Chaotic Systems with Absorption

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.