36 citations · 185 across the 21 of their papers we have counts for
14 papers · 1 filter
Dissipative outer billiards: a case study
Gianluigi Del Magno, José Pedro Gaivão, Eugene Gutkin
We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any tr…
Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems
Eugene Gutkin
This is an updated and expanded version of our earlier survey article \cite{Gut5}. Section introduces the subject matter. Sections expose the basic material followin…
Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces
Eugene Gutkin
We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on -periodic polygonal surfaces.…
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
Jean-Pierre Conze, Eugene Gutkin
We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of or . In the -periodic case, we establish cr…
Insecurity for compact surfaces of positive genus
Victor Bangert, Eugene Gutkin
A pair of points in a riemannian manifold is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is in…
The billiard ball problem and rotation numbers
Eugene Gutkin
We introduce the concepts of rotation numbers and rotation vectors for billiard maps. Our approach is based on the birkhoff ergodic theorem. We anticipate that it will be useful, i…
Secure two-dimensional tori are flat
Victor Bangert, Eugene Gutkin
A riemannian manifold is secure if the geodesics between any pair of points in the manifold can be blocked by a finite number of point obstacles. Compact, flat manifolds are secure…
Topological entropy and blocking cost for geodesics in riemannian manifolds
Eugene Gutkin
For a pair of points in a compact, riemannian manifold let (resp. ) be the number of geodesic segments with length joining these points (res…
Growth rates for geometric complexities and counting functions in polygonal billiards
Eugene Gutkin, Michal Rams
We introduce a new method for estimating the growth of various quantities arising in dynamical systems. We apply our method to polygonal billiards on surfaces of constant curvature…
Growth of the number of geodesics between points and insecurity for riemannian manifolds
Keith Burns, Eugene Gutkin
A Riemannian manifold is said to be uniformly secure if there is a finite number such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked b…
A few remarks on periodic orbits for planar billiard tables
Eugene Gutkin
I announce a solution of the conjecture about the measure of periodic points for planar billiard tables. The theorem says that if $\Om\subset\R^2$ is a compact domain with piecewis…
Connecting geodesics and security of configurations in compact locally symmetric spaces
Eugene Gutkin, Viktor Schroeder
A pair of points in a riemannian manifold makes a secure configuration if the totality of geodesics connecting them can be blocked by a finite set. The manifold is secure if every…
Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards
E. Gutkin, S. Tabachnikov
We introduce the class of piecewise convex transformations, and study their complexity. We apply the results to the complexity of polygonal billiards on surfaces of constant curvat…
Topological entropy of generalized polygon exchanges
Eugene Gutkin, Nicolai Haydn
We obtain geometric upper bounds on the topological entropy of generalized polygon exchange transformations. As an application of our results, we show that billiards in polygons an…