynamical system with an abundance of historic behavior (encompassing all systems with some hyperbolicity, particularly Axiom A systems), one can show the existence of a residual set with zero measure for every invariant probability measure. Hence, in principle, utilizing the classical ergodic theory to control the asymptotic topological/statistical behavior of generic orbits is not feasible. Nevertheless, the results presented here can also be applied to such a system, contributing to the study of generic orbits in systems with an abundance of historic behavior.

Ergodic Formalism for topological Attractors and historic behavior

Abstract

We introduce the concepts of Baire Ergodicity and Ergodic Formalism, employing them to study topological and statistical attractors. Specifically, we establish the existence and finiteness of such attractors and provide applications for maps of the interval, Viana maps, non-uniformly expanding maps, partially hyperbolic systems, strongly transitive dynamics, and skew-products. In a dynamical system with an abundance of historic behavior (encompassing all systems with some hyperbolicity, particularly Axiom A systems), one can show the existence of a residual set with zero measure for every invariant probability measure. Hence, in principle, utilizing the classical ergodic theory to control the asymptotic topological/statistical behavior of generic orbits is not feasible. Nevertheless, the results presented here can also be applied to such a system, contributing to the study of generic orbits in systems with an abundance of historic behavior.

Paper Structure

This paper contains 24 sections, 62 theorems, 97 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Statement of mains results
  3. Organization of the text
  4. Topological $\times$ Baire ergodicity
  5. Ergodicity for non-singular maps
  6. $u$-Baire ergodicity
  7. Topological and statistical attractors
  8. Baire ergodic components
  9. Topological attractors for Baire ergodic components
  10. $u$-Baire ergodic components and its topological attractors
  11. Statistical attractors for Baire and $u$-Baire ergodic components
  12. The statistical spectrum
  13. Applications of the ergodic formalism and proofs of mains theorems
  14. Maps with abundance of historic behavior
  15. Strongly transitive maps
  16. ...and 9 more sections

Key Result

Theorem A

Let $f:{\mathbb X}\circlearrowleft$ be a non-singular continuous map. If $f$ is transitive then, given a Borel measurable bounded function $\varphi:{\mathbb X}\to{\mathbb R}$, there exists $\gamma\in{\mathbb R}$ such that for a residual set of points $x\in{\mathbb X}$. As a consequence, for each Borel set $U\subset{\mathbb X}$, there exists $\theta\in[0,1]$ such that $\tau_x(U)=\theta$ for a resi

Figures (5)

  • Figure 1: The picture on the left shows the graph of the double map $S^1\ni[x]\mapsto[2x]\in S^1$. On the right side, the picture shows the graph of a $C^{\infty}$ map $f:S^1\circlearrowleft$, $S^1:={\mathbb R}/{\mathbb Z}$, that is conjugated to the double map and such that $\omega_f(x)=S^1$ and $\omega_f^{\star}([x])=\{[0]\}$ for Lebesgue almost every $[x]\in S^1$.
  • Figure 2: The Logistic map $f_t$ with parameter $0<t\le1$.
  • Figure 3: The picture above represents the graph of a map $f$ that is not continuous. Nevertheless the results of Section \ref{['SectionAttractors']} can be applied to $g:[0,1]\setminus\mathcal{C}\to[0,1]$, where $\mathcal{C}=\{c_1,c_2\}$ and $g=f|_{[0,1]\setminus\mathcal{C}}$, since $X_0:=[0,1]\setminus\mathcal{C}$ is an open and dense subset of $[0,1]$ and $g$ is a non-singular continuous map. As $\widetilde{X}:=\bigcap_{n\ge0}g^{-n}(X_0)$ is a residual set of $[0,1]$ and $f|_{\widetilde{X}}=g|_{\widetilde{X}}$, the generic behavior of a point $x\in[0,1]$ by $f$ can be analyzed by $g$.
  • Figure 4: A planar flow with divergent time averages attributed to Bowen.
  • Figure 5: The figure shows the ball $B_r(f^n(p))$ of radius $r>0$ contained in $W_f^s(V_n^c(p))\subset W_f^s(f^n(U))$.

Theorems & Definitions (144)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Proposition 3.0.1: Prop. 8.22, pp. 47 of KeBook
  • Proposition 3.0.2
  • proof
  • ...and 134 more