Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant Measures of Non-Uniformly Expanding Maps with Higher Order Critical Set

Ricardo Chicalé, Vanderlei Horita

TL;DR

The paper generalizes Viana-type maps to include higher-order critical points and proves the existence and uniqueness of absolutely continuous invariant measures for these non-uniformly expanding systems within a small $C^D$-neighborhood. By constructing a carefully designed partition into rectangles and an induced map with bounded distortion, it leverages the AL00 framework to obtain a finite, ergodic, ACIP and, via ALP05, super-polynomial decay of correlations and a Central Limit Theorem. The argument also yields topological mixing and, consequently, uniqueness of the SRB measure for the considered family. The results illuminate statistical properties of a broad class of skew-product-like maps with critical points and provide a rigorous route to quantitative ergodic conclusions in this setting.

Abstract

We prove existence and uniqueness of absolutely continuous invariant measures for generalizations of Viana maps admitting a higher order critical point introduced in arXiv:2312.00906. As a consequence of our approach, we obtain super-polynomial decay of correlations.

Invariant Measures of Non-Uniformly Expanding Maps with Higher Order Critical Set

TL;DR

The paper generalizes Viana-type maps to include higher-order critical points and proves the existence and uniqueness of absolutely continuous invariant measures for these non-uniformly expanding systems within a small -neighborhood. By constructing a carefully designed partition into rectangles and an induced map with bounded distortion, it leverages the AL00 framework to obtain a finite, ergodic, ACIP and, via ALP05, super-polynomial decay of correlations and a Central Limit Theorem. The argument also yields topological mixing and, consequently, uniqueness of the SRB measure for the considered family. The results illuminate statistical properties of a broad class of skew-product-like maps with critical points and provide a rigorous route to quantitative ergodic conclusions in this setting.

Abstract

We prove existence and uniqueness of absolutely continuous invariant measures for generalizations of Viana maps admitting a higher order critical point introduced in arXiv:2312.00906. As a consequence of our approach, we obtain super-polynomial decay of correlations.
Paper Structure (8 sections, 19 theorems, 136 equations, 2 figures)

This paper contains 8 sections, 19 theorems, 136 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary results and definitions
  3. Partition of $\mathbb{S}^1\times \mathcal{M}$
  4. Requirements for the elements of $\mathcal{R}_n$
  5. Construction of the partition
  6. Bounded Distortion
  7. Proof of Theorem A
  8. Proof of Theorem B

Key Result

Theorem A

For $d\geq 16$, $D\geq 2$ and $\alpha$ sufficiently small, the map $\varphi_{\alpha,D}$ has a finite invariant ergodic measure $\mu^{*}$ absolutely continuous with respect to the Lebesgue measure on $\mathbb{S}^1\times \mathcal{M}$ in every invariant component of the dynamics. Moreover, the same hol

Figures (2)

  • Figure 1: Map $h_D$ with a pre-periodic even critical point
  • Figure 2: Map $h_D$ with an odd critical point

Theorems & Definitions (37)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 27 more