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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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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.
Conformal and Harmonic Measures on Laminations Associated with Rational Maps
, 2002
"... The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination A and the associated hyperbolic 3lamination H endowed with an action of ..."
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The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination A and the associated hyperbolic 3lamination H endowed with an action of a discrete group of isomorphisms.
Dimension and product structure of hyperbolic measures
 ANNALS OF MATHEMATICS, 149 (1999), 755–783
, 1999
"... We prove that every hyperbolic measure invariant under a C 1+α diffeomorphism of a smooth Riemannian manifold possesses asymptotically “almost” local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. Th ..."
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Cited by 18 (0 self)
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We prove that every hyperbolic measure invariant under a C 1+α diffeomorphism of a smooth Riemannian manifold possesses asymptotically “almost” local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyperbolic sets. Using this property of hyperbolic measures we prove the longstanding EckmannRuelle conjecture in dimension theory of smooth dynamical systems: the pointwise dimension of every hyperbolic measure invariant under a C 1+α diffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide. This provides the rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures.
Hyperbolic Dynamical Systems
"... this paper of Hadamard. Birkhoff is among them and writes about "the symbols effectively introduced by Hadamard" [Bh3, p.184]. (It is an unresolved question just when symbol spaces began to be perceived as dynamical systems, rather than as a coding device.) Geodesic flows on negatively cur ..."
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this paper of Hadamard. Birkhoff is among them and writes about "the symbols effectively introduced by Hadamard" [Bh3, p.184]. (It is an unresolved question just when symbol spaces began to be perceived as dynamical systems, rather than as a coding device.) Geodesic flows on negatively curved surfaces were again studied in the 1920s and 1930s. For constant curvature, finite volume and finitely generated fundamental group the geodesic flow was shown to be topologically transitive [Kb, Lb], topologically mixing [Hl1], ergodic [Ho1], and mixing [Hl2]. (In the case of infinitely generated fundamental group the geodesic flow may be topologically mixing without being ergodic [Sd]). If the curvature is allowed to vary between two negative constants then finite volume implies topological mixing (Hedlund [Hl3] attributes this to [Gt]). Finally Hopf [Ho2] considered compact surfaces of nonconstant (predominantly) negative curvature and was able to show ergodicity and mixing of the Liouville measure (phase volume). This is interesting because despite the ergodicity paradigm central to statistical mechanics, the Boltzmann ergodic hypothesis (under which the time average of an observable, which is an experimentally measurable quantity, agrees with the space average, which is the corresponding quantity one can compute from theory) there was a dearth of examples of ergodic Hamiltonian systems. To this day the quintessential model for the ergodic hypothesis, the gas of hard spheres, resists attempts to prove ergodicity.
Stochasticlike behaviour in nonuniformly expanding maps Handbook of Dynamical Systems
, 2006
"... 1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3 ..."
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1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3
Ergodic Theory of Chaotic Dynamical Systems
 XIIth International Congress of Mathematical Physics (ICMP'97
, 1997
"... THEOREM ([41], [42]). Let f; ; m and R be as above. Then: (a) If R Rdm ! 1, then f admits an SRB measure . (b) If, additionally, gcdfR i g = 1, then (f; ) is mixing. (c) If mfR ? ng ! C` n for some ` ! 1, then 9 ~ ` ! 1 s.t. 8'; /; C n ('; /) ! C ~ ` n . (d) If mfR ? ng = O(n ..."
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THEOREM ([41], [42]). Let f; ; m and R be as above. Then: (a) If R Rdm ! 1, then f admits an SRB measure . (b) If, additionally, gcdfR i g = 1, then (f; ) is mixing. (c) If mfR ? ng ! C` n for some ` ! 1, then 9 ~ ` ! 1 s.t. 8'; /; C n ('; /) ! C ~ ` n . (d) If mfR ? ng = O(n \Gammaff ) for some ff ? 1, then C n ('; /) = O(n \Gammaff+1 ). (e) If R is as in (d) and ff ? 2, then the CLT holds for all '. Next, we argue that conceptually mfR ? ng is essentially the speed with which arbitrarily small pieces of unstable manifolds grow to a specified size. (This is not the same as Lyapunov exponents, which measure pointwise growth rates.) First we describe the picture. If f has good hyperbolic properties, then we can cover most of phase space with a finite number of sets \Gamma 1 ; \Delta \Delta \Delta ; \Gamma k with product structures (they look like W u \Theta W s trelises). If f is mixing, then in finite time, f n \Gamma i crosses over \Gamma j in the unstable di...
Ulam's Method for Random Interval Maps
 Nonlinearity
, 1999
"... We consider the approximation of absolutely continuous invariant measures (ACIM's) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulam's finite approximation scheme in the case of a single transformation was dealt with by Li [9]. We extend ..."
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We consider the approximation of absolutely continuous invariant measures (ACIM's) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulam's finite approximation scheme in the case of a single transformation was dealt with by Li [9]. We extend Ulam's construction to the situation where a family of piecewise monotonic transformations are composed according to either an iid or Markov law, and prove an analogous convergence result. In addition, we obtain a convergence rate for our approximations to the unique ACIM, and provide rigorous bounds for the L 1 error of the Ulam approximation. 1 Introduction We begin by giving a rough description of the setting and our results. Let fT k g r k=1 be a collection of mappings from the unit interval I into itself. Given an initial point x 2 I, and a (random) sequence (k 0 ; k 1 ; : : :) with kN 2 f1; : : : ; rg for N 0, we produce a (random) orbit by defining the N th point in the orbit to be xN =...
Dynamics of regular polynomial endomorphisms of C k
 Amer. J. Math
"... Let f = (f1,..., fk) : C k → C k be a mapping such that each fj is a polynomial of degree d. We consider the behavior of f as a dynamical system. That is, we consider the behavior of the iterates f n = f ◦ · · · ◦ f as n → ∞. The points of most interest are those whose forward orbits show recurr ..."
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Let f = (f1,..., fk) : C k → C k be a mapping such that each fj is a polynomial of degree d. We consider the behavior of f as a dynamical system. That is, we consider the behavior of the iterates f n = f ◦ · · · ◦ f as n → ∞. The points of most interest are those whose forward orbits show recurrent behavior. Thus we are led to focus on the set K of points
Entrainment and chaos in a pulsedriven HodgkinHuxley oscillator
, 2005
"... The HodgkinHuxley model describes action potential generation in certain types of neurons and is a standard model for conductancebased, excitable cells. Following the early work of Winfree and Best, this paper explores the response of a spontaneously spiking HodgkinHuxley neuron model to a period ..."
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The HodgkinHuxley model describes action potential generation in certain types of neurons and is a standard model for conductancebased, excitable cells. Following the early work of Winfree and Best, this paper explores the response of a spontaneously spiking HodgkinHuxley neuron model to a periodic pulsatile drive. The response as a function of drive period and amplitude is systematically characterized. A wide range of qualitatively distinct responses are found, including entrainment to the input pulse train and persistent chaos. These observations are consistent with a theory of kicked oscillators developed by Qiudong Wang and LaiSang Young. In addition to general features predicted by WangYoung theory, it is found that most combinations of drive period and amplitude lead to entrainment instead of chaos. This preference for entrainment over chaos is explained by the structure of the HodgkinHuxley phase resetting curve. 1
Rigorous Bounds On The Fast Dynamo Growth Rate Involving Topological Entropy
 COMM. MATH. PHYS
, 1994
"... The fast dynamo growth rate for a C k+1 map or flow is bounded above by topological entropy plus a 1=k correction. The proof uses techniques of random maps combined with a result of Yomdin relating curve growth to topological entropy. This upper bound implies the following antidynamo theorem: in ..."
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The fast dynamo growth rate for a C k+1 map or flow is bounded above by topological entropy plus a 1=k correction. The proof uses techniques of random maps combined with a result of Yomdin relating curve growth to topological entropy. This upper bound implies the following antidynamo theorem: in C 1 systems fast dynamo action is not possible without the presence of chaos. In addition topological entropy is used to construct a lower bound for the fast dynamo growth rate in the case Rm = 1.