Dimension estimates for iterated function systems and repellers. Part II

De-Jun Feng,Károly Simon

doi:10.1017/etds.2021.92

Abstract

AbstractThis is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of $C^1$ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of $C^1$ IFSs on ${\Bbb R}^d$ .

Highlights

The present paper is a continuation of our work in [24] for studying the dimension theory of C1 iterated function systems (IFSs) and repellers.One of the fundamental problems in fractal geometry and dynamical systems is to compute various fractal dimensions of attractors of IFSs and associated invariant measures.Downloaded from https://www.cambridge.org/core. 12 Jan 2022 at 09:57:06, subject to the Cambridge Core terms of use.D.-J
We introduce a generalized transversality condition (GTC) for parameterized families of C1 IFSs on Rd, and show that if the GTC is satisfied, for almost every (a.e.) parameter, the Hausdorff and box-counting dimensions of the IFS attractor are given by the singularity dimension, and the dimension of ergodic invariant measures on the attractor is given by its Lyapunov dimension
A natural question arises of how to verify the GTC for a parameterized family of C1 IFSs. We investigate this question for certain translational families of C1 IFSs

Summary

Introduction

The present paper is a continuation of our work in [24] for studying the dimension theory of C1 iterated function systems (IFSs) and repellers.One of the fundamental problems in fractal geometry and dynamical systems is to compute various fractal dimensions of attractors of IFSs and associated invariant measures.D.-J. We say that the family F t , t ∈ , satisfies a GTC with respect to η if there exist δ0 > 0 and a function ψ : (0, δ0) → [0, ∞) with limδ→0 ψ(δ) = 0 such that the following statement holds: for every t0 ∈ and every 0 < δ < δ0, there exists a constant C = C(t0, δ) > 0 such that for all distinct i, j ∈ and r > 0, η{t ∈ B(t0, δ) : | t (i) − t (j)| < r} ≤ Ce|i∧j|ψ(δ)Zit∧0 j(r), (1.5) Let t0 ∈ and 0 < δ < δ0, where δ0 is given as in Definition 1.1, the following properties hold.

Results

Conclusion