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78
Decay of Correlations
, 1993
"... this paper I describe a technique, originally due to G. Birkhoff [9], [10], that permits a direct study of the PerronFrobenius operator, and I show that its field of applicability is wider than that of Markov partitions. In essence, it is possible to construct systematically metrics (Hilbert metric ..."
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Cited by 95 (12 self)
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this paper I describe a technique, originally due to G. Birkhoff [9], [10], that permits a direct study of the PerronFrobenius operator, and I show that its field of applicability is wider than that of Markov partitions. In essence, it is possible to construct systematically metrics (Hilbert metrics) with respect to which the PerronFrobenius operator is a contraction. Such contraction allows to obtain the invariant measure (if not already known) by an elementary, and constructive, fixed point theorem, rather than by some compactness argument (this may please some idiosyncratic people, myself included), and automatically implies an exponential rate for the decay of correlations. I illustrate such an approach by applying it to several examples. For the sake of brevity and clarity the results are not presented in their full generality. DECAY OF CORRELATIONS 3 In particular, all the arguments used for twodimensional smooth maps can be extended to the ndimensional case. Results concerning more general systems (notably billiards and nonuniformly hyperbolic maps) will be published in separate papers. I also hope that the present exposition will prompt others to try to apply this method to the many cases where it could yield new results (e.g. dissipative systems, flows, etc.). The structure of the paper is as follows: section 1 describes the Hilbert metric and its properties. It is a brief review of the subject, intended to provide an easy reference for the reader. Section 2 shows how the technique works in the simplest example: a one dimensional uniformly hyperbolic map. It also mentions other consequences that can be obtained (e.g. Central Limit Theorem type results). In sections 3 I show how to extend the approach to the multidimensional casethe smooth case is trea...
Exponential Stability for Nonlinear Filtering
, 1996
"... We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficie ..."
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Cited by 58 (2 self)
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We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discretecountable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...
The PerronFrobenius theorem for homogeneous monotone functions
 Transacton of AMS
, 2004
"... Abstract. If A is a nonnegative matrix whose associated directed graph is strongly connected, the PerronFrobenius theorem asserts that A has an eigenvector in the positive cone, (R+) n. We associate a directed graph to any homogeneous, monotone function, f:(R+) n → (R+) n, and show that if the grap ..."
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Cited by 41 (10 self)
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Abstract. If A is a nonnegative matrix whose associated directed graph is strongly connected, the PerronFrobenius theorem asserts that A has an eigenvector in the positive cone, (R+) n. We associate a directed graph to any homogeneous, monotone function, f:(R+) n → (R+) n, and show that if the graph is strongly connected, then f has a (nonlinear) eigenvector in (R+) n. Several results in the literature emerge as corollaries. Our methods show that the PerronFrobenius theorem is “really ” about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.
Decay Of Correlations For Piecewise Expanding Maps
 JOURNAL OF STATISTICAL PHYSICS
, 1995
"... This paper investigates the decay of correlations in a large class of nonMarkov onedimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained. ..."
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Cited by 40 (5 self)
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This paper investigates the decay of correlations in a large class of nonMarkov onedimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained.
Ruelle's Transfer Operator for Random Subshifts of Finite Type
 Ergod. Th. & Dynam. Sys
, 1995
"... We consider a RuellePerronFrobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibb ..."
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Cited by 37 (6 self)
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We consider a RuellePerronFrobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibbs measure and equilibrium states. In order to do this we introduce the notion of bundle random dynamical systems and provide a theory for their invariant measures as well as give a precise definition of Gibbs measures. Key words: random dynamical system, random subshift of finite type, transfer operator, Gibbs measures, equilibrium states. 1991 Mathematics Subject Classification. Primary 58F03, 58F15; Secondary 60J10, 54H20. Introduction Random dynamical systems (RDS) generalize dynamical systems by allowing the dependence on an additional parameter evolving in time and describing a stochastic influence (cf. Arnold and Crauel [3]). The latter is modelled by an abstract dynamical system(\Ome...
Superiority and Complexity of the Spaced Seeds
 SODA
, 2006
"... Optimal spaced seeds were introduced by the theoretical computer science community to bioinformatics to effectively increase homology search sensitivity. They are now serving thousands of homology search queries daily. While dozens of papers have been published on optimal spaced seeds since their in ..."
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Cited by 34 (6 self)
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Optimal spaced seeds were introduced by the theoretical computer science community to bioinformatics to effectively increase homology search sensitivity. They are now serving thousands of homology search queries daily. While dozens of papers have been published on optimal spaced seeds since their invention, many fundamental questions still remain unanswered. In this paper, we settle several open questions in this area. Specifically, we prove that when the length of a nonuniformly spaced seed is bounded by an exponential function of the seed weight, the seed outperforms strictly the traditional consecutive seed in both (i) the average number of nonoverlapping hits and (ii) the asymptotic hit probability. Then, we study the computation of the hit probability of a spaced seed, solving three more open questions: (iii) hit probability computation in a uniform homologous region is NPhard and (iv) it admits a PTAS; (v) the asymptotic hit probability is computable in exponential time in seed length, independent of the homologous region length. 1
Evolutionary Formalism for Products of Positive Random Matrices
 Ann. Appl. Probab
, 1994
"... We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter we establish a random version of the PerronFrobenius theory which extends al ..."
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Cited by 31 (4 self)
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We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter we establish a random version of the PerronFrobenius theory which extends all known results and enables us to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. We develop a thermodynamic formalism for random dynamical systems, and in this framework prove that the top Lyapunov exponent is an analytic function of the generator of the cocycle. On this basis a fluctuation theory for products of positive random matrices can be developed which leads to an inequality in dynamical entropy that can be interpreted as a directionality principle for the mutation and selection process in evolutionary dynamics. Key words: evolutionary theory, random dynamical system, products of random matrices, PerronFrobenius theo...
Rigorous numerical investigation of the statistical properties of piecewise expanding maps  A feasibility study
, 2000
"... I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps 1 Introduction The aim of the present paper is to investigate the possibility of answering questions of ..."
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Cited by 26 (1 self)
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I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps 1 Introduction The aim of the present paper is to investigate the possibility of answering questions of the type: ffl Given a piecewise expanding map is it possible to decide if it is ergodic or mixing? ffl Is it possible to determine with arbitrary precision its absolutely continuous invariant measure? ffl If the map is mixing, is it possible to compute the exact rate of decay of correlations for a given function? Of course, the literature contains many papers in which some of these question are discussed either theoretically (especially, but not exclusively, as far as the invariant density is concerned) or numerically (e.g. [3], [4, 5, 6, 7], [8, 9], [14], [15], [18, 19], [21, 22], [23, 24, 25], [27, 28, 29, 30, 31, 32, 33], [34, 35], [38], [39, 40], [48], [49], [52], [55], [62], [66]). N...
Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics
 In: B. Hasselblatt and A. Katok (ed) Handbook of Dynamical Systems
, 2006
"... 1. Lyapunov exponents of dynamical systems 3 2. Examples of systems with nonzero exponents 6 3. Lyapunov exponents associated with sequences of matrices 18 ..."
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Cited by 23 (1 self)
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1. Lyapunov exponents of dynamical systems 3 2. Examples of systems with nonzero exponents 6 3. Lyapunov exponents associated with sequences of matrices 18