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In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve G in the plane. It is shown that there do not exist invariant circles near G when there is a point on G where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not $C^1$ there are examples with orbits that converge to a point of G. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dual billiards map. The appendix is a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards.
A polygon is called rational if the angle between each pair of sides is a rational multiple of $\pi$. The main theorem we will prove is
Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has "many" periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $\textbf{S}^1$.
We will also prove some refinements of Theorem 1: the "well distribution" of periodic orbits in the polygon and the residuality of the points $q \in Q$ with a dense set of periodic directions.
We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only if the geodesic is precompact, i.e. its image in the moduli space is contained in a compact set. Some applications are given, e.g. to stability properties of certain quasi-geodesics in Teichmüller space, and to estimates of translation distance for pseudo-Anosov maps.
We prove that for continuous maps on the interval, the existence of an n-cycle, implies the existence of n-1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
In $ \mathit {Ch91a}$ it was shown that the billiard ball map for the periodic Lorentz gas has infinite topological entropy. In this article we study the set of points with infinite Lyapunov exponents. Using the cell structure developed in $ \mathit {BSC90,Ku}$ we construct an ergodic invariant probability measure with infinite topological entropy supported on this set. Since the topological entropy is infinite this is a measure of maximal entropy. From the construction it is clear that there many such measures can coexist on a single component of topological transitivity. We also construct an ergodic invariant probability measure with finite entropy which is supported on this set showing that infinite exponents do not necessarily lead to infinite entropy.
For the polynomials $p_c(z)=z^d+c$, the periodic points of periods dividing $n$ are the roots of the polynomials $P_n(z)=p_c^{\circ n}(z)-z$, where any degree $d\geq 2$ is fixed. We prove that all periodic points of any exact period $k$ are roots of the same irreducible factor of $P_n$ over $\mathbb{C}(c)$. Moreover, we calculate the Galois groups of these irreducible factors and show that they consist of all permutations of periodic points which commute with the dynamics. These results carry over to larger families of maps, including the spaces of general degree-$d$-polynomials and families of rational maps. Main tool, and second main result, is a combinatorial description of the structure of the Mandelbrot set and its degree-$d$-counterparts in terms of internal addresses of hyperbolic components. Internal addresses interpret kneading sequences of angles in a geometric way and answer Devaney's question: "How can you tell where in the Mandelbrot a given rational external ray lands, without having Adrien Douady at your side?"
We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the relation of a hyperbolic 3-manifold to a Kleinian group. In order to construct the 3-lamination we analyze the natural extension of a rational map and the complex affine structure on the canonical 2-dimensional leaf space contained in it. In this paper the construction is carried out in full for post-critically finite maps. We show that the corresponding laminations have a compact convex core. As a first application we give a three-dimensional proof of Thurston's rigidity for post-critically finite mappings, via the "lamination extension" of the proofs of the Mostow and Marden rigidity and isomorphism theorems for hyperbolic 3-manifolds. An Ahlfors-type argument for zero measure of the Julia set is applied along the way. This approach also provides a new point of view on the Lattes deformable examples.
This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of $\textbf{C}^2$: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $\mu$ of the set $K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $\mu$ as the limit distribution of the periodic points of $f$.
Local formulae are given for the characteristic classes of a quasiconformal manifold using the subspace of exact forms in the Hilbert space of middle dimensional forms. The method applies to combinatorial manifolds and all topological manifolds except certain ones in dimension four.