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17
Dispersing billiards with moving scatterers. accepted by
 Communications in Mathematical Physics
, 2012
"... Abstract. We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argumen ..."
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Abstract. We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for nonstationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of timedependent dynamical systems. Acknowledgements. Stenlund is supported by the Academy of Finland; he also wishes to thank Pertti Mattila for valuable correspondence. Young is supported by NSF Grant DMS1101594, and Zhang is supported by NSF Grant DMS0901448. 1.
Nonstationary compositions of Anosov diffeomorphisms
 Nonlinearity
"... Abstract. Motivated by nonequilibrium phenomena in nature, we study dynamical systems whose timeevolution is determined by nonstationary compositions of chaotic maps. The constituent maps are topologically transitive Anosov diffeomorphisms on a 2dimensional compact Riemannian manifold, which are ..."
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Abstract. Motivated by nonequilibrium phenomena in nature, we study dynamical systems whose timeevolution is determined by nonstationary compositions of chaotic maps. The constituent maps are topologically transitive Anosov diffeomorphisms on a 2dimensional compact Riemannian manifold, which are allowed to change with time — slowly, but in a rather arbitrary fashion. In particular, such systems admit no invariant measure. By constructing a coupling, we prove that any two sufficiently regular distributions of the initial state converge exponentially with time. Thus, a system of the kind loses memory of its statistical history rapidly.
MEMORY LOSS FOR TIMEDEPENDENT PIECEWISE EXPANDING SYSTEMS IN HIGHER DIMENSION
"... We prove a counterpart of exponential decay of correlations for nonstationary systems. Namely, given two probability measures absolutely continuous with respect to a reference measure, their total variation distance decreases exponentially under action by compositions of arbitrarily chosen maps clo ..."
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We prove a counterpart of exponential decay of correlations for nonstationary systems. Namely, given two probability measures absolutely continuous with respect to a reference measure, their total variation distance decreases exponentially under action by compositions of arbitrarily chosen maps close to those that are both enveloping and piecewise expanding.
A vectorvalued almost sure invariance principle for Sinai billiards with random scatterers
, 2012
"... Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the to ..."
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Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of timedependent billiards on the one hand and in probability theory on the other, we prove a vectorvalued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and nonstationary. We provide an expression for the covariance matrix, which in the nonstationary case differs from the traditional one. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with timedependent observables, and improve the accuracy and generality of existing results.
AN ALMOST SURE ERGODIC THEOREM FOR QUASISTATIC DYNAMICAL SYSTEMS
"... Abstract. We prove almost sure ergodic theorems for a class of systems called quasistatic dynamical systems. These results are needed, because the usual theorem due to Birkhoff does not apply in the absence of invariant measures. We also introduce the concept of a physical family of measures for a ..."
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Abstract. We prove almost sure ergodic theorems for a class of systems called quasistatic dynamical systems. These results are needed, because the usual theorem due to Birkhoff does not apply in the absence of invariant measures. We also introduce the concept of a physical family of measures for a quasistatic dynamical system. These objects manifest themselves, for instance, in numerical experiments. We then verify the conditions of the theorems and identify physical families of measures for two concrete models, quasistatic expanding systems and quasistatic dispersing billiards.
EXPLICIT CORRELATION BOUNDS FOR EXPANDING CIRCLE MAPS USING THE COUPLING METHOD
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Some results on the entropy of nonautonomous dynamical systems
"... Abstract In this paper we advance the entropy theory of discrete nonautonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measuretheoretic entropy theory of general topological systems. We derive several conditions guaranteeing ..."
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Abstract In this paper we advance the entropy theory of discrete nonautonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measuretheoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the nonstationary subshifts of finite type, introduced by Fisher and Arnoux. In particular, we give sufficient conditions for the variational principle, relating the topological and measuretheoretic entropy, to hold. Keywords: Nonautonomous dynamical system; topological entropy; metric entropy; nonstationary subshift of finite type * Partly supported by DFG fellowship KA 3893/11. CK thanks Tomasz Downarowicz and Sergiy Kolyada for helpful comments.
Almost sure invariance principle for sequential and nonstationary dynamical systems
, 2014
"... 3 ASIP for sequential expanding maps of the interval. 7 4 ASIP for the shrinking target problem: expanding maps. 9 ..."
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3 ASIP for sequential expanding maps of the interval. 7 4 ASIP for the shrinking target problem: expanding maps. 9
QUASISTATIC DYNAMICS WITH INTERMITTENCY
"... Abstract. We study an intermittent quasistatic dynamical system composed of nonuniformly hyperbolic Pomeau–Manneville maps with timedependent parameters. We prove an ergodic theorem which shows almost sure convergence of time averages in a certain parameter range, and identify the unique physical ..."
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Abstract. We study an intermittent quasistatic dynamical system composed of nonuniformly hyperbolic Pomeau–Manneville maps with timedependent parameters. We prove an ergodic theorem which shows almost sure convergence of time averages in a certain parameter range, and identify the unique physical family of measures. The theorem also shows convergence in probability in a larger parameter range. In the process, we establish other results that will be useful for further analysis of the statistical properties of the model. Acknowledgements. This work was supported by the Jane and Aatos Erkko Foundation, and by Emil Aaltosen Säätiö. 1.
QUASISTATIC DYNAMICAL SYSTEMS
"... Abstract. We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) s ..."
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Abstract. We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Timeevolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the timeevolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behaviour as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a wellposed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the “obvious ” centering suggested by the initial distribution sometimes fails to yield the expected diffusion.