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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.
We prove that if $A$ is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $\mu$ is an arbitrary $f-$invariant measure with positive Lyapunov exponents on the boundary of $A$, then $\mu$-almost every point $q$ in the boundary of $A$ is accessible along a curve from $A$. In fact we prove the accessibility of every "good" $q$ i.e. such $q$ for which "small neighborhoods arrive at large scale" under iteration of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem on the accessibility of periodic sources.
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then periodic points in the boundary of A are dense in this boundary. To prove this in the non simply- connected or parabolic situations we prove a more abstract, geometric coding trees version.
We extend the work of Bielefeld, Fisher and Hubbard on Critical Portraits to the case of arbitrary postcritically finite polynomials. This determines an effective classification of postcritically finite polynomials as dynamical systems. This paper is the first in a series of two based on the author's thesis, which deals with the classification of postcritically finite polynomials. In this first part we conclude the study of critical portraits initiated by Fisher and continued by Bielefeld, Fisher and Hubbard.
One of the most striking early results in symplectic topology is Gromov's "Non-Squeezing Theorem", which says that it is impossible to embed a large ball symplectically into a thin cylinder of the form $\textbf{R}^{2n} \times B^2$, where $B^2$ is a $2$-disc. This led to Hofer's discovery of symplectic capacities, which give a way of measuring the size of subsets in symplectic manifolds. Recently, Hofer found a way to measure the size (or energy) of symplectic diffeomorphisms by looking at the total variation of their generating Hamiltonians. This gives rise to a bi-invariant (pseudo-)norm on the group $\textbf{Ham}(M)$ of compactly supported Hamiltonian symplectomorphisms of the manifold $M$. The deep fact is that this pseudo-norm is a norm; in other words, the only symplectomorphism on $M$ with zero energy is the identity map. Up to now, this had been proved only for sufficiently nice symplectic manifolds, and by rather complicated analytic arguments.
In this paper we consider a more geometric version of this energy, which was first considered by Eliashberg and Hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary. We prove, by a simple geometric argument, that both versions of energy give rise to genuine norms on all symplectic manifolds. Roughly speaking, we show that if there were a symplectomorphism of $M$ which had "too little" energy, one could embed a large ball into a thin cylinder $M \times B^2$. Thus there is a direct geometric relation between symplectic rigidity and energy.
The second half of the paper is devoted to a proof of the Non-Squeezing theorem for an arbitrary manifold $M$. We do not need to restrict to manifolds in which the theory of pseudo-holomorphic curves behaves well. This is of interest since most other deep results in symplectic topology are generalised from Euclidean space to other manifolds by using this theory, and hence are still not known to be valid for arbitrary symplectic manifolds.
We provide an effective classification of postcritically finite polynomials as dynamical systems by means of Hubbard Trees. This can be viewed as an application of the results developed in part 1 (ims93-5).