Showing papers by "Lai Sang Young published in 2012"
TL;DR: In this article, Li et al. proved the existence of geometric structures called horseshoes, which imply the presence of infinitely many periodic solutions for diffeomorphisms of compact manifolds.
Abstract: Two settings are considered: flows on finite dimensional Riemann- ian manifolds, and semiflows on Hilbert spaces with conditions consistent with those in systems defined by dissipative parabolic PDEs. Under certain assump- tions on Lyapunov exponents and entropy, we prove the existence of geometric structures called horseshoes; this implies in particular the presence of infinitely many periodic solutions. For diffeomorphisms of compact manifolds, analogous results are due to A. Katok. Here we extend Katok’s results to (i) continuous time and (ii) infinite dimensions. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012 E-mail address: lian@cims.nyu.edu Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012 E-mail address: lsy@cims.nyu.edu
43 citations
Posted Content•
TL;DR: In this article, the authors discuss the evolution of probability distributions for certain time-dependent dynamical systems and prove exponential loss of memory for expanding maps and for one-dimensional piecewise expanding maps with slowly varying parameters.
Abstract: This paper discusses the evolution of probability distributions for certain time-dependent dynamical systems. Exponential loss of memory is proved for expanding maps and for one-dimensional piecewise expanding maps with slowly varying parameters.
39 citations
TL;DR: In this paper, the relation between escape rates and pressure in general dynamical systems with holes is studied, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents.
Abstract: We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions, including Lebesgue and Sinai-Reulle-Bowen (SRB) measures. Lower bounds do not hold in such a generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon with holes.
33 citations
TL;DR: In this paper, a model of billiards with moving scatterers is proposed, where the locations and shapes of the scatterer may change by small amounts between collisions, and the main result is the exponential loss of memory of initial data at uniform rates.
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 non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.
29 citations
TL;DR: In this article, the mean energy along the chain is shown to be constant and equal to (1, 2 ) where TL and TR are the temperatures of the two heat baths.
Abstract: We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to \(\frac{1}{2} \sqrt{T_{L}T_{R}}\) where TL and TR are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier’s Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach.
8 citations
TL;DR: It is shown that horseshoes are abundant whenever the limit cycle is kicked to a specific region of the phase space and offer a geometric explanation for the stretch-and-fold behavior which ensues.
Abstract: This paper contains a numerical study of the periodically forced van der Pol system. Our aim is to determine the extent to which chaotic behavior occurs in this system as well as the nature of the chaos. Unlike previous studies, which used continuous forcing, we work with instantaneous kicks, for which the geometry is simpler. Our study covers a range of parameters describing nonlinearity, kick sizes, and kick periods. We show that horseshoes are abundant whenever the limit cycle is kicked to a specific region of the phase space and offer a geometric explanation for the stretch-and-fold behavior which ensues.
2 citations