Dehn filling Dehn twists

Let $\Sigma _{g,p}$ be the genus–g oriented surface with p punctures, with either g > 0 or p > 3. We show that $MCG(\Sigma _{g,p})/DT$ is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group $MCG(\Sigma _{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma _{g,p}$ for suitable K. Moreover, we show that in low complexity $MCG(\Sigma _{g,p})/DT$ is in fact hyperbolic. In particular, for 3g − 3 + p ⩽ 2, we show that the mapping class group $MCG(\Sigma _{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma _{g,p})$ is separable. The aforementioned results follow from general theorems about composite rotating families, in the sense of [13], that come from a collection of subgroups of vertex stabilizers for the action of a group G on a hyperbolic graph X. We give conditions ensuring that the graph X/N is again hyperbolic and various properties of the action of G on X persist for the action of G/N on X/N.

About the Orbit Structure of Sequences of Maps of Integers

Motivated by connections to the study of sequences of integers, we study, from a dynamical systems point of view, the orbit structure for certain sequences of maps of integers. We find sequences of maps for which all individual orbits are bounded and periodic and for which the number of periodic orbits of fixed period is finite. This allows the introduction of a formal ζ -function for the maps in these sequences, which are actually polynomials. We also find sequences of maps for which the orbit structure is more complicated, as they have both bounded and unbounded orbits, both individual and global. Most of our results are valid in a general numeration base.

Pinwheels and Quarter Turns

This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ‎ denote the second iterate of the outer billiards map defined on R 2 − P. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ‎, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ‎. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the X-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.

Foliations and conjugacy: Anosov structures in the plane

In a non-compact setting, the notion of hyperbolicity, together with the associated structure of stable and unstable manifolds (for unbounded orbits), is highly dependent on the choice of metric used to define it. We consider the simplest version of this, the analogue for the plane of Anosov diffeomorphisms, studied earlier by White and Mendes. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations. We show that if the structure of stable and unstable manifolds is required to be preserved by these conjugacies, the number of distinct equivalence classes of Anosov diffeomorphisms in the plane becomes infinite.

Complete orbits for twist maps on the plane: the case of small twist

In this article we consider twist maps that are non-periodic (and hence are defined on the plane rather than on the cylinder) and have small twist at infinity. Under natural assumptions the existence of infinitely many bounded orbits is established, and furthermore it is proved that unbounded orbits follow bounded orbits for long times. An application is given to the Fermi–Ulam ping-pong model with a non-periodic moving wall.

PERIODICITY AND RECURRENCE IN PIECEWISE ROTATIONS OF EUCLIDEAN SPACES

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for almost every piecewise isometry. By contrast, if n≥2 is even then periodic points are stable for almost every piecewise isometry whose set of defining isometries are not orientation reversing. If, in addition, the defining isometries satisfy an incommensurability condition then all unbounded orbits must be irrationally coded.