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

TL;DR

This paper establishes that for typical $C^1$ iterated function systems satisfying a generalized transversality condition, the upper box-counting and packing dimensions of attractors and measures match their respective singularity and Lyapunov dimensions.

Contribution

It introduces a generalized transversality condition (GTC) for parametrized $C^1$ IFSs and proves that under GTC, the upper bounds on dimensions are exact for typical systems.

Findings

GTC holds for certain parametrized families of $C^1$ IFSs.

Upper bounds on dimensions are sharp under GTC.

The results extend the dimension theory of $C^1$ IFSs and repellers.

Abstract

This is the second part of our study of the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on ${\Bbb R}^d$. In the first part we proved that the upper box-counting dimension of the attractor of any $C^1$ 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. Here we introduce a generalized transversality condition (GTC) for parametrized families of $C^1$ IFSs, and show that these upper bounds give actually the dimensions for "typical" $C^1$ IFSs under this transversality condition. Moreover we verify the GTC for some parametrized families of $C^1$ IFSs on ${\Bbb R}^d$.