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

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 $f\in {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{v}\mathrm{o}\mathrm{l}}^{\mathrm{\infty }}({\mathbb{T}}^3)$ with a contracting center direction: $f$ is $C^{\mathrm{\infty }}$-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 $W_f^u$ in the base, then for any $F$-invariant section, if $\mathrm{\Phi }$ has no a fractal graph, then it is smooth along $W_f^u$ 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.