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On the Regularity of the Dimension of Cookie-Cutter-Like Sets

Victor Kleptsyn, Alexandro Luna

TL;DR

The paper proves that for cookie-cutter-like sets generated by rarely switching sequences of expanding maps, the Hausdorff dimension equals the minimum of the constituent cookie-cutter dimensions and the upper box-counting dimension equals the maximum. It develops a non-stationary thermodynamic formalism, showing that the non-stationary pressure can be well-approximated by weighted stationary pressures, which yields exact dimension formulas under mild switching and frequency conditions. It further shows that parameter-dependent families can exhibit non-differentiable dimension functions due to the min–max structure, contrasting with analytic dependence in stationary or quadratic-irrational Sturmian cases. The work connects these dimension regularity results to the spectra of Sturmian Hamiltonians and provides a concrete lower-bound example illustrating the necessity of the rare-switching assumption.

Abstract

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the corresponding dimensions of the cookie-cutter sets generated by the individual expanding maps. In the case of one-parameter families of such systems, this provides a simple mechanism for producing non-differentiable fractal dimensions as functions of the parameter. This supports a conjecture that the Hausdorff dimension of the spectrum of a Sturmian Hamiltonian, in general, does not have to be differentiable as a function of the coupling constant. This is in drastic contrast to the analytic dependence of the dimensions of such spectra with quadratic irrational frequencies, e.g. the Fibonacci Hamiltonian, previously shown by M. Pollicott.

On the Regularity of the Dimension of Cookie-Cutter-Like Sets

TL;DR

The paper proves that for cookie-cutter-like sets generated by rarely switching sequences of expanding maps, the Hausdorff dimension equals the minimum of the constituent cookie-cutter dimensions and the upper box-counting dimension equals the maximum. It develops a non-stationary thermodynamic formalism, showing that the non-stationary pressure can be well-approximated by weighted stationary pressures, which yields exact dimension formulas under mild switching and frequency conditions. It further shows that parameter-dependent families can exhibit non-differentiable dimension functions due to the min–max structure, contrasting with analytic dependence in stationary or quadratic-irrational Sturmian cases. The work connects these dimension regularity results to the spectra of Sturmian Hamiltonians and provides a concrete lower-bound example illustrating the necessity of the rare-switching assumption.

Abstract

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the corresponding dimensions of the cookie-cutter sets generated by the individual expanding maps. In the case of one-parameter families of such systems, this provides a simple mechanism for producing non-differentiable fractal dimensions as functions of the parameter. This supports a conjecture that the Hausdorff dimension of the spectrum of a Sturmian Hamiltonian, in general, does not have to be differentiable as a function of the coupling constant. This is in drastic contrast to the analytic dependence of the dimensions of such spectra with quadratic irrational frequencies, e.g. the Fibonacci Hamiltonian, previously shown by M. Pollicott.

Paper Structure

This paper contains 17 sections, 10 theorems, 81 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results and plan of the paper
  4. Acknowledgments
  5. Main results
  6. Rarely switching systems
  7. Implications for parameter-dependent systems
  8. Discussion: dimensions of the spectra of Sturmian Hamiltonians
  9. Moran formula and thermodynamic formalism for cookie-cutters
  10. Cookie-cutter sets
  11. Thermodynamic formalism for cookie-cutter-like sets
  12. Distortion control
  13. Proofs
  14. Proofs
  15. Controlling additivity: main proposition
  16. ...and 2 more sections

Key Result

Theorem 2.2

Let $\mathcal{F}=\{F_1,\dots,F_k\}$ be a system of $C^{1+\tau}$ expanding cookie-cutter maps, where $\tau>0$. Assume that the sequence $\overline{c}=(c_n)$ is rarely switching, $\kappa_n=o(n)$, and satisfies the frequencies condition eq:freq. Then the Hausdorff and upper box-counting dimensions of t

Figures (4)

  • Figure 1: Left: cookie-cutter map $F$; right: the maps $f_j$, forming the corresponding IFS, and the intervals $I_j=f_j(I)$.
  • Figure 2: Left: basic set $\Lambda$ of a horseshoe map $\Phi$ with its stable and unstable laminations, and a transverse interval $I$; right: an iteration of this interval and its projection along the stable foliation that forms a cookie-cutter
  • Figure 3: Intersecting graphs of the dimensions $\dim_H(J(F_j^{\mathbf{a}}))$ and the corresponding non-differentiable graphs of $\dim_H$ and $\overline{\dim}_B$.
  • Figure 4: Graphs of $F_0$,$F_1$

Theorems & Definitions (24)

  • Definition
  • Example 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4: D. Ruelle, R1982
  • Corollary 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 3.1: M. Pollicott, P2015
  • Conjecture 3.2
  • ...and 14 more