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Suppose $I$ is an interval on a Cantor tree with $m$ children $I'$, each of length $\varepsilon$. I have that $\sum_{\hat I\text{ is a child of }I}|\hat I |^s=|I|^s \Rightarrow s=\frac{\log(m)}{-\log\varepsilon}$. Is $s$ the "local dimension" of the tree?

Is there a relationship between the the Hausdorff dimension of a Cantor tree and its local dimension?

The notation I'm using is similar to that found in Falconer's "Fractal Geometry":

By the "$k$th level of the Cantor tree", I am referring to whatever remains of the original interval after $k-1$ steps of removing intervals.

If I have a tree for which the number of children at each level is constant, say $M_k$ is the number of children of each interval in level $k-1$ and $D_k$ is the supremum of all the intervals in the $kth$ level, what does the upper bound for the Hausdorff dimension given by $\frac{\log(M_1...M_{k-1})}{-\log(D_k)}$ represent? I think it is the local dimension as well, but I am not sure.

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up vote 1 down vote accepted

There are many different definitions of local dimension of a set, but the one I've seen the most often occurring together with the ordinary Hausdorff dimension is the following:

Let $\cal C$ be the Cantor set obtained by at each step removing some closed interval from each interval from the last step. The size of this interval need not be fixed neither between steps nor for the intervals in each step and it need not be centered. Now pick any point $\xi \in \cal C$. Then $\xi$ can be written as the intersection of all intervals from the construction containing $\xi$. Let this sequence of intervals be $\{ I_j \}$ where $I_j$ is an interval from the $n$th step of the construction. Let $I_j^1$, $I_j^2$, ..., $I_j^m$ be the $m$ children of $I_j$. Then the local Hausdorff dimension of $\cal C$ at $\xi$ is defined as

$$\sup \{ s: \prod \Bigl( \frac{|I_j^1|}{|I_j|}\Bigr)^s + ... + \Bigl(\frac{|I_j|^m}{|I_j|} \Bigr)^s <\infty\}$$

In your case, since all children have the same length, this would simplify to

$$\sup \{ s: \prod \Bigl( m\frac{|I_j^1|}{|I_j|}\Bigr)^s <\infty\}$$

If defined in this way, the Hausdorff dimension of the set (for nice enough sets) is the supremum of its local dimensions.

The local dimension is thus always defined at a point, and not in a certain step, which implies that we must use the information in infinitely many steps to be able to say something at all.

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