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Local scaling of a set means that in a neighborhood of a point the structure of the set can be mapped into a finer scale structure of the set. These scaling transformations are compact sets of locally affine (that is: with uniformly $\alpha$-Hölder continuous derivatives) contractions. In this setting, without any assumption on the spacing of these contractions such as the open set condition, we show that the measure of the set is an upper semi-continuous of the scaling transformation in the $C^0$-topology. With a restriction on the 'non-conformality' (see below) the Hausdorff dimension is lower semi-continous function in the $C^{1}$-topology. We include some examples to show that neither of these notions is continuous.
We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upper- and a lower-bound for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the moduli of the eigenvalues of the underlying affine contraction are all equal (this includes Jordan blocks). The tiles we discuss play an important role in the theory of wavelets. We calculate the dimension for a number of examples.
We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on $T^2$ which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
In this paper we establish $C^2$ a-priori bounds for the scaling ratios of critical circle mappings in a form that gives also a compactness property for the renormalization operator.
The existence of smooth families of Lorenz maps exhibiting all possible dynamical behavior is established and the structure of the parameter space of these families is described.
The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmüller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov.
In a certain sense this hyperbolicity is an explanation of why the Teichmüller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmüller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmüller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface.
(revised version of January 1998)
We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. These products were observed by J. Milnor in computer experiments which inspired Lavaurs' proof of non local-connectivity for the cubic connectedness locus. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction is an $\textit {intertwining surgery}$ on two quadratics. The idea of intertwining first appeared in a collection of problems edited by Bielefeld. Using quasiconformal surgery techniques of Branner and Douady, we show that any two quadratics may be intertwined to obtain a cubic polynomial. The proof of continuity in our two-parameter setting requires further considerations involving ray combinatorics and a pullback argument.
A S. N. Bernstein problem is solved under a natural irreducibility condition. Earlier this result was obtained only in some special case.
We show that for any unimodal polynomial $f$ with real coefficients, all conformal measures for $f$ are ergodic.
We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Möbius transformation by its critical points.