Vector-valued Ruelle operator with weakly contractive IFS

The vector-valued Ruelle operator defined by contractive iterated function systems (IFS) was discussed by the author [Y.L. Ye, Vector-valued Ruelle operator, J. Math. Anal. Appl. 299 (2004) 341-356]. In this paper we consider vector-valued Ruelle operators defined by weakly contractive IFS. And, a vector-valued analogue of the Ruelle-Perron-Frobenius theorem for the scalar Ruelle operator is set up. Our theorem gives a sufficient condition for the vector-valued Ruelle operator to be quasi-compact. Under this sufficient condition, we prove that the rate of convergence of the iterated operators is exponential.     

Ruelle operator with nonexpansive IFS on the line

Ruelle operator defined by weakly contractive iterated function systems (IFS) satisfying the open set condition was discussed in the paper [K.S. Lau, Y.L. Ye, Ruelle operator with nonexpansive IFS, Studia Math. 148 (2001) 143-169]. There, one of our theorems gave a sufficient condition for the possession of the Perron-Frobenius property. In this paper we consider Ruelle operator defined by nonexpansive IFS on the line instead of by weakly contractive one. And we prove, under the same condition, that the newly defined Ruelle operator has the Perron-Frobenius property. It extends the Ruelle-Perron-Frobenius theorem partially to the nonexpansive IFS.     

Pollicott–Ruelle Resonances for Open Systems

We define Pollicott–Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane.     

The analyticity of a generalized Ruelle’s operator

In this work we propose a generalization of the concept of Ruelle’s operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle’s operator. Our operator generalizes both the Ruelle operator proposed in [2] and the Perron Frobenius operator defined in [7]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle’s operator and present some examples.     

The Ruelle Transfer Operator in the Context of Orthogonal Polynomials

We redefine the Ruelle transfer operator, a classical tool from dynamical systems theory, in terms of orthogonal polynomial sequences. This transfer operator will be given via the preimages of the Chebyshev polynomials of the first kind and we will show that function spaces determined by the Chebyshev polynomials of the first kind are left invariant while function spaces determined by various other orthogonal polynomial sequences are not.     

On the uniqueness of Gibbs states in some dynamical systems

By applying Grothendieck theory and Ruelle thermodynamic formalism, we prove that, for expansive dynamical systems and interaction potentials satisfying certain conditions of analyticity, the associated Gibbs states are unique. This allows us to draw an analogy between some quantities in classical thermodynamics and abstract dynamics in the spirit of the previous work of the authors [13]. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Produits aléatoires d'opérateurs matrices de transfert

Résumé Nous étudions le comportement asymptotique de produits aléatoires d'opérateurs de Ruelle-Perron-Frobenius. Nous étendons le travail de Ruelle obtenu dans le cas homogène, au cas aléatoire.

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Summary We study the asymptotic behaviour of random products of Ruelle-Perron-Frobenius operators. We extend the previous work of Ruelle in the homogeneous case to the random case.

     

Solution of the basin problem for Hénon-like attractors

For a large class of non-uniformly hyperbolic attractors of dissipative diffeomorphisms, we prove that there are no “holes” in the basin of attraction: stable manifolds of points in the attractor fill-in a full Lebesgue measure subset. Then, Lebesgue almost every point in the basin is generic for the SRB (Sinai-Ruelle-Bowen) measure of the attractor. This solves a problem posed by Sinai and by Ruelle, for this class of systems. Oblatum 30-IX-1999 & 8-VI-2000?Published online: 18 September 2000     

The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps

In this paper, we study the Ruelle zeta function and the Selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps. In particular, we show that they have meromorphic extensions to \mathbbC{\mathbb{C}} and satisfy functional equations. We also derive the order of the singularity of the Ruelle zeta function at the origin. To prove these results, we completely analyze the weighted unipotent orbital integrals on the geometric side of the Selberg trace formula when test functions are defined for the fundamental representations.     

The stable manifold theorem for non-linear stochastic systems with memory: II. The local stable manifold theorem

We state and prove a Local Stable Manifold Theorem (Theorem 4.1) for non-linear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary trajectories of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary trajectory. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle, together with interpolation arguments.     

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.     

The Ruelle-Sullivan map for actions of ℝ n

The Ruelle Sullivan map for an ℝⁿ-action on a compact metric space with invariant probability measure is a graded homomorphism between the integer Cech cohomology of the space and the exterior algebra of the dual of ℝⁿ. We investigate flows on tori to illuminate that it detects geometrical structure of the system. For actions arising from Delone sets of finite local complexity, the existence of canonical transversals and a formulation in terms of pattern equivariant functions lead to the result that the Ruelle Sullivan map is even a ring homomorphism provided the measure is ergodic.     

Flows of stochastic dynamical systems: ergodic theory

We present a version of the Multiplicative Ergodic (Oseledec) Theorem for the flow of a nonlinear stochastic system definedon a smooth compact manifold. This theorem establishes the existence of a Lyapunov spectrum for the flow, which characterises the asymptotic behaviour of the derivative flow. Then we establish the existence of stable manifolds for the flow (on which trajectories cluster) associated with the Lyapunov spectrum. This work is a generalisation of that of Ruelle who deals with ordinary dynamical systems. Finally we give an example of a stochastic system for which the flow is calculated explicitly, and which illustrates the behaviour predicted by the abstract results.     

Expanding maps on compact metric spaces

The notions of locally expansive, positively expansive, expanding in the sense of Ruelle and expanding in the sense of Duvall and Husch are equivalent in a quite general setting.     

EXISTENCE OF NONNEGATIVE SOLUTIONS FOR DEGENERATE ECOLOGICAL MODELS

In this paper,nonnegative solutions for the degenerate elliptic systems are considered.First,nonnegative solutions for scalar equation with spatial discontinuities are studied. Then themethod developed for scalar equation is applied to study elliptic systems. At last,the existence criteria of nonnegative solutions of elliptic systems are given.     

Ruelle operators and decay of correlations for contact Anosov flows

We prove strong spectral estimates for Ruelle transfer operators for arbitrary $C^2$ contact Anosov flows. As a consequence of this we obtain: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error; (c) exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure.     

On three types of dynamics and the notion of attractor

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor–repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.     

A route to chaos in a conservative system of three interacting Langmuir waves

This paper presents the results of numerical calculations of a route to chaos in a conservative Hamiltonian system of three Langmuir waves interacting with each other through three-wave couplings. The route is investigated by studying time series, power spectra, phase space portraits and Lyapnov exponents of wave variables for several combinations of wave vectors. The results show that the system follows a route which is very similar to the Ruelle–Takens–Newhouse scenario observed in dissipative systems, and widths and shifts of peaks in power spectra appeared due to the three moderate strength wave interactions. The breaks of tori in the system are also numerically investigated by studying the dependency of Maximum Lyapnov exponents for wave-variables on a parameter which represents the nonlinearity of the system.     

Ibragimov-type invariants for a system of two linear parabolic equations

We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented.     

Cuntz�CKrieger Algebras and Wavelets on Fractals

We consider representations of Cuntz–Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron–Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.