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On exponential separation of analytic self-conformal sets on the real line

Balázs Bárány, István Kolossváry, Sascha Troscheit

Abstract

In a recent article, Rapaport showed that there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of such exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the $\mathcal{C}^2$ topology. Moreover, we give a sufficient condition for the IFS to be exponentially separated which allows us to construct explicit examples which are exponentially separated. The key technical tool is the introduction of the \emph{dual IFS} which we believe has significant interest in its own right. As an application we also characterise when an analytic IFS can be conjugated to a self-similar IFS.

On exponential separation of analytic self-conformal sets on the real line

Abstract

In a recent article, Rapaport showed that there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of such exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the topology. Moreover, we give a sufficient condition for the IFS to be exponentially separated which allows us to construct explicit examples which are exponentially separated. The key technical tool is the introduction of the \emph{dual IFS} which we believe has significant interest in its own right. As an application we also characterise when an analytic IFS can be conjugated to a self-similar IFS.

Paper Structure

This paper contains 15 sections, 21 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Acknowledgements.
  3. Notations and main results
  4. Discussion
  5. Dimension theoretic consequences
  6. Concrete examples
  7. Conjugation to self-similar systems
  8. The key idea: the dual IFS induced by analytic functions
  9. Basic definitions and properties
  10. Further analysis of the dual natural projection H
  11. Proof of the sufficient condition for SESC
  12. Existence of open and dense set of IFSs with SESC
  13. Preliminaries
  14. Proof of \ref{['thm:ESCOpenDense']}
  15. Conjugation to self-similar IFS

Key Result

Theorem 1.4

[example]thm:ESCOpenDense The set of IFSs $\{\Phi \colon \Phi\text{ satisfies SESC}\}\subseteq \mathfrak{S}_N$ contains an open and dense subset in the $\mathcal{C}^2$ topology.

Figures (1)

  • Figure 1: Illustration of disjoint cylinders on the left and ones which are not disjoint on the right.

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Rapaport_SelfConfESC25arXiv
  • Corollary 1.7
  • Proposition 1.8
  • proof
  • Definition 1.9
  • ...and 37 more