31 Progress of Theoretical Physics, Vol. 115, No. 1, January 2006 Transport Properties of a Piecewise Linear Transformation and Deterministic L´ evy Flights Tomoshige Miyaguchi ∗) Department of Applied Physics, Faculty of Science and Engineering, Waseda University, Tokyo 169-8555, Japan (Received September 6, 2005) The transport properties of a 1-dimensional piecewise linear dynamical system are inves- tigated through the spectrum of its Frobenius-Perron operator. For a class of initial densities, eigenvalues and eigenfunctions of the Frobenius-Perron operator are obtained explicitly. It is also found that in the long length wave limit, this system exhibits normal diffusion and super diffusion called L´ evy flight. The diffusion constant and stable index are derived from the eigenvalues. §1. Introduction In the past decades, chaotic dynamics have been found to be essential for under- standing non-equilibrium behavior such as relaxation and transport processes. 1), 2) For this reason, elucidation of the microscopic origins of stochastic behavior is im- portant for studies of irreversible phenomena. In dynamical system theory, the baker and multi-baker transformations are well-known models of fully chaotic systems and have been studied intensively. These models are conservative in the sense that they are area-preserving, and area preservation is a general property of the Poincar´ e maps of Hamiltonian systems with two degrees of freedom. Furthermore, baker and multi- baker maps are hyperbolic, and this property leads to the exponential decay of correlation functions and normal transport. 3) – 5) Although hyperbolic systems like the baker and multi-baker maps have played important roles for understanding irreversible processes, the power law decay of correlation functions and anomalous diffusion are frequently observed in Hamil- tonian systems if their phase spaces consist of integrable (torus) and non-integrable components (chaos). 6) – 11) Therefore, generic Hamiltonian systems seem to be non- hyperbolic in this sense, and 1-dimensional non-hyperbolic systems have been the subject of several studies. 12) – 18) Those studies revealed the existence of long time correlations and anomalous diffusion in the models investigated. However, there has yet been no extension to 2-dimensional area-preserving maps of these 1-dimensional non-hyperbolic systems. In a previous paper, 11) we showed numerically that a Poincar´ e map of a generic Hamiltonian system with two degrees of freedom exhibits L´ evy flight type 19) – 22) transport phenomena (note that the original Hamiltonian flow exhibits “L´ evy walk” type transport. 9) ), and in another paper, 23) we introduced a 1-dimensional map (call the L´ evy flight map in what follows) that generates L´ evy flight. However, ∗) E-mail: tomo-m@aoni.waseda.jp