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?


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.)



Hausdorff dimension of a subset of Cantor set
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.


The condition that $x\_n = 1 \Rightarrow x\_{n+1} = 0$ is a *Markov* 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 <http://en.wikipedia.org/wiki/Subshift_of_finite_type> for more details.


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 *[topological entropy](http://en.wikipedia.org/wiki/Topological_entropy)* of the SFT.


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.


We wrote a more detailed description of this in Pesin & Climenhaga, "Lectures on fractal geometry and dynamical systems", or you can find many parts of it in most standard textbooks on dynamical systems.


