Hyperbolic graphs: Critical regularity and box dimension

Díaz, L.; Gelfert, K.; Gröger, M.; Jäger, T.

doi:10.1090/tran/7454

Hyperbolic graphs: Critical regularity and box dimension
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by L. J. Díaz, K. Gelfert, M. Gröger and T. Jäger PDF

Trans. Amer. Math. Soc. 371 (2019), 8535-8585 Request permission

Abstract:

We study fractal properties of invariant graphs of hyperbolic and partially hyperbolic skew product diffeomorphisms in dimension three. We describe the critical (either Lipschitz or at all scales Hölder continuous) regularity of such graphs. We provide a formula for their box dimension given in terms of appropriate pressure functions. We distinguish three scenarios according to the base dynamics: Anosov, one-dimensional attractor, or Cantor set. A key ingredient for the dimension arguments in the latter case will be the presence of a so-called fibered blender.

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