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

TL;DR

This paper explores the regularity and fractal properties of invariant distributions in partially hyperbolic systems, revealing phase transitions and rigidity phenomena linked to fractal geometry and Hölder exponents.

Contribution

It introduces a new approach connecting fractal geometry to dynamics, establishing sharp phase transition results and a non-fractal invariance principle for partially hyperbolic diffeomorphisms.

Findings

Hölder exponents exceeding thresholds imply higher regularity and integrability.

Invariant distributions with non-excessive Hölder regularity have fractal graphs.

A non-fractal invariance principle links fiber expansion rates to smoothness of invariant sections.

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 H\"older exponents exceeding the expected threshold. Specifically, we prove: If the H\"older 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 H\"older 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 H\"older 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 $\Phi$ 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.