Plane curve with continuously increasing Hausdorff dimension

Question

In a recent paper, we required the following fact.

Proposition 1. There exists a simple closed curve $\gamma\subset\mathbb{C}$ with the following property. If $\phi$ is a biholomorphic map, defined on a domain $U$ intersecting $\gamma$, such that $\phi(U\cap \gamma) \subset \phi(U)\cap \gamma$, then $\phi$ is the identity.

There are of course many ways to construct such a curve $\gamma$, but we did not find a reference in the literature. In the end, we settled on using the following fact.

Proposition 2. Given any closed interval $[t_1,t_2]\subset [1,2]$, there exists an arc $\alpha\colon [t_1,t_2]\to \mathbb{C}$ such that the local Hausdorff dimension of $\alpha$ at $\alpha(t)$ is equal to $t$ for all $t\in[t_1,t_2]$.

Here by the local Hausdorff dimension we mean the infimum of the Hausdorff dimensions of relative neighbourhoods of $\gamma(t)$ in $\gamma$.

Proposition 2 implies Proposition 1. Indeed, biholomorphic maps are bi-Lipschitz, and therefore preserve Hausdorff dimension. So if we form a Jordan curve $\gamma$ from $\alpha$ by identifying $\alpha(t_1)$ and $\alpha(t_2)$ in a suitable way (e.g. using a map that is conformal on a neighbourhood of $(t_1,t_2)$), then $\gamma$ has the desired property.

Again, we did not find a reference for Proposition 2, but it is easy to give a construction e.g. using a modified version of the von Koch curve.

However, surely these facts are well-known, and I would be surprised if such examples have not previously appeared in published papers.

Question. Does anyone know of a reference (ideally a classical reference) for Proposition 1 (or Proposition 2) in the published literature?

Many thanks for your help!

Answer 1

I do not have a reference but can propose a proof of Proposition 1 which is independent of and simpler than Proposition 2.

Let us first construct a non-closed curve, namely a graph of a convex function on $[0,1]$. It is known that for a convex function $f$ the left and right derivatives exist everywhere, and these limits coincide except at a countable set. Make this countable set dense on $[0,1]$, and arrange so that the jumps of the derivative (the difference between the right and left derivative) at the points of this countable set are all pairwise distinct. To construct such a function, choose a convergent series $$\sum_{k=1}^\infty a_k<\infty,\quad a_k>0$$ with all $a_k$ pairwise distinct, and choose a countable dense set $\{ x_k\}\in(0.1)$. Then define $f$ by $$f''=\sum_{k=1}^\infty a_k\delta(x-x_k)$$ in the sense of distributions. The graph of this function has the desired property, since at every point $P_k=(x_k,f(x_k))$ the left and right tangents exist and the angles between these left and right tangents are distinct so any conformal map must fix each $P_k$.

To obtain a closed curve, choose an analogous concave function $g$ on $[0,1]$ with $g(0)=f(0),g(1)=f(1)$ and your curve $\gamma$ is the union of their graphs.