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In this paper we shall show that there exists $l_0$ such that for each even integer $l \geq l_0$ there exists $c_1 \in \mathbb{R}$ for which the Julia set of $z \mapsto z^l + c_1$ has positive Lebesgue measure. This solves an old problem.
Editor's note: In 1997, it was shown by Xavier Buff that there was a serious flaw in the argument, leaving a gap in the proof. Currently (1999), the question of polynomials with a positive measure Julia sets remains open.
In this paper we shall show that there exists a polynomial unimodal map $f: [0,1] \mapsto [0,1]$ which is
1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$ is equal to an interval)
2) for which $\omega(c)$ is a Cantor set
3) for which $\omega(x)=\omega(c)$ for Lebesgue almost all x.
So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor.
Let $H: \mathbb{C}^2 \to \mathbb{C}^2$ be the Hénon mapping given by $$ \begin{bmatrix}x\\y\end{bmatrix} \mapsto \begin{bmatrix}p(x) - ay\\x\end{bmatrix}. $$ The key invariant subsets are $K_\pm$, the sets of points with bounded forward images, $J_\pm = \partial K_\pm$ their boundaries, $J = J_+ \cap J_-$, and $K = K_+ \cap K_-$. In this paper we identify the topological structure of these sets when $p$ is hyperbolic and $|a|$ is sufficiently small, \ie, when $H$ is a small perturbation of the polynomial $p$. The description involves projective and inductive limits of objects defined in terms of $p$ alone.
According to Sullivan, a space $\mathcal{E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional Teichmüller space. This is a basic idea in Sullivan's approach to the Renormalization Conjecture. One of its principle ingredients is to supply $\mathcal{E}$ with the Teichmüller metric. To have such a metric one has to know, first of all, that all maps of $\mathcal{E}$ are quasi-symmetrically conjugate. This was proved [Ji] and [JS] for some classes of non-renormalizable maps (when the critical point is not too recurrent). Here we consider a space of non-renormalizable unimodal maps with in a sense fastest possible recurrence of the critical point (called Fibonacci). Our goal is to supply this space with the Teichmüller metric.
This is an outline of work in progress. We study the conjecture that the topological entropy of a real cubic map depends "monotonely" on its parameters, in the sense that each locus of constant entropy in parameter space is a connected set. This material will be presented in more detail in a later paper.
This paper investigates dynamics that persist under isotopy in classes of orientation-preserving homeomorphisms of orientable surfaces. The persistence of periodic points with respect to periodic and strong Nielsen equivalence is studied. The existence of a dynamically minimal representative with respect to these relations is proved and necessary and sufficient conditions for the isotopy stability of an equivalence class are given. It is also shown that most the dynamics of the minimal representative persist under isotopy in the sense that any isotopic map has an invariant set that is semiconjugate to it.
A key problem in holomorphic dynamics is to classify complex quadratics $z\mapsto z^2+c$ up to topological conjugacy. The Rigidity Conjecture would assert that any non-hyperbolic polynomial is topologically rigid, that is, not topologically conjugate to any other polynomial. This would imply density of hyperbolic polynomials in the complex quadratic family (Compare Fatou [F, p. 73]). A stronger conjecture usually abbreviated as MLC would assert that the Mandelbrot set is locally connected.
A while ago MLC was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at most finitely renormalizable parameter values. One of our goals is to prove MLC for some infinitely renormalizable parameter values. Loosely speaking, we need all renormalizations to have bounded combinatorial rotation number (assumption C1) and sufficiently high combinatorial type (assumption C2).
For real quadratic polynomials of bounded combinatorial type the complex a priori bounds were obtained by Sullivan. Our result complements the Sullivan's result in the unbounded case. Moreover, it gives a background for Sullivan's renormalization theory for some bounded type polynomials outside the real line where the problem of a priori bounds was not handled before for any single polynomial. An important consequence of a priori bounds is absence of invariant measurable line fields on the Julia set (McMullen) which is equivalent to quasi-conformal (qc) rigidity. To prove stronger topological rigidity we construct a qc conjugacy between any two topologically conjugate polynomials (Theorem III). We do this by means of a pull-back argument, based on the linear growth of moduli and a priori bounds. Actually the argument gives the stronger combinatorial rigidity which implies MLC.
We complete the paper with an application to the real quadratic family. Here we can give a precise dichotomy (Theorem IV): on each renormalization level we either observe a big modulus, or essentially bounded geometry. This allows us to combine the above considerations with Sullivan's argument for bounded geometry case, and to obtain a new proof of the rigidity conjecture on the real line (compare McMullen and Swiatek).
We prove that non-hyperbolic non-renormalizable quadratic polynomials are expansion inducing. For renormalizable polynomials a counterpart of this statement is that in the case of unbounded combinatorics renormalized mappings become almost quadratic. Technically, this follows from the decay of the box geometry. Specific estimates of the rate of this decay are shown which are sharp in a class of S-unimodal mappings combinatorially related to rotations of bounded type. We use real methods based on cross-ratios and Schwarzian derivative complemented by complex-analytic estimates in terms of conformal moduli.
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).
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.