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Towers and Bratteli-Vershik systems in Fibonacci-like unimodal maps

Jorge Olivares-Vinales, Semin Yoo

Abstract

For a class of Fibonacci-like unimodal maps, the restriction to the $ω$-limit set of the unique turning point defines a minimal Cantor system. We construct these Cantor sets geometrically using a nested sequence of finite covers with a tower structure. From this tower structure, we recover the associated Bratteli-Vershik model determined by the cutting times and obtain an explicit formula for the unique ergodic invariant probability measure supported on the $ω$-limit set. We conclude with applications illustrating the scope of the construction.

Towers and Bratteli-Vershik systems in Fibonacci-like unimodal maps

Abstract

For a class of Fibonacci-like unimodal maps, the restriction to the -limit set of the unique turning point defines a minimal Cantor system. We construct these Cantor sets geometrically using a nested sequence of finite covers with a tower structure. From this tower structure, we recover the associated Bratteli-Vershik model determined by the cutting times and obtain an explicit formula for the unique ergodic invariant probability measure supported on the -limit set. We conclude with applications illustrating the scope of the construction.
Paper Structure (16 sections, 18 theorems, 197 equations, 1 figure)

This paper contains 16 sections, 18 theorems, 197 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Covers and Bratteli diagrams
  3. Applications of the tower system
  4. Strategy and organization
  5. Preliminaries
  6. Kneading theory of unimodal maps
  7. The Fibonacci-like maps
  8. The set $\omega_f(c)$
  9. Bratteli-Vershik system associated with the collection $\{M_{d,k}\}_{k \geq 0}$
  10. Bratteli diagrams.
  11. Bratteli-Vershik systems.
  12. Incidence matrices
  13. The Bratteli-Vershik system associated with the collection $\{M_{d,k}\}_{k \ge 0}$
  14. Projecting $X_{B_d}$ onto $\omega_f(c)$.
  15. The push-forward measure.
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists a collection $\{M_{d,k}\}_{k \ge 0}$ of compact sets satisfying the following.

Figures (1)

  • Figure 1: First $d$ levels of the Bratteli diagram associated with the collection $\{M_{d,k}\}_{k \geq 0}$ of covers of $T_d|_{\omega(c)}$. Each collection of vertices is arranged in increasing order. For example, from left to right, the vertices in $V_3$ are $v_3(1)$, $v_3(d-2)$, $v_3(d-1)$, and $v_3(d)$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 27 more