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On nonlinear iterated function systems with overlaps

Boris Solomyak

TL;DR

The paper addresses overlaps in nonlinear iterated function systems on the line by constructing a concrete three-map IFS from linear-fractional transformers with a common fixed point and proving that the attractor's Hausdorff dimension equals the conformal dimension, i.e., no dimension drop occurs. It achieves this via a 1-parameter analytic family and the SolTak/Hochman framework to establish strong exponential separation on a full-measure set of parameters, yielding $\dim_H(\Lambda_t)=\dim_{conf}(\mathcal{F}_t)$ for almost all $t>0$, with $\dim_{conf}(\mathcal{F}_t)<1$. The work also analyzes the natural measure, showing a gap between the local and conformal dimensions, and poses open questions about SESC for parameter families and about algebraic parameter values ensuring freeness. These results demonstrate that dimension drop can be avoided in certain nonlinear IFS with overlaps and offer avenues for further exploration of exact overlaps in nonlinear, non-self-similar settings.

Abstract

We construct an example of an iterated function system on the line, consisting of linear fractional transformations, such that two of the maps share a fixed points, but the dimension of the attractor equals the conformal dimension, so that there is no ``dimension drop''.

On nonlinear iterated function systems with overlaps

TL;DR

The paper addresses overlaps in nonlinear iterated function systems on the line by constructing a concrete three-map IFS from linear-fractional transformers with a common fixed point and proving that the attractor's Hausdorff dimension equals the conformal dimension, i.e., no dimension drop occurs. It achieves this via a 1-parameter analytic family and the SolTak/Hochman framework to establish strong exponential separation on a full-measure set of parameters, yielding for almost all , with . The work also analyzes the natural measure, showing a gap between the local and conformal dimensions, and poses open questions about SESC for parameter families and about algebraic parameter values ensuring freeness. These results demonstrate that dimension drop can be avoided in certain nonlinear IFS with overlaps and offer avenues for further exploration of exact overlaps in nonlinear, non-self-similar settings.

Abstract

We construct an example of an iterated function system on the line, consisting of linear fractional transformations, such that two of the maps share a fixed points, but the dimension of the attractor equals the conformal dimension, so that there is no ``dimension drop''.
Paper Structure (6 sections, 11 theorems, 67 equations)

This paper contains 6 sections, 11 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Example
  3. Concluding remarks
  4. On the dimension of the natural measure
  5. Questions
  6. Appendix: on the freeness of the semigroup $\{A,B\}^+$

Key Result

Theorem 1.2

If a self-similar IFS $\Phi = \{x\mapsto r_i x + a_i\}_{i\in {\mathcal{I}}},\ x\in \mathbb R$, with $|r_i|\in (0,1)$, satisfies the SESC along a subsequence, then

Theorems & Definitions (23)

  • Conjecture 1: Exact coincidence conjecture
  • Definition 1.1
  • Theorem 1.2: Hochman Hochman2014
  • Theorem 1.3: SolTak
  • Remark 1.4
  • Theorem 2.1
  • Definition 1: SolTak and HS2017
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 13 more