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Register a new userFixed points of the Ruelle-Thurston operator and the Cauchy transform
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Genadi Levin
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We give necessary and sufficient conditions for a function in a naturally appearing functional space to be a fixed point of the Ruelle-Thurston operator associated to a rational function, see Lemma 2.1. The proof uses essentially a recent [13]. As an immediate consequence, we revisit Theorem 1 and Lemma 5.2 of [11], see Theorem 1 and Lemma 2.2 below.
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Let $Gamma$ be a co-compact Fuchsian group of isometries on the Poincare disk $DD$ and $Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $Delta$, equivariant by $Gamma$ with real eigenvalue $lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution $dd_{f,s}$ (the Helgason distribution) which is equivariant by $Gamma$ and supported at infinity $partialDD=SS^1$. The geodesic flow on the compact surface $DD/Gamma$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:SS^1toSS^1$, the so-called Bowen-Series transformation. Let $ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-sln |T|$. M. Pollicott showed that $dd_{f,s}$ is an eigenfunction of the dual operator $ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $psi_{f,s}$ of $ll_s$ for the eigenvalue 1, given by an integral formula [ psi_{f,s} (xi)=int frac{J(xi,eta)}{|xi-eta|^{2s}} dd_{f,s} (deta), ] oindent where $J(xi,eta)$ is a ${0,1}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface $DD/Gamma$.
In 1980s, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurstons Theorem.
We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $tgeq 0$, taking values on S. We denote by $Omega$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($Omega, {cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${cal L}^infty (P)$. From the probability $P$ we define a Ruelle operator ${cal L}^t, tgeq 0$, acting on functions $f:Omega to mathbb{R}$ of ${cal L}^infty (P)$. Given $V:Omega to mathbb{R}$, such that is constant in sets of the form ${X_0=c}$, we define a modified Ruelle operator $tilde{{cal L}}_V^t, tgeq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $ u_V$ on $Omega$ associated to $tilde{{cal L}}^t_V, tgeq 0$. We also show the following property for the probability $ u_V$: for any integrable $gin {cal L}^infty (P)$ and any real and positive $t$ $$ int e^{-int_0^t (V circ Theta_s)(.) ds} [ (tilde{{cal L}}^t_V (g)) circ theta_t ] d u_V = int g d u_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras).
Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.
At each time $ninmathbb{N}$, let $bar{Y}^{(n)}=(y_{1}^{(n)},y_{2}^{(n)},cdots)$ be a random sequence of non-negative numbers that are ultimately zero in a random environment $xi=(xi_{n})_{ninmathbb{N}}$ in time, which satisfies for each $ninmathbb{N}$ and a.e. $xi,~E_{xi}[sum_{iinmathbb{N}_{+}}y_{i}^{(n)}(xi)]=1.$ The existence and uniqueness of the non-negative fixed points of the associated smoothing transform in random environments is considered. These fixed points are solutions of the distributional equation for $a.e.~xi,~Z(xi)overset{d}{=}sum_{iinmathbb{N}_{+}}y_{i}^{(0)}(xi)Z_{i}(Txi),$ where when given the environment $xi$, $Z_{i}(Txi)~(iinmathbb{N}_{+})$ are $i.i.d.$ non-negative random variables, and distributed the same as $Z(xi)$. As an application, the martingale convergence of the branching random walk in random environments is given as well. The classical results by Biggins (1977) has been extended to the random environment situation.
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