Intrinsic Ergodicity of Open dynamical systems for the doubling map

TL;DR

This paper establishes conditions under which open dynamical systems for the doubling map are intrinsically ergodic, showing that such systems are typically ergodic with full measure, and identifies exceptions.

Contribution

It provides new sufficient conditions for intrinsic ergodicity in open doubling map systems and demonstrates that this property holds for almost all parameters in a specified range.

Findings

Full measure set of parameters yields intrinsically ergodic attractors.

Constructs examples where intrinsic ergodicity fails.

Provides criteria for intrinsic ergodicity in open dynamical systems.

Abstract

We give sufficient conditions for intervals $(a,b)$ such that the associated open dynamical system for the doubling map is intrinsically ergodic. We also show that the set of parameters $(a,b) \in (\frac{1}{4}, \frac{1}{2}) \times (\frac{1}{2},\frac{3}{4})$ such that the attractor $(\Lambda_{(a,b)}, f_{(a,b)})$ is intrinsically ergodic has full Lebesgue measure and we construct a set of points where intrinsic ergodicity does not hold.