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

TL;DR

This paper constructs and analyzes minimal Cantor systems arising from Fibonacci-like unimodal maps, providing explicit models and measures, with applications demonstrating the approach's versatility.

Contribution

It introduces a geometric construction of Cantor sets for Fibonacci-like unimodal maps and derives explicit Bratteli-Vershik models and invariant measures.

Findings

Explicit formula for the unique ergodic measure

Construction of minimal Cantor systems via tower structures

Applications illustrating the construction's scope

Abstract

For a class of Fibonacci-like unimodal maps, the restriction to the $\omega$-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 $\omega$-limit set. We conclude with applications illustrating the scope of the construction.