A New Method for Computing Topological Pressure

The topological pressure introduced by Ruelle and similar quantities describe dynamical multifractal properties of dynamical systems. These are important characteristics of mesoscopic systems in the classical regime. Original definition of these quantities are based on the symbolic description of the dynamics. It is hard or impossible to find symbolic description and generating partition to a general dynamical system, therefore these quantities are often not accessible for further studies. Here we present a new method by which the symbolic description can be omitted. We apply the method for a mixing and an intermittent system. Typeset using REVTEX 1 In recent years the application of the thermodynamic formalism [1] (TF) in analyzing dynamical multifractal properties [2–4] has been widely accepted. Besides giving an illuminating analogy with the statistical mechanics, it provides a deeper understanding of nonanalytic behavior in the scaling properties of trajectories in dynamical systems which can be interpreted as phase transitions [5]. The topological pressure (TP) plays a central role in the TF. The computation of the TP is difficult in general, since the knowledge of the symbolic dynamics [6] of the system and its generating partition is inevitable. There is no general theory at present, which can provide a systematic and numerically realizable method to construct a generating partition to a general dynamical system [7]. Therefore the TP has been computed mostly for low dimensional maps and billiard systems, where the symbolic dynamics is accessible, or has been computed with averaging the generalized Lyapunov exponents [8]. In mesoscopic devices like dots, antidots and wells Hamiltonian dynamical systems with smooth potentials play significant role. It would be very enlighting to apply TF for such non-trivial systems. In this Letter we introduce a new technique based on a suitably defined correlation function [9] to measure the TP. The advantage of the method is that the detailed knowledge of the system is not required and the calculation can be made with low computational demand for either hyperbolic or mixing systems. The new method makes possible to measure TP by following one single trajectory of an ergodic system. The TP for maps can be defined as the logarithm P (q) = logz0(q) of the leading zero z0(q) of the Ruelle zeta function 1/ζ(z, q) = ∏