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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.
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 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.
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.
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 $f$.
The Milnor problem on one-dimensional attractors is solved for $S-$unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $\omega(c)$ of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynamics.
A distortion theory is developed for $S-$unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of $S-$unimodal maps is classified according to a distortion property, called the Markov-property.
We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (non-uniform) hyperbolic behavior.
This paper concentrates on optical Hamiltonian systems of $T*\mathbb{T}^n$, i.e. those for which $H_{pp}$ is a positive definite matrix, and their relationship with symplectic twist maps. We present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems. The novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system. We also present a theorem of suspension by Hamiltonian systems for the class of symplectic twist map that emerges in our study. Finally, we extend our results to manifolds of negative curvature.
This will is an expository description of quadratic rational maps.
- Sections 2 through 6 are concerned with the geometry and topology of such maps.
- Sections 7-10 survey of some topics from the dynamics of quadratic rational maps. There are few proofs.
- Section 9 attempts to explore and picture moduli space by means of complex one-dimensional slices.
- Section 10 describes the theory of real quadratic rational maps.
For convenience in exposition, some technical details have been relegated to appendices:
- Appendix A outlines some classical algebra.
- Appendix B describes the topology of the space of rational maps of degree $d$.
- Appendix C outlines several convenient normal forms for quadratic rational maps, and computes relations between various invariants.
- Appendix D describes some geometry associated with the curves $Per_n(\mu)\subset M$.
- Appendix E describes totally disconnected Julia sets containing no critical points.
- Appendix F, written in collaboration with Tan Lei, describes an example of a connected quadratic Julia set for which no two components of the complement have a common boundary point.