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Rare events, escape rates and quasistationarity: some exact formulae
 Journal Statistical Physics
"... Abstract. We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given. 1. ..."
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Cited by 42 (2 self)
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Abstract. We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given. 1.
Escape rates and conditionally invariant measures
, 2005
"... We consider dynamical systems on domains that are not invariant under the dynamics – for example, a system with a hole in the phase space – and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, ..."
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Cited by 41 (6 self)
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We consider dynamical systems on domains that are not invariant under the dynamics – for example, a system with a hole in the phase space – and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, we are led quickly to conditionally invariant measures that are absolutely continuous with respect to Lebesgue (a.c.c.i.m.). Comparisons with SRB measures are inevitable, yet there are important differences. Via informal discussions and examples, this paper seeks to clarify the ideas involved. It includes also a brief review of known results and possible directions of further work in this developing subject.
Stability of statistical properties in twodimensional piecewise hyperbolic maps Trans.
 Am. Math. Soc.
, 2008
"... Abstract. We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of twodimensional maps with uniformly bounded second derivative. For ..."
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Cited by 34 (17 self)
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Abstract. We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of twodimensional maps with uniformly bounded second derivative. For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.
Invariant Measures and Their Properties. A Functional Analytic Point of View
, 2002
"... In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the ..."
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Cited by 28 (2 self)
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In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the breadth of the method.
Markov extensions for dynamical systems with holes: an application to expanding maps of the interval,
 Israel J. of Math.
, 2005
"... Abstract We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (a ..."
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Cited by 20 (7 self)
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Abstract We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. for C 1+α expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.
Existence and convergence properties of physical measures for certain dynamical systems with holes
 ERGODIC THEORY DYNAM. SYS
, 2010
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Where to place a hole to achieve a maximal escape rate
, 2008
"... A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space ..."
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Cited by 17 (1 self)
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A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period) which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.
Markov extensions and conditionally invariant measures for certain logistic maps with small holes Ergod. Theory Dyn.
 Syst.
, 2005
"... Abstract We study the family of quadratic maps fa(x) = 1 − ax 2 on the interval [−1, 1] with 0 ≤ a ≤ 2. When small holes are introduced into the system, we prove the existence of an absolutely continuous conditionally invariant measure using the method of Markov extensions. The measure has a densit ..."
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Cited by 16 (5 self)
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Abstract We study the family of quadratic maps fa(x) = 1 − ax 2 on the interval [−1, 1] with 0 ≤ a ≤ 2. When small holes are introduced into the system, we prove the existence of an absolutely continuous conditionally invariant measure using the method of Markov extensions. The measure has a density which is bounded away from zero and is analogous to the density for the corresponding closed system. These results establish the exponential escape rate of Lebesgue measure from the system, despite the contraction in a neighborhood of the critical point of the map. We also prove convergence of the conditionally invariant measure to the SRB measure for fa as the size of the hole goes to zero.
Escape rates and physically relevant measures for billiards with small holes
 Communications Math. Phys
"... We study the billiard map corresponding to a periodic Lorentz gas in 2dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, ..."
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Cited by 13 (4 self)
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We study the billiard map corresponding to a periodic Lorentz gas in 2dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map. This paper is about leaky dynamical systems, or dynamical systems with holes. Consider a dynamical system defined by a map or a flow on a phase space M, and let H ⊂ M be a hole through which orbits escape, that is to say, once an orbit enters H, we stop considering it from that point on. Starting from an initial probability distribution µ0 on M, mass will leak out of the system as it evolves. Let µn denote the distribution remaining at time n. The most basic question one can ask about a leaky system is its rate of escape, i.e. whether
QUASIINVARIANT MEASURES, ESCAPE RATES AND THE EFFECT OF THE HOLE
, 906
"... perturbation of T into an interval map with a hole. Given a number ℓ, 0 < ℓ < 1, we compute an upperbound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than −ln(1 − ℓ). The two main ingredients o ..."
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Cited by 11 (2 self)
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perturbation of T into an interval map with a hole. Given a number ℓ, 0 < ℓ < 1, we compute an upperbound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than −ln(1 − ℓ). The two main ingredients of our approach are Ulam’s method and an abstract perturbation result of Keller and Liverani. 1.