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Metric properties of mean wiggly continua

Abstract : We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The main tech- nical ingredient is a construction, for every continuum K, of a Borel probabilistic measure μ with the property that on every ball B(x,r), x ∈ K, the measure is bounded by a universal constant multiple of r exp(−g(x, r)), where g(x, r) ≥ 0 is an explicit function. The continuum K is mean wiggly at exactly those points x ∈ K where g(x,r) has a logarithmic growth to ∞ as r→0. The theory of mean wiggly continua leads, via the product formula for dimensions, to new esti- mates of the Hausdorff dimension for Cantor sets. We prove also that asymptotically flat sets are of Hausdorff dimension 1 and that asymp- totically non-porous continua are of the maximal dimension. Another application of the theory is geometric Bowen's dichotomy for Topolog- ical Collet-Eckmann maps in rational dynamics. In particular, mean wiggly continua are dynamically natural as they occur as Julia sets of quadratic polynomials for parameters from a generic set on the bound- ary of the Mandelbrot set M.
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Contributor : Nicolae Mihalache <>
Submitted on : Tuesday, March 5, 2013 - 3:07:41 PM
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  • HAL Id : hal-00796896, version 1

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Jacek Graczyk, Peter Jones, Nicolae Mihalache. Metric properties of mean wiggly continua. 2012. ⟨hal-00796896⟩

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