2024/11/29 by Disheng Xu, Xu, Disheng, Jiesong Zhang +1
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2411.19665
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 ∈ Diff^∞vol(\mathbbT3) with a contracting center direction: f is C^∞-rigid if and only if both Es and Ec exhibit Hölder exponents exceeding the expected threshold. Specifically, we prove: If the Hölder exponent of Es exceeds the expected value, then Es is C1+ and Eu ⊕ Es is jointly integrable. If the Hölder exponent of Ec exceeds the expected value, then Wc forms a C1+ foliation. If Es (or Ec) 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 Wuf in the base, then for any F-invariant section, if Φ has no a fractal graph, then it is smooth along Wuf 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.