Related papers: Rotation number values at a discontinuity

For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of…

Identity-homotopic self-homeomorphisms of a space of non-periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation numbers for such maps. In constrast to…

For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and…

We apply set-valued numerical methods to compute an accurate enclosure of the rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points, which is used to check the rationality of the…

We study properties of an array of numbers, called "the triangle," in which each row is formed by rotating all the numbers in the previous row to the left by $m$ positions in cyclical fashion, then appending a number to the end of the row.…

We study discrete-time random dynamical systems where each fibre map is an orientation-preserving homeomorphism of the circle. We prove that the existence of a random periodic cycle with period at least two implies that the random rotation…

For a continuous map on a topological graph containing a unique loop S, it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and…

In this paper, we study rotation numbers of random dynamical systems on the circle. We prove the existence of rotation numbers and the continuous dependence of rotation numbers on the systems. As an application, we prove a theorem on…

Given an integer m>1 we consider Zm-equivariant and orientation preserving homeomorphisms in the plane with an asymptotically stable fixed point at the origin. We present examples without periodic points and having some complicated…

In this article we present an efficient algorithm to compute rotation intervals of circle maps of degree one. It is based on the computation of the rotation number of a monotone circle map of degree one with a constant section. The main…

This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation…

We describe the relation between the dynamical properties of a quasiperiodically forced orientation-preserving circle homeomorphism and the behavior of the fibered rotation number with respect to strictly monotone perturbations. Despite the…

The parameter dependence of the rotation number in families of circle maps which are perturbations of rational rotations is described. We show that if, at a critical parameter value, the map is a (rigid) rotation $x\to x+\frac{p}{q}~({\rm…

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we…

Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to $r$-ball…

We consider an orientation preserving homeomorphism $h$ of $S^2$ which admits a repellor denoted $\infty$ and an attractor $-\infty$, which is not a North-South map, such that the basins of $\infty$ and $-\infty$ intersect. We study various…

We show how the small perturbations of a linear cocycle have a relative rotation number associated with an invariant measure of the base dynamics an with a $2$-dimensional bundle of the finest dominated splitting (provided that some…

We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $f_{\p}$ are parametrized by a quintuple $\p$ of real numbers satisfying inequations. Viewing…

We consider the rotation number $\rho(t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^\infty$ diffeomorphism of the circle $S^1$. We shall show that if $\rho(t)$ is…

We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on the circle. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincar\'{e} lifts) approach does depend on…