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* Starred papers have appeared in the journal cited.
It will be shown that the smooth conjugacy class of an $S-$unimodal map which does not have a periodic attractor neither a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M.Shub and D.Sullivan for smooth expanding maps of the circle.
We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries about hyperbolic components of the Mandelbrot set.
Along the way, we introduce partitions of dynamical and parameter planes which are of independent interest, and we interpret the Mandelbrot set as a symbolic parameter space of kneading sequences and internal addresses.
New insights into the combinatorial structure of the the Mandelbrot set are given by 'Correspondence' and 'Translation' Principles both conjectured and partially proved by E. Lau and D. Schleicher. We provide complete proofs of these principles and discuss results related to them.
Note: The 'Translation' and 'Correspondence' Principles given earlier turned out to be false in the general case. In April 1999, an errata was added to discuss which parts of the two statements are incorrect and which parts remain true.
The aim of this work is to describe the equivalence relations in $\mathbb{Q/Z}$ that arise as the rational lamination of polynomials with all cycles repelling. We also describe where in parameter space one can find a polynomial with all cycles repelling and a given rational lamination. At the same time we derive some consequences that this study has regarding the topology of Julia sets.
We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $\alpha$ depends only on the rate of convergence. We also give examples of $C^\infty$ critical circle maps with the same rotation number that are not $C^{1+\beta}$ conjugate for any $\beta>0$.
We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$.
We show that the Feigenbaum-Cvitanović equation can be interpreted as a linearizing equation, and the domain of analyticity of the Feigenbaum fixed point of renormalization as a basin of attraction. There is a natural decomposition of this basin which enables to recover a result of local connectivity by Jiang and Hu for the Feigenbaum Julia set.
This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler-Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry-Mather theory of twist maps and the Hedlund-Morse-Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.
This paper surveys various results concerning stability for the dynamics of Lagrangian (or Hamiltonian) systems on compact manifolds. The main, positive results state, roughly, that if the configuration manifold carries a hyperbolic metric, $\textit {i.e.}$ a metric of constant negative curvature, then the dynamics of the geodesic flow persists in the Euler-Lagrange flows of a large class of time-periodic Lagrangian systems. This class contains all time-periodic mechanical systems on such manifolds. Many of the results on Lagrangian systems also hold for twist maps on the cotangent bundle of hyperbolic manifolds. We also present a new stability result for autonomous Lagrangian systems on the two torus which shows, among other things, that there are minimizers of all rotation directions. However, in contrast to the previously known $\textit{hedlund}$ case of just a metric, the result allows the possibility of gaps in the speed spectrum of minimizers. Our negative result is an example of an autonomous mechanical Lagrangian system on the two-torus in which this gap actually occurs. The same system also gives us an example of a Euler-Lagrange minimizer which is not a Jacobi minimizer on its energy level.
In this work we will show that the Teichmüller distance for all elements of a certain class of generalized polynomial-like maps (the class of off-critically hyperbolic generalized polynomial-like maps) is actually a distance, as in the case of real polynomials with connected Julia set, as studied by Sullivan. This class contains several important classes of generalized polynomial-like maps, namely: Yoccoz, Lyubich, Sullivan and Fibonacci. In our proof we can not use external arguments (like external classes). Instead we use hyperbolic sets inside the Julia sets of our maps. Those hyperbolic sets will allow us to use our main analytic tool, namely Sullivan's rigidity Theorem for non-linear analytic hyperbolic systems. Lyubich has constructed a measure of maximal entropy measure $m$ on the Julia set of any rational function $f$. Zdunik classified exactly when the Hausdorff dimension of $m$ equals the Hausdorff dimension of the Julia set. We show that the strict inequality holds if $f$ is off-crititcally hyperbolic, except for Chebyshev polynomials. This result is a particular case of Zdunik's result if we consider $f$ as a polynomial, but is an extension of Zdunik's result if $f$ is a generalized polynomial-like map. The proof follows from the non-existence of invariant affine structure.