Content-Type: multipart/mixed; boundary="-------------0108240723839" This is a multi-part message in MIME format. ---------------0108240723839 Content-Type: text/plain; name="01-308.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-308.keywords" Intermittency, Transfer Operator, Spectral Analysis ---------------0108240723839 Content-Type: application/x-tex; name="text.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="text.tex" %\documentclass[amssymb,aps,prl,preprint,groupedaddress,showpacs]{revtex4} \documentclass[amssymb,aps,prl,twocolumn,groupedaddress,showpacs]{revtex4} \usepackage{graphicx} \newcommand{\C}{{\cal C}} \renewcommand{\H}{{\cal H}} \newcommand{\K}{{\cal K}} \renewcommand{\L}{{\cal L}} \newcommand{\M}{{\cal M}} \renewcommand{\P}{{\cal P}} \renewcommand{\S}{{\cal S}} \newcommand{\p}{\varphi} \begin{document} \title{Complete Determination of the Spectrum of a Transfer Operator associated with Intermittency} \author{Thomas Prellberg} \email{thomas.prellberg@tu-clausthal.de} \affiliation{Institut f\"ur Theoretische Physik, Technische Universit\"at Clausthal, Arnold-Sommerfeld Stra\ss e 6, D-38678 Clausthal-Zellerfeld, Germany} \date{\today} \begin{abstract} It is well established that the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point. We present here for the first time a complete spectral analysis for an example of such an intermittent map, the Farey map. We give a simple proof that the transfer operator is self-adjoint on a suitably defined Hilbert space and show that its spectrum decomposes into a continuous part (the interval $[0,1]$) and isolated eigenvalues of finite multiplicity. Using a suitable first-return map, we present a highly efficient numerical method for the determination of all the eigenvalues, including the ones embedded in the continuous spectrum. \end{abstract} \pacs{02.30.Sa, 02.30.Tb, 02.70.Hm, 05.10.-a, 05.45.Ac} \maketitle \section{\label{intro}Introduction} Intermittency, one of the main routes from order to chaos \cite{schuster1995a-a}, is characterized by the loss of stability of a fixed point of the dynamics. The dynamics directly at the transition is determined by a marginally unstable fixed point near which trajectories are slowed down severely, leading to the for intermittency characteristic interplay of chaotic and regular dynamics. Such a behavior can easily be modeled by a map $f$ of the unit interval $[0,1]$ which is uniformly expanding everywhere except near an indifferent fixed point at zero, where $f(x)\sim x+cx^{r+1}$ as $x\rightarrow 0$ with exponent $1+r>1$. A typical example is the Manneville map $f(x)=x+x^{1+r}\mbox{mod} 1$ \cite{manneville1980a-a}. There have been many theoretical approaches to the description of intermittency such as renormalization group analysis \cite{hu1982a-a}. An approach suited to rigorous treatment is given by the thermodynamic formalism \cite{ruelle1978a-a} and leads to the spectral analysis of transfer operators. In contrast with uniformly expanding maps, however, the indifferent fixed point induces non-Gibbsian equilibrium states or, more precisely, weakly Gibbsian states \cite{maes2000a-a}. Therefore, deeper understanding based on rigorous analysis has been hard to come by. Some progress was made by studying a suitably defined piecewise linear interval map \cite{gaspard1988a-a}, albeit at the cost of simplifying the dynamics by severely reducing correlations. In \cite{prellberg1991a-a,prellberg1992a-a} we argued the disappearance of a spectral gap for the Perron-Frobenius operator, implying loss of an exponential decay of correlations for the dynamics. The decay of correlations has later been shown to follow a power law; in the analytic case ($r=1$) a numerical estimate of $t^{-2}$ \cite{lambert1993a-a} has been complemented by a rigorous upper bound of $t^{-2}\log t$, obtained by random perturbation techniques \cite{liverani1999a-a}. Only very recently has the Perron-Frobenius operator for a particular intermittent map, the Farey map, been shown to be self-adjoint on an appropriate Hilbert space \cite{isola2001a-a} with continuous spectrum on the interval $[0,1]$. There is also numerical work available describing just how the continuous spectrum emerges when approaching the intermittency transition \cite{kaufmann1996a-a}. Much less is known about the spectrum of the Ruelle-Perron-Frobenius (RPF) operator, which generalizes the Perron-Frobenius operator within the framework of the thermodynamic formalism. In \cite{prellberg1991a-a,prellberg1992a-a} we showed that for a general class of intermittent maps this operator is quasi-compact with essential spectral radius equal to one, and that the leading eigenvalue undergoes a phase transition characteristic for intermittent dynamics. Only recently was it shown that for a class of piecewise analytic maps the continuous spectrum is restricted to the interval $[0,1]$ on an appropriate function space \cite{rugh1999a-a}. The present work gives for the first time a complete spectral analysis of the RPF operator for an intermittent map, the Farey map. A promising conceptual approach to the study of intermittency is the study of a first-return map (or induced map) with respect to a domain of phase space away from the intermittent region. In a pioneering study \cite{prellberg1991a-a,prellberg1992a-a} we introduced a suitably modified transfer operator for the induced map and related its spectral properties to the ones of the transfer operator of the intermittent system. This approach has since been extended to the study of other quantities such as regularized Fredholm determinants and dynamical zeta functions \cite{dodds1993a-a,dodds1993a-b,isola1995a-a,isola2001a-a} and multi-dimensional systems \cite{pollicott2001a-a}. One essential idea employed is that one can understand a general transfer operator $\P$ of an intermittent map by first considering its ``intermittent part'' $\P_0$ and then viewing the ``chaotic remainder'' $\P_1=1-\P$ as a perturbation. In this way, many of the results presented here can in principle be extended to more general intermittent maps, although we shall focus our attention on the {\it Farey map} of the interval $[0,1]$ onto itself, which is defined as \begin{equation} f(x)=\left\{\begin{array}{c} f_0(x)=x/(1-x)\;,\quad\mbox{if $0\leq x\leq1/2$,}\\ f_1(x)=(1-x)/x\;,\quad\mbox{if $1/21$ by an amount $2\pi i\log^{s-1}(x)/\Gamma(s)$.}. Therefore $\lambda_n(z)=1$ for real $z>1$ can only be satisfied when $\beta=-N/2$, and no eigenvalues of $\P$ embedded in the continuous spectrum exist for other values of $\beta$. Thus, in what follows we shall mainly be concerned with real $z\leq1$. For $\beta>1$ we find that all the eigenvalues of $\M_z$ are strictly less than one in modulus, so that $\P$ has continuous spectrum $[0,1]$ and no non-zero eigenvalues at all. As $\beta$ decreases the eigenvalues of $\M_z$ increase in value. For $\beta>1$, we find that the leading eigenvalue branch of $\M_z$ intersects $\lambda=1$, leading to the emergence of a simple leading eigenvalue of $\P$ for $\beta<1$ \footnote{In \cite{prellberg1991a-a,prellberg1992a-a} it was proved that this leading eigenvalue decreases to $1$ like $-(1-\beta)/\log(1-\beta)$ as $\beta$ approaches $1$ from below.}. The $z$-dependence of the spectrum of $\M_z$ is shown in Figure \ref{fig1} for $\beta=1/2$. Only the largest eigenvalue intersects $\lambda=1$, implying that $\P$ has only one non-zero eigenvalue. At $\beta=-3/2$ a second eigenvalue branch begins to intersect $\lambda=1$ at large negative $z$, implying that $\P$ has two non-zero eigenvalues. The second eigenvalue of $\P$ is negative and becomes in modulus equal to the essential spectral radius at $\beta=-2$, and therefore determines for $\beta<-2$ the spectral gap which controls the decay of correlations. Figure \ref{fig2} illustrates the leading spectrum of $\M_z$ for $\beta=-3$. At this special value of $\beta$ we can extend the eigenvalue branches beyond $z=1$ and find a total of four eigenvalue branches which cross $\lambda=1$. One of these crossings is at $z=13.101$, corresponding to an eigenvalue of $\P$ embedded in the continuous spectrum. Proceeding in this fashion, we can numerically determine the complete spectrum of $\P$ for arbitrary real $\beta$. Figure \ref{fig3} shows the spectrum obtained in this way for $-2.5\leq\beta\leq1.5$. In summary, we have abstractly characterized the spectrum of the transfer operator for the Farey map and presented a highly efficient method for the explicit computation of its eigenvalues. Of special interest is the interplay of sub-dominant eigenvalues and the continuous spectrum. 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