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Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a contractible set.
We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon.
Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the one-sided shift space of $d$ symbols. We give exotic examples of maps of an arbitrary degree $d$ with a non-simply connected, completely invariant basin of attraction and arbitrary number $k\ge 2$ of critical points in the basin. For such a map $f\in H^d$ with $k < d$ there is no polynomial in $H(f)$. Finally we describe a computer experiment joining an exotic example to a Newton's method (for a polynomial) rational function with a 1-parameter family of rational maps.
Let $G$ be a non-elementary, finitely generated Kleinian group, $\Lambda(G)$ its limit set and $\Omega(G) = \overline {\mathbb C} \backslash \Lambda(G)$ its set of discontinuity. Let $\delta(G)$ be the critical exponent for the Poincarè series and let $\Lambda_c$ be the conical limit set of $G$. Suppose $\Omega_0$ is a simply connected component of $\Omega(G)$. We prove that
- $\delta(G) = \dim(\Lambda_c)$
- A simply connected component $\Omega$ is either a disk or $\dim(\partial \Omega)>1$
- $ \Lambda(G)$ is either totally disconnected, a circle or has dimension $>1$
- $G$ is geometrically infinite iff $\dim(\Lambda)=2$
- If $G_n \to G$ algebraically then $\dim(\Lambda)\leq \liminf \dim(\Lambda_n)$
- The Minkowski dimension of $\Lambda$ equals the Hausdorff dimension
- If $\text{area}(\Lambda)=0$ then $\delta(G) =\dim(\Lambda(G))$
The proof also shows that $\dim(\Lambda(G)) >1$ iff the conical limit set has dimension $>1$ iff the Poincarè exponent of the group is $>1$. Furthermore, a simply connected component of $\Omega(G)$ either is a disk or has non-differentiable boundary in the the sense that the (inner) tangent points of $\partial \Omega$ have zero $1$-dimensional measure. almost every point (with respect to harmonic measure) is a twist point.
We show that the conjugacy class of an eventually expanding continuous piecewise affine interval map is contained in a smooth codimension 1 submanifold of parameter space. In particular conjugacy classes have empty interior. This is based on a study of the relation between induced Markov maps and ergodic theoretical behavior.
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.