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 with a contracting center direction: f \in \Diff^\infty_{vol}(\T^3) is C^\infty-rigid if and only if both E^s and E^c exhibit Hölder exponents exceeding the expected threshold. Moreover, for f \in \Diff^2_{vol}(\T^3), 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.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.
Joint work with Jiesong Zhang.