Chaotic Systems with Absorption

TL;DR

This paper develops a dynamical-system framework to analyze absorption in chaotic systems, deriving formulas for escape rates, multifractality, and a generalized dimension relation, validated through billiard simulations.

Contribution

It introduces an operator formalism for absorption in chaotic systems, providing new formulas for escape rates and fractal dimensions, and extends existing theoretical relations.

Findings

Derived a general escape rate formula using conditionally-invariant measures.

Showed increased multifractality due to absorption effects.

Validated theoretical results with cardioid billiard simulations.

Abstract

Motivated by applications in optics and acoustics we develop a dynamical-system approach to describe absorption in chaotic systems. We introduce an operator formalism from which we obtain (i) a general formula for the escape rate $\kappa$ in terms of the natural conditionally-invariant measure of the system; (ii) an increased multifractality when compared to the spectrum of dimensions $D_q$ obtained without taking absorption and return times into account; and (iii) a generalization of the Kantz-Grassberger formula that expresses $D_1$ in terms of $\kappa$, the positive Lyapunov exponent, the average return time, and a new quantity, the reflection rate. Simulations in the cardioid billiard confirm these results.