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Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem

Vitor Araujo, Vilton Pinheiro

TL;DR

This work addresses the basin problem and statistical properties for non-uniformly hyperbolic systems with a dominated splitting, focusing on hyperbolic cu-Gibbs states. It extends Gibbs-Markov-Young (GMY) structures from partial hyperbolicity to dominated splittings by leveraging improved hyperbolic blocks built from hyperbolic times and coherent schedules. The authors prove that the geometric, ergodic, and topological basins coincide modulo Lebesgue measure and establish robust exponential mixing for physical measures without requiring a uniformly expanding or contracting subbundle; mixing rates are governed by the tail of the hyperbolic-time expansion function $h(x)$. Additionally, they develop a full GMY framework for non-uniformly hyperbolic attracting sets, ensuring the existence of physical measures with integrable return times and enabling precise correlation-decay estimates, with broad applicability to partially hyperbolic and dominated-splitting systems.

Abstract

We show that the ergodic, topological and geometric basins coincide for hyperbolic dominated ergodic $cu$-Gibbs states, solving the ``basin problem'' for a wide class of non-uniformly hyperbolic systems. We obtain robust examples of exponential mixing physical measures for systems with multidimensional nonuniform hyperbolic dominated splitting, without uniformly expanding or contracting subbundles. Both results are a consequence of extending the construction of Gibbs-Markov-Young structures from partial hyperbolic systems to systems with only a dominated splitting, using the existence of an ``improved hyperbolic block'', with respect to Pesin's Nonuniform Hyperbolic Theory, for hyperbolic dominated measures of smooth maps, obtained through hyperbolic times and associated ``coherent schedules'' introduced by one of the coauthors.

Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem

TL;DR

This work addresses the basin problem and statistical properties for non-uniformly hyperbolic systems with a dominated splitting, focusing on hyperbolic cu-Gibbs states. It extends Gibbs-Markov-Young (GMY) structures from partial hyperbolicity to dominated splittings by leveraging improved hyperbolic blocks built from hyperbolic times and coherent schedules. The authors prove that the geometric, ergodic, and topological basins coincide modulo Lebesgue measure and establish robust exponential mixing for physical measures without requiring a uniformly expanding or contracting subbundle; mixing rates are governed by the tail of the hyperbolic-time expansion function . Additionally, they develop a full GMY framework for non-uniformly hyperbolic attracting sets, ensuring the existence of physical measures with integrable return times and enabling precise correlation-decay estimates, with broad applicability to partially hyperbolic and dominated-splitting systems.

Abstract

We show that the ergodic, topological and geometric basins coincide for hyperbolic dominated ergodic -Gibbs states, solving the ``basin problem'' for a wide class of non-uniformly hyperbolic systems. We obtain robust examples of exponential mixing physical measures for systems with multidimensional nonuniform hyperbolic dominated splitting, without uniformly expanding or contracting subbundles. Both results are a consequence of extending the construction of Gibbs-Markov-Young structures from partial hyperbolic systems to systems with only a dominated splitting, using the existence of an ``improved hyperbolic block'', with respect to Pesin's Nonuniform Hyperbolic Theory, for hyperbolic dominated measures of smooth maps, obtained through hyperbolic times and associated ``coherent schedules'' introduced by one of the coauthors.

Paper Structure

This paper contains 41 sections, 28 theorems, 63 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Non-uniform expansion and/or contraction, Gibbs states and physical measures
  4. Non-uniform hyperbolicity
  5. Hyperbolic and dominated invariant probability measures
  6. Attracting sets
  7. Gibbs states
  8. Ergodic and geometric basin coincide Lebesgue modulo zero
  9. Average expansion times and mixing for hyperbolic dominated Gibbs state
  10. Robust non-uniformly hyperbolic exponentially mixing class
  11. GMY structure for hyperbolic dominated $cu$-Gibss states
  12. Existence of physical measures and GMY structure
  13. Consequences for hyperbolic dominated measures
  14. Organization of the text, comments and conjectures
  15. Comments and conjectures
  16. ...and 26 more sections

Key Result

Theorem A

Let $g:M\circlearrowleft$ be a $C^{1+\eta}$ diffeomorphism, for some $\eta\in(0,1]$, with a dominated splitting $T_AM=E^{cs}_A\oplus E^{cu}_A$ over an attracting set $A$ on a trapping neighborhood $U\subset M$, and an ergodic hyperboliic dominated $cu$-Gibbs state $\mu$ for $g$ with $\operatorname{s If $A$ is an attractor (i.e., transitive), then $\operatorname{supp}\mu=A$ and we obtain modulo ze

Figures (8)

  • Figure 1: Sketch of the "figure 8" attracting set $A$ given by the double homoclinic connection $W^s(P)$ associated to tha hyperbolic saddle fixed point $P$ and an attracting neighborhood $U$.
  • Figure 2: Sketch of the deformation of the linear Anosov diffeomorphism around the hyperbolic fixed point $p$ with stable index $s$ in the left hand side. In the center figure two new saddles appear with the same stable index $s$ while the stable index of $p$ becomes $s-1$. In the right hand side, the saddle $q$ becomes an attracting center along the stable direction. The strong unstable direction $E^{uu}$ depicted above has dimension $u\ge2$.
  • Figure 3: Backward contractions at hyperbolic times.
  • Figure 4: Forward contractions at inverse hyperbolic times above versus forward contractions at reverse hyperbolic times below.
  • Figure 5: Forward contractions from $f^{-n}x$ to $x$.
  • ...and 3 more figures

Theorems & Definitions (66)

  • Remark 2.1: Hyperbolic dominated Gibbs states and non-uniform hyperbolicity
  • Theorem A
  • Remark 2.2: wild attractors
  • Theorem B
  • Remark 2.3
  • Corollary C
  • Theorem D
  • Theorem 2.4
  • Corollary E
  • Remark 2.5
  • ...and 56 more