Hausdorff dimension of a subset of Cantor set - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:51:02Z http://mathoverflow.net/feeds/question/88576 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88576/hausdorff-dimension-of-a-subset-of-cantor-set Hausdorff dimension of a subset of Cantor set MichaelNgelo 2012-02-15T23:17:52Z 2012-02-16T00:24:35Z <p>What is the Hausdorff dimension of the subset $$F := \{ x = \sum^\infty_{n=1} \frac{2 x_n}{3^n} \in [0,1] : x_n \in \{ 0 , 1 \} , x_n = 1 \Rightarrow x_{n+1}=0 \}$$ of the Cantor set? Is it known already?</p> <p>As far as I know, this set can be corresponded to a binary tree related to the Fibonacci sequence. (I don't know how to call that tree.)</p> http://mathoverflow.net/questions/88576/hausdorff-dimension-of-a-subset-of-cantor-set/88579#88579 Answer by Vaughn Climenhaga for Hausdorff dimension of a subset of Cantor set Vaughn Climenhaga 2012-02-16T00:24:35Z 2012-02-16T00:24:35Z <p>The comments by Andreas and Anton give you the answer already to your specific question. Let me give a more general answer, since your question is very representative of a whole class of examples.</p> <p>The condition that $x_n = 1 \Rightarrow x_{n+1} = 0$ is a <em>Markov</em> condition: the value of $x_{n+1}$ is restricted by the value of $x_n$. In your case you are considering all sequences in $\{0,1\}^\mathbb{N}$ such that the symbol $1$ cannot follow itself; one could also consider more symbols and more complicated restrictions, such as "every occurrence of $2$ can only be followed by $0$ or $2$, but not $1$". See <a href="http://en.wikipedia.org/wiki/Subshift_of_finite_type" rel="nofollow">http://en.wikipedia.org/wiki/Subshift_of_finite_type</a> for more details.</p> <p>Subshifts of finite type (abbreviated SFTs) are also called topological Markov chains, and can be presented in terms of a transition matrix, as described in that Wikipedia article. The logarithm of the largest eigenvalue of the transition matrix is an important quantity called the <em><a href="http://en.wikipedia.org/wiki/Topological_entropy" rel="nofollow">topological entropy</a></em> of the SFT.</p> <p>When you construct a subset of the Cantor set as in your question, the topological entropy turns out to be directly related to the Hausdorff dimension: namely Hausdorff dimension is topological entropy divided by $\log \lambda$, where $\lambda$ is the contraction ratio at each step of the construction of the Cantor set.</p> <p>We wrote a more detailed description of this in Pesin &amp; Climenhaga, "Lectures on fractal geometry and dynamical systems", or you can find many parts of it in most standard textbooks on dynamical systems.</p>