Dimension estimates for $C^1$ iterated function systems and repellers. Part I

TL;DR

This paper establishes upper bounds for the box-counting and packing dimensions of attractors and invariant measures in $C^1$ iterated function systems and repellers, linking geometric and dynamical dimensions.

Contribution

It proves that the upper box-counting dimension of attractors is bounded by the singularity dimension, and the upper packing dimension of measures is bounded by the Lyapunov dimension in $C^1$ systems.

Findings

Upper box-counting dimension ≤ singularity dimension for $C^1$ IFS attractors.

Upper packing dimension ≤ Lyapunov dimension for ergodic invariant measures.

Results extend to repellers of $C^1$ expanding maps on manifolds.

Abstract

This is the first article in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.