# Hyperbolic graphs: critical regularity and box dimension

## TL;DR

This paper investigates the fractal and regularity properties of invariant graphs in hyperbolic skew product systems, providing formulas for their box dimension and analyzing different dynamical scenarios.

## Contribution

It introduces a formula for the box dimension of invariant graphs in hyperbolic systems and characterizes their regularity based on the base dynamics and the presence of fibered blenders.

## Key findings

- Derived a formula for box dimension using pressure functions.
- Identified three dynamical scenarios affecting regularity and dimension.
- Established the role of fibered blenders in dimension analysis.

## 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 H\"older 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.