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Maciej P. Wojtkowski

Bio: Maciej P. Wojtkowski is an academic researcher from University of Arizona. The author has contributed to research in topics: Lyapunov exponent & Hamiltonian system. The author has an hindex of 15, co-authored 31 publications receiving 1047 citations. Previous affiliations of Maciej P. Wojtkowski include University of Warmia and Mazury in Olsztyn & University of Szczecin.

Papers
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Journal ArticleDOI
TL;DR: In this paper, a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents was introduced, where the exponents have non-vanishing exponents.
Abstract: We introduce a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents.

237 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that preservation of cones leads to non-vanishing of Lyapunov exponents of billiards, and that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.
Abstract: We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view

174 citations

Book ChapterDOI
TL;DR: The Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (nonuniform) hyperbolic behavior is discussed.
Abstract: We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (nonuniform) hyperbolic behavior.

115 citations

Journal ArticleDOI
TL;DR: In this article, a generalization of the Hamiltonian formalism is studied and the symmetry of the Lyapunov spectrum established for the resulting systems, applied to the Gausssian isokinetic dynamics of interacting particles with hard core collisions and other systems.
Abstract: A generalization of the Hamiltonian formalism is studied and the symmetry of the Lyapunov spectrum established for the resulting systems. The formalism is applied to the Gausssian isokinetic dynamics of interacting particles with hard core collisions and other systems.

90 citations

Journal ArticleDOI
TL;DR: In this paper, a generalization of the Hilbert projective metric to the space of positive deenite matrices is introduced, where the action of any monotone map on the manifold of positive Lagrangian subspaces contracts the metric of the Riemannian Grassmannian.
Abstract: We introduce a generalization of the Hilbert projective metric to the space of positive deenite matrices which we view as part of the Lagrangian Grass-mannian. x0. Introduction. In his treatment of Kalman Bucy lters Bougerol B1], B2] uses the Riemannian metric on the set of positive deenite matrices considered as a Riemannian symmetric space. Graphs of symmetric linear maps from R n to R n are Lagrangian subspaces in the standard linear symplectic space R n R n. We call a Lagrangian subspace positive if it is a graph of a positive deenite linear map. Further we call a linear symplectic map monotone if it maps positive Lagrangian subspaces onto positive Lagrangian subspaces. Bougerol discovered that the symplectic matrices in Kalman ltering theory are monotone. He shows that the action of any monotone map on the manifold of positive Lagrangian subspaces contracts the metric of the Riemannian symmetric space. It is the only (up to scale) Riemannian metric which has this property. The goal of this paper is to introduce a natural Finsler metric in the manifold of positive deenite matrices which in addition to being contracted by the action of any monotone map has striking geometric properties. In particular we obtain that the coeecient of least contraction is equal to the hyperbolic tangent of one half of the diameter of the image. This is the same relation which was obtained by Birkhoo Bir] for the Hilbert projective metric. In the case of the positive orthant the Hilbert metric is also only Finsler (cf. W2]) which reeects the nonsmoothness of the cone. It is natural that the generalization of the Hilbert metric to the space of positive deenite matrices is not smooth because its boundary in the Lagrangian Grassmannian is not smooth. After this paper was written we learned that this metric was discussed earlier by Vesentini Ves] from a completely diierent point of view which is applicable also The second author gratefully acknowledges the hospitality of Forschungsinstitut f ur Mathe-matik at ETH Z urich, where this paper was written.

60 citations


Cited by
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Journal ArticleDOI
TL;DR: In this article, the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces is discussed.
Abstract: This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces. The main results are for discrete time systems. To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 1970’s (see [S1], [B], [R2]). Since then much progress has been made on two fronts: there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero Lyapunov exponents ([O], [P1], [Ka], [LY]), and there are detailed analyses of specific kinds of dynamical systems including, for example, billiards, 1-dimensional and Henon-type maps ([S2], [BSC]; [HK], [J]; [BC2], [BY1]). Statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero Lyapunov exponents. The goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than Axiom A. This class will not be defined explicitly, but it includes some of the much studied examples. By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation. Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times. These conditions can be checked in concrete situations, giving a unified way of proving a number of results, some new and some old. Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Henon-type maps.

875 citations

Book
01 Jan 1995
TL;DR: In this paper, the Hadamard Cartan Theorem and the Hopf-Rinow Theorem for rank rigidity of geodesic flows on metric spaces are discussed.
Abstract: I. On the interior geometry of metric spaces.- 1. Preliminaries.- 2. The Hopf-Rinow Theorem.- 3. Spaces with curvature bounded from above.- 4. The Hadamard-Cartan Theorem.- 5. Hadamard spaces.- II. The boundary at infinity.- 1. Closure of X via Busemann functions.- 2. Closure of X via rays.- 3. Classification of isometries.- 4. The cone at infinity and the Tits metric.- III. Weak hyperbolicity.- 1. The duality condition.- 2. Geodesic flows on Hadamard spaces.- 3. The flat half plane condition.- 4. Harmonic functions and random walks on ?.- IV. Rank rigidity.- 1. Preliminaries on geodesic flows.- 2. Jacobi fields and curvature.- 3. Busemann functions and horospheres.- 4. Rank, regular vectors and flats.- 5. An invariant set at infinity.- 6. Proof of the rank rigidity.- Appendix. Ergodicity of geodesic flows.- 1. Introductory remarks.- Measure and ergodic theory preliminaries.- Absolutely continuous foliations.- Anosov flows and the Ho continuity of invariant distributions.- Proof of absolute continuity and ergodicity.

614 citations

Journal ArticleDOI
TL;DR: A survey of time-reversal symmetry in dynamical systems can be found in this paper, where the relation of time reversible dynamical sytems to equivariant and Hamiltonian systems is discussed.

483 citations

Journal ArticleDOI
David Ruelle1
TL;DR: In this article, the authors review various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics, and adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution.
Abstract: This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures In terms of these one can prove the Gallavotti–Cohen fluctuation theorem One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations At equilibrium one recovers in particular the Onsager reciprocity relations Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss The results just mentioned hold so far only for classical systems; they do not involve large size, ie, they hold without a thermodynamic limit

436 citations