Related papers: Rotation number values at a discontinuity

Let $f$ be an orientation preserving homeomorphism of $S^2$ which has a (nontrivial) continuum $X$ as a minimal set. Then there are exactly two connected components of $S^2\setminus X$ which are left invariant by $f$ and all the others are…

A family of plane oriented continuous paths depending on a fixed real positive number $R$ is considered. For any point $x$ on the path, the previous points lie out of any circle of radius $R$ having at $x$ interior normal in a suitable…

Let $X$ be a closed invariant subset of the half--open annulus $\mathbb{A}$ such that $\mathbb{A} \setminus X$ is homeomorphic to $\mathbb{A}$. We prove that either the rotation number of all forward semi--orbits of accessible points of $X$…

We study cocycles of homeomorphisms of $\T$ in the isotopy class of the identity over shift spaces, using as a tool a novel definition of rotation sets inspired in the classical work of Miziurewicz and Zieman. We discuss different notions…

The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant…

We construct two examples of invariant manifolds that despite being locally unstable at every point in the transverse direction are globally stable. Using numerical simulations we show that these invariant manifolds temporarily repel nearby…

Let $f$ be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When $f$…

We study the set of periods of degree 1 continuous maps from sigma into itself, where sigma denotes the space shaped like the letter sigma (i.e., a segment attached to a circle by one of its endpoints). Since the maps under consideration…

This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slipping at the point of contact and no spinning about the vertical. A reduction of the equations of…

We study extensions of the measure of maximal entropy to suitable compactifications of the parameter space and the moduli space of rational maps acting on the Riemann sphere. For parameter space, we consider a space which resolves the…

For two-parameter families of dissipative twist maps, we investigate the dynamics of invariant graphs as well as the thresholds for their existence and breakdown. Our main results are as follows: (1) For arbitrarily small $C^r$…

In the paper, we consider bivariate isotropic dilation matrices that are similar (up to constant factors) to rotation matrices; and we show that, in this case, the two-scale relation can be considered also as the relation between not only…

Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z^d for the initial configuration of all rotors aligned. If n particles in turn perform rotor walks starting from the origin, we…

We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and…

Every simple drawing of a graph in the plane naturally induces a rotation system, but it is easy to exhibit a rotation system that does not arise from a simple drawing in the plane. We extend this to all surfaces: for every fixed surface…

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the…

We give a formula on the rotation number of a sequence of primitive vectors, which is a generalization of the formula on the rotation number of a unimodular sequence in \cite{Higashitani and Masuda}.

We study periodic orbits for area-preserving surface diffeomorphisms, particularly some global properities related to the action function and rotation numbers. We generalize recent works of Machel Hutchings [4], proving the existence of…

We consider a rational map f:S->S of a complex projective surface together with an invariant meromorphic two form. Under a mild topological assumption on the map, we show that the zeroes of the invariant form can be eliminated by birational…

We explore the triangle outer billiards map in points at infinity in the hyperbolic plane, focusing on the rotation number. Building on Dogru and Tabachnikov's work, which established the conditions for triangles where the rotation number…