Simultaneous Action of Finitely Many Interval Maps: Some Dynamical and Statistical Properties

TL;DR

This paper investigates the statistical and dynamical properties of finitely many interval maps acting simultaneously on the unit interval, focusing on ergodicity, recurrence, decay of correlations, and invariance principles.

Contribution

It introduces a framework using symbolic dynamics and Ruelle operators to analyze the complex behavior of multiple interval maps acting together.

Findings

Established ergodicity for the combined dynamics.

Derived rates of recurrence and decay of correlations.

Proved invariance principles leading to the central limit theorem.

Abstract

In this paper, we consider finitely many interval maps simultaneously acting on the unit interval $I = [0, 1]$ in the real line $\mathbb{R}$; each with utmost finitely many jump discontinuities and study certain important statistical properties. Even though we use the symbolic space on $N$ letters to reduce the case of simultaneous dynamics to maps on an appropriate space, our aim in this paper remains to resolve ergodicity, rates of recurrence, decay of correlations and invariance principles leading upto the central limit theorem for the dynamics that evolves through simultaneous action. In order to achieve our ends, we define various Ruelle operators, normalise them by various means and exploit their spectra.