A geometric approach to semi-dispersing billiards (Survey)

1998 ◽  
Vol 18 (2) ◽  
pp. 303-319 ◽  
Author(s):  
D. BURAGO ◽  
S. FERLEGER ◽  
A. KONONENKO
Keyword(s):  
Topological Entropy ◽  
Riemannian Geometry ◽  
Geometric Approach ◽  
Open Problems ◽  
Periodic Trajectories ◽  
Uniform Estimates ◽  
Dispersing Billiards

We summarize the results of several recent papers, together with a few new results, which rely on a connection between semi-dispersing billiards and non-regular Riemannian geometry. We use this connection to solve several open problems about the existence of uniform estimates on the number of collisions, topological entropy and periodic trajectories of such billiards.

Topological entropy of semi-dispersing billiards

1998 ◽  
Vol 18 (4) ◽  
pp. 791-805 ◽  
Author(s):  
D. BURAGO ◽  
S. FERLEGER ◽  
A. KONONENKO
Keyword(s):  
Sectional Curvature ◽  
Topological Entropy ◽  
Riemannian Geometry ◽  
Lorentz Gas ◽  
Periodic Points ◽  
Billiard Table ◽  
Positive Sectional Curvature ◽  
Periodic Trajectories ◽  
Exponential Estimates ◽  
Dispersing Billiards

In this paper we continue to explore the applications of the connections between singular Riemannian geometry and billiard systems that were first used in [6] to prove estimates on the number of collisions in non-degenerate semi-dispersing billiards.In this paper we show that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. Also, we prove exponential estimates on the number of periodic points (for the first return map to the boundary of a simple-connected billiard table) and the number of periodic trajectories (for the billiard flow). In \S5 we prove some estimates for the topological entropy of Lorentz gas.

Milnor's conjecture on monotonicity of topological entropy: Results and questions

2014 ◽  
Author(s):  
Sebastian van Strien
Keyword(s):  
Topological Entropy ◽  
State Of The Art ◽  
Rigidity Theorem ◽  
Real Polynomial ◽  
Open Problems ◽  
Polynomial Maps ◽  
History Of ◽  
The Subject ◽  
Quadratic Case ◽  
Real Polynomial Maps

This chapter discusses Milnor's conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof. It discusses the history of this conjecture, gives an outline of the proof in the general case, and describes the state of the art in the subject. The proof makes use of an important result by Kozlovski, Shen, and van Strien on the density of hyperbolicity in the space of real polynomial maps, which is a far-reaching generalization of the Thurston Rigidity Theorem. (In the quadratic case, density of hyperbolicity had been proved in studies done by M. Lyubich and J. Graczyk and G. Swiatek.) The chapter concludes with a list of open problems.

scholarly journals A Survey on Differential Geometry of Riemannian Maps Between

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Bayram Şahin
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Research Area ◽  
Riemannian Submersions ◽  
Open Problems ◽  
Geometric Objects ◽  
Riemannian Map

AbstractThe main aim of this paper is to state recent results in Riemannian geometry obtained by the existence of a Riemannian map between Riemannian manifolds and to introduce certain geometric objects along such maps which allow one to use the techniques of submanifolds or Riemannian submersions for Riemannian maps. The paper also contains several open problems related to the research area.

Riemann - Cartan Geometry of Trivializable Gauge Fields

1989 ◽  
Vol 44 (3) ◽  
pp. 222-238 ◽  
Author(s):  
M. Mattes ◽  
M. Sorg
Keyword(s):  
Gauge Field ◽  
Riemannian Geometry ◽  
Geometric Approach ◽  
Gauge Fields ◽  
Riemannian Structure ◽  
Cartan Geometry ◽  
Yang Mills ◽  
Integrability Conditions

A Riemann-Cartan structure can be associated to any SO (4) trivializable gauge field. Under certain integrability conditions, this non-Riemannian geometry may be replaced by a strictly Riemannian one. The Yang-Mills equations guarantee the existence of such a Riemannian structure. The general SO(4) trivializable solution for the SO(3) Yang-Mills equations is discussed within the geometric approach.

CHAOTIC PROPERTIES OF SUBSHIFTS GENERATED BY A NONPERIODIC RECURRENT ORBIT

2000 ◽  
Vol 10 (05) ◽  
pp. 1067-1073 ◽  
Author(s):  
XIN-CHU FU ◽  
YIBIN FU ◽  
JINQIAO DUAN ◽  
ROBERT S. MACKAY
Keyword(s):  
Periodic Point ◽  
Topological Entropy ◽  
Metric Spaces ◽  
Initial Conditions ◽  
Open Problems ◽  
Recurrent Point ◽  
Orbit Closures ◽  
Nonwandering Point ◽  
Sensitive Dependence ◽  
The One

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain nonperiodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a nonperiodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

scholarly journals A Geometric Approach to Voiculescu-Brown Entropy

2004 ◽  
Vol 47 (4) ◽  
pp. 553-565
Author(s):  
David Kerr
Keyword(s):  
Topological Entropy ◽  
Basic Problem ◽  
Geometric Approach ◽  
Topological Dynamics ◽  
Positive Entropy ◽  
Algebraic Dynamics

AbstractA basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are “chaotic.” While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of C*-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author’s talk at theWinter 2002Meeting of the CanadianMathematical Society in Ottawa, Ontario.

Uniform Estimates on the Number of Collisions in Semi-Dispersing Billiards

1998 ◽  
Vol 147 (3) ◽  
pp. 695 ◽  
Author(s):  
D. Burago ◽  
S. Ferleger ◽  
A. Kononenko
Keyword(s):  
Uniform Estimates ◽  
Dispersing Billiards
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