Rotation Sets of Billiards with One Obstacle

Rotation Sets for orbits of Degree One Circle Maps

Let F be the lifting of a circle map of degree one. In [?] a notion of F -rotation interval of a point x ∈ S was given. In this paper we define and study a new notion of a rotation set of a point which preserves more of the dynamical information contained in the sequences {F(y)}n=0 than the preserved by the one from [?]. In particular, we characterize dynamically the endpoints of these sets and...

Obstructing Visibilities with One Obstacle

Obstacle representations of graphs have been investigated quite intensely over the last few years. We focus on graphs that can be represented by a single obstacle. Given a (topologically open) polygon C and a finite set P of points in general position in the complement of C, the visibility graph GC(P ) has a vertex for each point in P and an edge pq for any two points p and q in P that can see ...

Rotation Sets for Some Non–continuous Maps of Degree One

Iteration of liftings of non necessarily continuous maps of the circle into itself are considered as discrete dynamical systems of dimension one. The rotation set has proven to be a powerful tool to study the set of possible periods and the behaviour of orbits for continuous and old heavy maps. An extension of the class of maps for which the rotation set maintains this power is given.

On smooth relaxations of obstacle sets

We present and discuss a method to relax sets described by finitely many smooth convex inequality constraints by the level set of a single smooth convex inequality constraint. Based on error bounds and Lipschitz continuity, special attention is paid to the maximal approximation error and a guaranteed safety margin. Our results allow to safely avoid the obstacle by obeying a single smooth constr...

Directional Entropy of Rotation Sets