[Paper Review] Existence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps

Shiping Cao, Malte Hassler+3 more|arXiv (Cornell University)|Aug 17, 2020

Mathematical Dynamics and Fractals24 references4 citations

TL;DR

This paper establishes the existence and uniqueness of balanced resistance forms—key for constructing diffusions—on Julia sets of Misiurewicz-Sierpinski maps, a class of finitely ramified fractals. Using Sabot’s theory of preserved relations and dynamical properties of the maps, the authors prove that such forms exist and are unique when the parameters satisfy $\frac{1}{m} + \frac{1}{n} > \frac{1}{2}$, with a renormalization constant $\eta = \frac{2n+1}{n+1}$ for $m=1$.

ABSTRACT

We study the balanced resistance forms on the Julia sets of Misiurewicz-Sierpinski maps, which are self-similar resistance forms with equal weights. In particular, we use a theorem of Sabot to prove the existence and uniqueness of balanced forms on these Julia sets. We also provide an explorative study on the resistance forms on the Julia sets of rational maps with periodic critical points.

Motivation & Objective

• Address the fundamental problem of constructing diffusion processes on a new class of finitely ramified fractals: Julia sets of Misiurewicz-Sierpinski maps.
• Establish the existence and uniqueness of balanced resistance forms on these fractals, which are essential for defining diffusions.
• Overcome the challenge of non-symmetric symmetry groups by leveraging the dynamics of the rational maps rather than geometric symmetries.
• Extend Sabot’s uniqueness framework beyond nested fractals to a broader class of self-similar, post-critically finite fractals.
• Provide a systematic analysis of preserved relations and their spectral radii to determine existence conditions.

Proposed method

• Apply Sabot’s theorem on preserved relations to analyze resistance forms on the Julia sets of Misiurewicz-Sierpinski maps.
• Use the graph-directed structure induced by the iterated function system (IFS) and the dynamics of the rational map $R_{\lambda,n,m}(z) = z^n + \lambda/z^m$.
• Define rotationally symmetric resistance forms on level-1 cells and reduce the problem to solving a functional equation involving mappings $\Psi_{k,l}$.
• Compute the exact values of the spectral radii $\underline{\rho}$ and $\overline{\rho}$ for non-trivial preserved relations $\mathcal{J}_1$ and $\mathcal{J}_2$ on the boundary vertex set.
• Apply the criterion that existence of a solution depends on whether $\overline{\rho} < \underline{\rho}^{-1}$ for each preserved relation.
• Use Hilbert’s projective metric and Brouwer’s fixed point theorem to prove existence of rotationally symmetric solutions under the condition $\frac{1}{m} + \frac{1}{n} > \frac{1}{2}$.

Experimental results

Research questions

• RQ1 Under what conditions does a balanced resistance form exist on the Julia set of a Misiurewicz-Sierpinski map?
• RQ2 Is the resistance form unique when it exists, and does this uniqueness hold without requiring symmetry of the form?
• RQ3 How do the dynamical properties of the rational map $R_{\lambda,n,m}$ influence the existence and uniqueness of resistance forms?
• RQ4 What role do preserved relations play in determining the existence of graph-directed invariant resistance forms?
• RQ5 Can the renormalization constant $\eta$ for the resistance form be explicitly computed, and how does it depend on the parameters $m$ and $n$?

Key findings

• The existence and uniqueness of a balanced resistance form on the Julia set of a Misiurewicz-Sierpinski map is established when $\frac{1}{m} + \frac{1}{n} > \frac{1}{2}$.
• For the case $m=1$, the renormalization constant is explicitly given by $\eta = \frac{2n+1}{n+1}$, which matches the Sierpinski gasket's constant when $n=2$.
• There is no graph-directed invariant resistance form when $\frac{1}{m} + \frac{1}{n} < \frac{1}{2}$, including the cases $(m,n) = (3,6), (6,3), (4,4)$.
• The critical case $\frac{1}{m} + \frac{1}{n} = \frac{1}{2}$ is unresolved, but numerical experiments suggest non-existence for $(m,n) = (3,6)$ and $(6,3)$.
• The unique solution is rotationally symmetric, and its existence is proven via Hilbert’s projective metric and Brouwer’s fixed point theorem.
• Only two non-trivial preserved relations $\mathcal{J}_1$ and $\mathcal{J}_2$ exist on the boundary vertex set $\{p_0, p_1, q_0, q_1\}$, with exact spectral radii: $\underline{\rho}_{\mathcal{J}_1} = \overline{\rho}_{\mathcal{J}_1} = \frac{1}{2}$, $\overline{\rho}_{\mathcal{J}_2} = \frac{1}{n}$, and $\underline{\rho}_{V/\mathcal{J}_2} = \frac{mn}{m+n}$.