[Paper Review] Metric properties of mean wiggly continua

Jacek Graczyk, Peter W. Jones+1 more|arXiv (Cornell University)|Mar 29, 2012

Holomorphic and Operator Theory14 references3 citations

TL;DR

This paper introduces the concept of mean wiggly continua—continua that wiggle sufficiently often at small scales—to establish lower bounds on Hausdorff dimension using integral estimates of $β$-numbers. It constructs a Borel probability measure with controlled decay to prove that such continua have Hausdorff dimension strictly greater than 1, extending prior uniform wigglyness results to non-uniform, mean-based conditions.

ABSTRACT

We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The main technical ingredient is a construction, for every continuum K, of a Borel probabilistic measure μwith the property that on every ball B(x,r), with x in 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 in K where g(x, r) has a logarithmic growth to infinity as r goes to 0. The theory of mean wiggly continua leads, via the product formula for dimensions, to new estimates of the Hausdorff dimension for Cantor sets. We prove also that asymptotically flat sets are of Hausdorff dimension 1 and that asymptotically non-porous continua are of the maximal dimension. Another application of the theory is geometric Bowen's dichotomy for Topological 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 boundary of the Mandelbrot set.

Motivation & Objective

• To generalize uniform wigglyness-based dimension estimates to non-uniform, mean-based conditions.
• To develop a measure-theoretic tool that controls local mass decay in terms of wiggliness at all scales.
• To establish sharp dimension estimates for continua with oscillatory geometry at positive density of scales.
• To apply the theory to dynamical systems, particularly Julia sets of quadratic polynomials on the boundary of the Mandelbrot set.
• To prove that asymptotically flat sets have dimension 1 and asymptotically non-porous continua achieve maximal dimension.

Proposed method

• Construct a Borel probability measure $\mu$ on a wiggly subset $Z \subset K$ such that $\mu(B(x,r)) \leq c' r \exp(-c \int_r^{\mathrm{diam}\,K} \beta_K^2(x,t) \frac{dt}{t})$ for all $x \in Z$, $r > 0$.
• Use corona-type decomposition techniques to control the measure's decay based on the $\beta$-number integral.
• Define mean wiggly continua as those where $\liminf_{r \to 0} \frac{1}{-\log r} \int_r^{\mathrm{diam}\,K} \beta_K^2(x,t) \frac{dt}{t} > 0$.
• Apply the product formula for Hausdorff dimension to estimate dimension of Cantor sets.
• Use the measure construction to prove that mean wiggly continua have $\dim_H(K) > 1$.
• Apply the framework to prove geometric Bowen's dichotomy for Topological Collet-Eckmann maps and show Julia sets of quadratic polynomials on $\partial\mathcal{M}$ are mean wiggly.

Experimental results

Research questions

• RQ1 Can lower bounds on Hausdorff dimension be established for continua that wiggle only on average across scales, rather than uniformly?
• RQ2 What is the sharp relationship between the integral of $\beta_K^2(x,t)$ and the Hausdorff dimension of a continuum?
• RQ3 Can the theory of mean wiggly continua be applied to Julia sets in complex dynamics, particularly those on the boundary of the Mandelbrot set?
• RQ4 Do asymptotically flat sets necessarily have Hausdorff dimension 1, and do asymptotically non-porous continua achieve maximal dimension?
• RQ5 How does the measure-theoretic construction of $\mu$ with controlled decay relate to the geometric structure of the underlying continuum?

Key findings

• For any continuum $K$ of diameter 1, there exists a Borel probability measure $\mu$ supported on a wiggly subset $Z \subset K$ such that $\mu(B(x,r)) \leq c' r \exp(-c \int_r^{\mathrm{diam}\,K} \beta_K^2(x,t) \frac{dt}{t})$ for all $x \in Z$, $r > 0$.
• If $\liminf_{r \to 0} \frac{1}{-\log r} \int_r^{\mathrm{diam}\,K} \beta_K^2(x,t) \frac{dt}{t} \geq \beta_0^2 > 0$ for all $x \in K \setminus E$ with $\mathcal{H}^1(E) < \infty$, then $\dim_H(K) \geq 1 + c\beta_0^2$ for a universal constant $c > 0$.
• The dimension estimate remains valid under the weaker condition $\liminf_{r \to 0} \frac{1}{-\log r} \int_r^{\mathrm{diam}\,K} \beta_K^2(x,t) \frac{dt}{t} > 0$, implying $\dim_H(K) > 1$.
• Asymptotically flat sets have Hausdorff dimension exactly 1.
• Asymptotically non-porous continua have maximal Hausdorff dimension.
• Julia sets of quadratic polynomials for parameters on the boundary of the Mandelbrot set $\partial\mathcal{M}$ are mean wiggly, and thus have dimension greater than 1, confirming their dynamical naturality.