Topological dynamics of the doubling map with asymmetrical holes

TL;DR

This paper investigates the topological dynamics of the doubling map with asymmetrical holes, characterizing entropy, transitivity, and the structure of parameter spaces, revealing dense subsets with finite type dynamics and connections to other dynamical systems.

Contribution

It provides a detailed description of the topological entropy and the structure of parameter sets for the doubling map with asymmetrical holes, linking it to subshifts of finite type and other systems.

Findings

The set of parameters with subshift of finite type dynamics is open and dense.

The paper establishes connections between these systems, intermediate β-expansions, and Lorenz maps.

It analyzes topological transitivity and the specification property for these maps.

Abstract

We study the dynamics of the attractor of the doubling map with an asymmetrical hole by associating to each hole an element of the lexicographic world. A description of the topological entropy function is given. We show that the set of parameters $(a,b)$ such that the dynamics of the mentioned attractor corresponds to a subshift of finite type is open and dense. Using the connections between this family of open dynamical systems, intermediate $\beta$-expansions and Lorenz maps we study the topological transitivity and the specification property for such maps.