UCLA

We study certain approximately self-similar metric spaces that arise from expanding Thurston maps $f: \mathbb{S}^2 \to \mathbb{S}^2$ called visual spheres. It is known [HP09, Theorem 4.2.11][BM17, Theorem 18.1(ii')]that the quasisymmetry class of a visual sphere of $f$ is related to the rationality of $f$. We prove that a visual sphere is indeed approximately self-similar if $f$ does not have periodic critical points. This is done by picking a nice visual metric of an expanding Thurston map. Using the nice metric, we study the solenoid of $f$. We put a specific metric on the leaves of $f$ and show that the leaves and weak tangents are almost the same thing. We then study the visual spheres of expanding Thurston maps with an emphasis on a quasisymmetric invariant, called the Ahlfors regular conformal dimension. We show that the Ahlfors regular conformal dimension of any weak tangent of a visual sphere is the same as the Ahlfors regular conformal dimension of the visual sphere itself, and that the Ahlfors regular conformal dimension of any weak tangent is attainable if and only if Ahlfors regular conformal dimension of the visual sphere is attainable. We show by an example that the same does not hold for more general metric spaces. Finally, we show that a visual sphere has $p$-thick curve family supported on an $f$-invariant porous subset if $f$ has a irreducible $p$-thick $f$-stable multicurve. We give an example to show that when $p = 2$, the condition of a $p$-thick $f$-stable multicurve is sharp.