Questions tagged [hausdorff-dimension]

Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension. To be used with [tag:fractals] or [tag:dimension-theory].

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5
votes
3answers
200 views

The Hausdorff dimension of the zero set of a real analytic function

Let $n>1$, and let $f:\mathbb{R}^n \to \mathbb{R}$ be a real-analytic function which is not identically zero. Does $\dim_{\mathcal H}(f^{-1}(0)) \le n-1$? here $\dim_{\mathcal H}$ refers to the ...
6
votes
1answer
394 views

Hausdorff dimension of Hamiltonian orbit closure and symplectic leaves

Let $\dot{x} = \Pi \cdot \nabla H$ be a smooth Hamiltonian-Poisson system on $\mathbb{R}^n$. $H: \mathbb{R}^n \to \mathbb{R}$ is the Hamiltonian and $\Pi = (\Pi^{ij})$ is a skew-symmetric matrix of ...
10
votes
1answer
361 views

Fractal dimensional analysis?

I know how to use a ruler to approximate a length of an object (like a wire or a stick) in meters. I could also use the ruler to approximate a two dimensional area (like a table top or a parking lot) ...
3
votes
0answers
43 views

Last step in proof of countable stability of Hausdorff dimension

In part of Kenneth Falconer's proof of the countable stability of Hausdorff dimension on p. 49, sect 3.2 of Fractal Geometry, I understand him to say that $$\dim_H \bigcup_{i=1}^{\infty}F_i\leq \sup_{...
1
vote
0answers
84 views

Growth rate of the Hausdorff measure

Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows: $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf ...