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Blocking: new examples and properties of products

Published online by Cambridge University Press:  01 April 2009

PILAR HERREROS
PILAR HERREROS*
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA (email: [email protected])
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Abstract

We say that a pair of points x and y is secure if there exists a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena in both the manifold and the billiard table settings. In approaching this, we study whether a product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security which requires that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we construct a compact C1 surface which contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, and generates billiard tables with similar blocking behavior.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 569 - 578
Copyright
Copyright © Cambridge University Press 2008

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