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Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity

Disheng Xu, Jiesong Zhang

TL;DR

This work develops a non-fractal invariance principle linking fractal geometry to the regularity of invariant distributions in partially hyperbolic systems. It proves a phase-transition-type dichotomy: excessive Hölder regularity yields smoothness and holonomy-invariance of invariant distributions, while failure to meet the thresholds yields fractal graphs and reduced regularity. In the 3-torus setting with a contracting center, the authors establish a sharp dichotomy for $E^s$ and $E^c$ (leading to $W^c$ smoothness or fractality) and derive a rigidity criterion: $f$ is $C^{\infty}$-rigid iff $E^s$ and $E^c$ are $\theta_s{+}$ and $\theta_c{+}$ Hölder. The framework unifies fractal graph analysis with invariant-section regularity, introduces stable-fractal conjectures, and suggests broad extensions to higher dimensions and non-uniform settings.

Abstract

We introduce a novel approach linking fractal geometry to partially hyperbolic dynamics, revealing several new phenomena related to regularity jumps and rigidity. One key result demonstrates a sharp phase transition for partially hyperbolic diffeomorphisms $f \in \mathrm{Diff}^\infty_{\mathrm{vol}}(\mathbb{T}^3)$ with a contracting center direction: $f$ is $C^\infty$-rigid if and only if both $E^s$ and $E^c$ exhibit Hölder exponents exceeding the expected threshold. Specifically, we prove: If the Hölder exponent of $E^s$ exceeds the expected value, then $E^s$ is $C^{1+}$ and $E^u \oplus E^s$ is jointly integrable. If the Hölder exponent of $E^c$ exceeds the expected value, then $W^c$ forms a $C^{1+}$ foliation. If $E^s$ (or $E^c$) does not exhibit excessive Hölder regularity, it must have a fractal graph. These and related results originate from a general non-fractal invariance principle: for a skew product $F$ over a partially hyperbolic system $f$, if $F$ expands fibers more weakly than $f$ along $W^u_f$ in the base, then for any $F$-invariant section, if $Φ$ has no a fractal graph, then it is smooth along $W^u_f$ and holonomy-invariant. Motivated by these findings, we propose a new conjecture on the stable fractal or stable smooth behavior of invariant distributions in typical partially hyperbolic diffeomorphisms.

Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity

TL;DR

This work develops a non-fractal invariance principle linking fractal geometry to the regularity of invariant distributions in partially hyperbolic systems. It proves a phase-transition-type dichotomy: excessive Hölder regularity yields smoothness and holonomy-invariance of invariant distributions, while failure to meet the thresholds yields fractal graphs and reduced regularity. In the 3-torus setting with a contracting center, the authors establish a sharp dichotomy for and (leading to smoothness or fractality) and derive a rigidity criterion: is -rigid iff and are and Hölder. The framework unifies fractal graph analysis with invariant-section regularity, introduces stable-fractal conjectures, and suggests broad extensions to higher dimensions and non-uniform settings.

Abstract

We introduce a novel approach linking fractal geometry to partially hyperbolic dynamics, revealing several new phenomena related to regularity jumps and rigidity. One key result demonstrates a sharp phase transition for partially hyperbolic diffeomorphisms with a contracting center direction: is -rigid if and only if both and exhibit Hölder exponents exceeding the expected threshold. Specifically, we prove: If the Hölder exponent of exceeds the expected value, then is and is jointly integrable. If the Hölder exponent of exceeds the expected value, then forms a foliation. If (or ) does not exhibit excessive Hölder regularity, it must have a fractal graph. These and related results originate from a general non-fractal invariance principle: for a skew product over a partially hyperbolic system , if expands fibers more weakly than along in the base, then for any -invariant section, if has no a fractal graph, then it is smooth along and holonomy-invariant. Motivated by these findings, we propose a new conjecture on the stable fractal or stable smooth behavior of invariant distributions in typical partially hyperbolic diffeomorphisms.

Paper Structure

This paper contains 30 sections, 34 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Partially hyperbolic systems and associated invariant distributions
  3. Low regularity of invariant distributions vs bunching property vs rigidity phenomena
  4. Fractal geometry and dimension theory of graphs (or invariant sections)
  5. Non-fractal invariance principle for invariant sections
  6. Applications to partially hyperbolic Anosov systems: regularity bootstrap
  7. Applications to partially hyperbolic Anosov systems: global rigidity
  8. Stable $C^1$ or stable fractal conjectures
  9. More historical remarks and problems
  10. Organization of the paper
  11. Notations and the statement of non-fractal invariance principle
  12. Variants of non-fractal invariance principle
  13. Quantitative version of non-fractal invariance principle
  14. Applications: distributions of partially hyperbolic Anosov systems
  15. Preliminaries
  16. ...and 15 more sections

Key Result

Theorem 1.5

Suppose $f \in {\rm Diff}^{2}_{\mathrm{vol}}(\mathbb{T}^3)$ is a partially hyperbolic Anosov diffeomorphism. Then:

Figures (1)

  • Figure 1: The space of smooth partially hyperbolic systems: the bunching region is open, the fractal region is conjectured to be open, and the critical part and rigid part are conjectured to have no interior.

Theorems & Definitions (59)

  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • Theorem 1.11
  • ...and 49 more