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147
Iterated random functions
 SIAM Review
, 1999
"... Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys ..."
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Cited by 232 (3 self)
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Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. 1. Introduction. The
Sets of matrices all infinite products of which converge. Linear Algebra and its Applications
, 1992
"... An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing ..."
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Cited by 114 (0 self)
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An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing selfsimilar objects like von Koch’s snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set X to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices. Necessary and sufficient conditions are given for a finite set Z to be an RCP set having a limit function M,(d) = rIT = lAd,, where d = (d,,., d,,..>, which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of columnstochastic matrices are completely characterized. Some results are given on the problem of algorithmically
Simplicity of Lyapunov spectra: proof of the ZorichKontsevich conjecture
, 2005
"... We prove the ZorichKontsevich conjecture that the nontrivial Lyapunov exponents of the Teichmüller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the ..."
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Cited by 49 (8 self)
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We prove the ZorichKontsevich conjecture that the nontrivial Lyapunov exponents of the Teichmüller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.
Random walks on infinite graphs and groups  a survey on selected topics
 Bull. London Math. Soc
, 1994
"... 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 ..."
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Cited by 42 (2 self)
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2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4
Généricité d'exposants de Lyapunov nonnuls pour des produits déterministes de matrices
 Ann. Inst. H. Poincaré Anal. Non Linéarié
, 2000
"... We propose a geometric sucient criterium \a la Furstenberg" for the existence of nonzero Lyapunov exponents for certain linear cocycles over hyperbolic transformations: nonexistence of probability measures on the bers invariant under the cocycle and under the holonomies of the stable and u ..."
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Cited by 37 (10 self)
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We propose a geometric sucient criterium \a la Furstenberg" for the existence of nonzero Lyapunov exponents for certain linear cocycles over hyperbolic transformations: nonexistence of probability measures on the bers invariant under the cocycle and under the holonomies of the stable and unstable foliations of the transformation. This criterium applies to locally constant and to dominated cocycles over hyperbolic sets endowed with an equilibrium state. As a consequence, we get that nonzero exponents exist for an open dense subset of these cocycles, which is also of full Lebesgue measure in parameter space for generic parametrized families of cocycles. This criterium extends to continuous time cocycles obtained by lifting a hyperbolic ow to a projective ber bundle, tangent to some foliation transverse to the bers. Again, nonzero Lyapunov exponents are implied by nonexistence of transverse measures invariant under the holonomy of the foliation. We apply this last re...
Random Fibonacci sequences and the number 1.13198824...
 MATHEMATICS OF COMPUTATION
, 1999
"... For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n>2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 =1.61803398.... But for a simple modification with both additions and subtractions — the random Fibonacci sequences define ..."
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Cited by 36 (2 self)
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For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n>2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 =1.61803398.... But for a simple modification with both additions and subtractions — the random Fibonacci sequences defined by t1 = t2 =1, and for n>2, tn = ±tn−1 ± tn−2, where each ± sign is independent and either + or − with probability 1/2 — it is not even obvious if tn  should increase with n. Our main result is that n tn  →1.13198824... as n →∞ with
What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals
 JOURNAL OF FUNCTIONAL ANALYSIS 166, 197217 (1999)
, 1999
"... We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of 2 if f is continuous and 2f is defined as a continuous function. We show that if f is a nonconstant function in the domain of 2, then f 2 is n ..."
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Cited by 35 (19 self)
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We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of 2 if f is continuous and 2f is defined as a continuous function. We show that if f is a nonconstant function in the domain of 2, then f 2 is not in the domain of 2. We give two proofs of this fact. The first is based on the analog of the pointwise identity 2f 2&2f 2f ={f  2, where we show that {f 2 does not exist as a continuous function. In fact the correct interpretation of 2f 2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of 2, at a junction point x0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for  f (x) & f (x0)  in terms of d(x, x0); for a certain value;, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of 2.
Lyapunov exponents: How frequently are dynamical systems hyperbolic?
 IN ADVANCES IN DYNAMICAL SYSTEMS. CAMBRIDGE UNIV
, 2004
"... Lyapunov exponents measure the asymptotic behavior of tangent vectors under iteration, positive exponents corresponding to exponential growth and negative exponents corresponding to exponential decay of the norm. Assuming hyperbolicity, that is, that no Lyapunov exponents are zero, Pesin theory p ..."
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Cited by 32 (2 self)
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Lyapunov exponents measure the asymptotic behavior of tangent vectors under iteration, positive exponents corresponding to exponential growth and negative exponents corresponding to exponential decay of the norm. Assuming hyperbolicity, that is, that no Lyapunov exponents are zero, Pesin theory provides detailed geometric information about the system, that is at the basis of several deep results on the dynamics of hyperbolic systems. Thus, the question in the title is central to the whole theory. Here we survey and sketch the proofs of several recent results on genericity of vanishing and nonvanishing Lyapunov exponents. Genericity is meant in both topological and measuretheoretical sense. The results are for dynamical systems (diffeomorphisms) and for linear cocycles, a natural generalization of the tangent map which has an important role in Dynamics as well as in several other areas of Mathematics and its applications. The first section contains statements and a detailed discussion of main results. Outlines of proofs follow. In the last section and the appendices we prove a few useful related results.
Delocalization in Random Polymer Models
 Commun. Math. Phys
"... A random polymer model is a onedimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer ..."
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Cited by 31 (7 self)
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A random polymer model is a onedimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have purepoint spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.
exponents and Hodge theory
 Mathematical Beauty of Physics”, Saclay
, 1996
"... Abstract. Claude Itzykson was fascinated (among other things) by the mathematics of integrable billiards (see [AI]). This paper is devoted to new results about the chaotic regime. It is an extended version of the talk of one of us (M.K.) on the conference “The Mathematical Beauty of Physics”, Saclay ..."
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Cited by 26 (4 self)
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Abstract. Claude Itzykson was fascinated (among other things) by the mathematics of integrable billiards (see [AI]). This paper is devoted to new results about the chaotic regime. It is an extended version of the talk of one of us (M.K.) on the conference “The Mathematical Beauty of Physics”, Saclay, June 1996, dedicated to the memory of C. Itzykson. We started from computer experiments with simple onedimensional ergodic dynamical systems, and quite unexpectedly ended with topological string theory. The result is a formula connecting fractal dimensions in one dimensional “conformal field theory ” and explicit integrals over certain moduli spaces. Also a new analogy arose between ergodic theory and complex algebraic geometry. We will finish the preface with a brief summary of what is left behind the scene. Our moduli spaces are close relatives of those arising in SeibergWitten approach to the supersymmetric YangMills theory. The integrals in the main formula can also be considered as correlators in a topological string theory with c = 1. Probably, there is way to calculate them in terms of a matrix model and an integrable