Can there be a set whose Hausdorff dimension gradually changes?
For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, and changes without jumps?
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Sign up to join this communityCan there be a set whose Hausdorff dimension gradually changes?
For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, and changes without jumps?
I assume you want a set $A\subseteq [0,1]$ such that $\dim (A\cap [0,x])=x$ for all $x$. We can define $A_1$ by taking the union of a (Borel) subset of dimension $0$ of $[0,1/2]$ with a subset of dimension $1/2$ of $[1/2,1]$.To obtain $A_2$, we then make our sets larger on $[1/4,1/2]$, $[3/4,1]$ and again, we make the dimension on each current interval equal to its left endpoint. Then $A_1\subseteq A_2\subseteq \ldots$, and $A=\bigcup A_n$ works: by monotone convergence, $h^d(I\cap A)=\lim h^d(I\cap A_n)$, so $A\cap [0,x]$ has the right dimension at each $x=k2^{-n}$ and thus everywhere.
For an example of a different sort, you can consider the record set $R$ of a fractional Brownian motion with varying Hurst parameter $H(t)$. In this short paper it is shown that when $H(t)$ is constant, it equals the Hausdorff dimension of $R$. The properties of the fBM with varying Hurst parameter $H(t)$ are given here for measurable functions $H(t)$ taking values in $(\tfrac12,1)$. One expects that the Hausdorff dimension of $R\cap (t-\epsilon,t+\epsilon)$ converges to $H(t)$ as $\epsilon\to 0$.