Welcome to the preprint server of the Institute for Mathematical Sciences at Stony Brook University.
The IMS preprints are also available from the mathematics section of the arXiv e-print server, which offers them in several additional formats. To find the IMS preprints at the arXiv, search for Stony Brook IMS in the report name.
PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
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.
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).
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.
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.
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.
A general construction for $\sigma-$finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of $f^n_*(\lambda)$ will imply the existence of a $\sigma-$finite invariant measure for the map $f$ which is absolutely continuous with respect to $\lambda$, a measure on the phase space describing the sets of measure zero.
Furthermore we will discuss sufficient conditions for the existence of $\sigma-$finite invariant absolutely continuous measures for real 1-dimensional dynamical systems.
Circle maps with a flat spot are studied which are differentiable, even on the boundary of the flat spot. Estimates on the Lebesgue measure and the Hausdorff dimension of the non-wandering set are obtained. Also, a sharp transition is found from degenerate geometry similar to what was found earlier for non-differentiable maps with a flat spot to bounded geometry as in critical maps without a flat spot.
We consider polynomial maps $f:C\to C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $C$ to itself which have degree two or more on each copy. In any space $\mathcal{p}^{S}$ of suitably normalized maps of this type, the post-critically bounded maps form a compact subset $\mathcal{C}^{S}$ called the connectedness locus, and the hyperbolic maps in $\mathcal{C}^{S}$ form an open set $\mathcal{H}^{S}$ called the hyperbolic connectedness locus. The various connected components $H_\alpha\subset \mathcal{H}^{S}$ are called hyperbolic components. It is shown that each hyperbolic component is a topological cell, containing a unique post-critically finite map which is called its center point. These hyperbolic components can be separated into finitely many distinct "types", each of which is characterized by a suitable reduced mapping schema $S(f)$. This is a rather crude invariant, which depends only on the topology of $f$ restricted to the complement of the Julia set. Any two components with the same reduced mapping schema are canonically biholomorphic to each other. There are similar statements for real polynomial maps, or for maps with marked critical points.
The group $SL(n,{\bf Z})$ acts linearly on $\mathcal{R}^n$, preserving the integer lattice $\mathcal{Z}^{n} \subset \mathcal{R}^{n}$. The induced (left) action on the n-torus $\mathcal{T}^{n} = \mathcal{R}^{n}/\mathcal{Z}^{n}$ will be referred to as the ``standard action''. It has recently been shown that the standard action of $SL(n,\mathcal{Z})$ on $\mathcal{T}^n$, for $n \geq 3$, is both topologically and smoothly rigid. That is, nearby actions in the space of representations of $SL(n,\mathcal{Z})$ into ${\rm Diff}^{+}(\mathcal{T}^{n})$ are smoothly conjugate to the standard action. In fact, this rigidity persists for the standard action of a subgroup of finite index. On the other hand, while the $\mathcal{Z}$ action on $\mathcal{T}^{n}$ defined by a single hyperbolic element of $SL(n,\mathcal{Z})$ is topologically rigid, an infinite dimensional space of smooth conjugacy classes occur in a neighborhood of the linear action. The standard action of $SL(2, \mathcal{Z})$ on $\mathcal{T}^2$ forms an intermediate case, with different rigidity properties from either extreme. One can construct continuous deformations of the standard action to obtain an (arbritrarily near) action to which it is not topologically conjugate. The purpose of the present paper is to show that if a nearby action, or more generally, an action with some mild Anosov properties, is conjugate to the standard action of $SL(2, \mathcal{Z})$ on $\mathcal{T}^2$ by a homeomorphism $h$, then $h$ is smooth. In fact, it will be shown that this rigidity holds for any non-cyclic subgroup of $SL(2, \mathcal{Z})$.
A semigroup generated by two dimensional $C^{1+\alpha}$ contracting maps is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives of generators and the smoothness $\alpha$ of the generators satisfy a compatibility condition $K< 1/l^{\alpha}$. We prove that the shape of the image of the core of a ball under any element of a regular semigroup is good (bounded geometric distortion like the Koebe $1/4$-lemma [1]). And we use it to show a lower and a upper bounds of the Hausdorff dimension of the limit set of a regular semigroup. We also consider a semigroup generated by higher dimensional maps.