Computing Eigenvalues of Symbolic Kinetic Equation

\def\crlf{\par\vskip-\parskip} Methods of symbolic dynamics provide a novel way of reconstructing the coarse grained phase space dynamics from the analysis of a single turbulent fluctuating variable X(t). Symbolic Kinetic Equation (SKE) described the time evolution of a coarsed grained phase space probability distribution function P$_{\ell ,n }$[1,2]. $ P_{\ell ^\prime,n+1 }=\sum\limits_\ell P_{\ell ,n} \times \Gamma (\ell \rightarrow \ell ^\prime)$; Here n = 0,1,2,$\ldots$,N is a decretized time, index $\ell$ labels different coarse grained phase space volumes and $\Gamma (\ell \rightarrow \ell ^\prime $) is the probability of a transition from state $\ell $ to the state $\ell^\prime $. Stationary solution of SKE is called invariant distribution function P$_ {\ell }$. It corresponds to the largest eigenvalue $\lambda$ = 1. In this paper we are computing the whole spectrum of eigenvalues $\lambda_{i }$ of SKE, which describe the approach to the stationary state. We would like to demonstrate that $\lambda_{i}$ are invariant of the dynamics, that is they are the same for different fluctuating variables. We are using fluctuating variables X$_{i}$(t) generated by analytic model of drift wave turbulence[3] and real tokamak experimental data.\crlf \noindent [1] A.B. Rechester and R.B. White, Phys. Lett. \textbf{A 156}, 419 (1991). \crlf \noindent [2] M. Lehrman and A.B. Rechester, Phys. Rev. Lett. \textbf {87}, 164501 (2001). \crlf \noindent [3] L. Chen, Z. Lin, R. White, Physics of Plasmas {\bf 7}, 3129 (2000).