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

Abstract: In this talk, we will 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 $f$ with a contracting center direction on $T^3$: $f$ is $C^\infty$-rigid if and only if both $E^s$ and $E^c$ exhibit Hölder exponents exceeding the expected threshold. 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 $\Phi$, if $\Phi$ has no fractal graph, then it is smooth along $W^u_f$ and holonomy-invariant. This is based on a joint work with Disheng Xu.