Example of a geodesic metric space with non-integer Hausdorff dimension

Question

If one googles metric spaces with a finite non-integer Hausdorff dimension, one usually finds fractals like the Cantor set or the Koch snowflake.

These two examples are not so well-behaved in the sense that they are not geodesic. A metric space $(X,d)$ is called geodesic if for any points $a,b\in X$ with $d(a,b)=r>0$ there is a distance preserving map $\gamma : [0,r] \to X$ with $\gamma(0)=a$ and $\gamma(r)=b$. This is stronger than but implies that any two points can be joined by a rectifyable curve which rules out the two above examples.

So, my question is: Is there a geodesic metric space with a finite non-integer Hausdorff dimension?

Answer 1

Laakso's Wormhole Spaces

In this paper, Laakso constructs spaces of arbitrary dimension $Q$ greater than one which are path connected.

The basic idea is to start with a Cantor set $\mathscr{C}$ and cross it with an interval. That is, the "base space" is $\mathscr{C}\times[0,1]$. Then certain points are identified with each other, creating "wormholes" which connect different parts of the Cantor set to each other. The idea is that the points are chosen at each "level" of the Cantor set in such a way that it can be guaranteed that it is always possible to construct a path from one point to another, but in such a way that the topology isn't totally trashed and the dimension is not changed.

The relevant theorem in the paper is

Theorem 2.6. The space $F$ is compact, Ahlfors $Q$-regular and every pair of points can be connected by pencils with constants independent of points. Moreover, pencils can be chosen to contain only geodesics

Here, Ahlfors $Q$-regularity means that balls scale "nicely"—roughly speaking, the volume of a ball is proportional to $r^Q$, where $r$ is the radius of that ball. One of the nice properties of $Q$-regular spaces is that the Hausdorff dimension of such a space is $Q$.

In addition, the paper describes, in the construction of this kind of space, how to build a geodesic between any two points, and how to measure the length of that geodesic.

Sierpinski Triangle

I did not start with this example, as it is more difficult to provide an explicit path between two points (it requires working through sequences of approximations). However, it is (I think) a more intuitive kind of example, as it is relatively easy to see how the paths should work (again, thinking of approximations).

Start with a complete graph with three vertices, and assume that the length of each edge is one unit. That is, think of an equilateral triangle as a graph. Make three copies of this graph, each half the size of the original, and "glue them together" so that they fit nicely into the original graph. Repeat this process infinitely many times—the Sierpinski triangle is the limiting object, and looks something like

This image is from a paper by Freiberg, Kigami, and Ruiz. This paper is reasonably readable, and gives a somewhat more explicit construction.

The metric on this space is induced by the sequence of approximating graphs. For pairs of "rational" points of the Sierpinski triangle (i.e. those points which are vertices of one of the approximating graphs), there is a path (not necessarily unique) which joins the two points (and the distance between those points is the length of the path). For paths which include an "irrational" endpoint, that endpoint can be approximated by a sequence of rational points, and the distance between that point and any other point can be approximated in the "obvious" manner (i.e. by considering paths on the approximating graphs of finer and finer scale).

The Hausdorff dimension of the Sierpinski triangle is $\log_3(4)$.