Hausdorff Dimension of this set

In summary, the conversation discusses the definition of $$\mathbb{Y} = C \times C^{c} \subset \mathbb{R}^{2}$$ where ##C## is the Cantor set and ##C^{c}## is its complement in ##[0,1]##. It is mentioned that ##\mathbb{Y}## is neither open nor closed, and the Hausdorff dimension of ##C## is ##\Large \frac{log2}{log3}##. The conversation then moves on to discussing how to compute the Hausdorff dimension of the Cartesian product of sets, using a theorem from Wikipedia. This leads to the conclusion that the Hausdorff dimension
  • #1
Bachelier
376
0
Define:

$$\mathbb{Y} = C \times C^{c} \subset \mathbb{R}^{2}$$
where ##C## is the Cantor set and ##C^{c}## is its complement in ##[0,1]##

First I think ##\mathbb{Y}## is neither open nor closed.

Second, the Hausdorff dimension of ##C## is ##\Large \frac{log2}{log3}##. How do we compute the ##HD## of the Cartesian product of sets? For instance ##HD(ℝ^{k})= k## hence can we compute ##HD(\mathbb{Y})?##
 
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  • #3
micromass said:
Wikipedia seems to have a nice theorem about this: http://en.wikipedia.org/wiki/Hausdorff_dimension#Self-similar_sets

This theorem implies that the Hausdorff dimension is the solution of ##6\left(\frac{1}{3}\right)^s = 1##.

So that gives the ##Hausdorff \ dim## of ##C^c## to be 1. makes sense.
 

What is the Hausdorff dimension of a set?

The Hausdorff dimension of a set is a measure of its geometric complexity. It describes how much space the set occupies in higher dimensions.

How is the Hausdorff dimension different from other dimensions?

Unlike the traditional integer dimensions (such as length, width, and height), the Hausdorff dimension can be a non-integer value. This allows for a more precise characterization of complex sets with irregular shapes.

How is the Hausdorff dimension calculated?

The Hausdorff dimension is calculated using the Hausdorff measure, which is a way to measure the size of a set in terms of its "covering" by smaller sets. The dimension is then defined as the infimum of all values of s for which the Hausdorff measure is zero.

What is the significance of the Hausdorff dimension in mathematics?

The Hausdorff dimension is an important concept in fractal geometry, which studies the geometric properties of irregular, self-similar structures. It also has applications in fields such as dynamical systems, chaos theory, and probability theory.

Can the Hausdorff dimension be greater than three?

Yes, the Hausdorff dimension can take on values greater than three. In fact, there are many sets with Hausdorff dimension between 3 and 4, such as the Koch curve and the Sierpinski triangle, which have a fractal-like structure that cannot be fully described by traditional integer dimensions.

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