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Kuramoto Model on Sierpinski Gasket I: Harmonic Maps
Journal article   Open access   Peer reviewed

Kuramoto Model on Sierpinski Gasket I: Harmonic Maps

Georgi Medvedev and Matthew S Mizuhara
Studies in Applied Mathematics, v 156(5), e70233
May 2026
DOI: https://doi.org/10.1111/sapm.70233
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https://doi.org/10.1111/sapm.70233View
Published, Version of Record (VoR) Open Access via Drexel Libraries Read and Publish Program 2026 Open CC BY-NC-ND V4.0

Abstract

Motivated by the study of attractors in the Kuramoto model (KM) on graphs, approximating the Sierpinski gasket (SG), we revisit the problem of harmonic maps (HMs) from SG to the circle, first considered by Strichartz. We provide a geometric proof of Strichartz's theorem, which states that for a prescribed degree and suitable boundary conditions, there exists a unique HM from SG to the circle. Furthermore, we extend this result to HMs on post-critically finite (p.c.f.) fractals. For continuous functions on SG, we define a degree given by a vector of integers of arbitrary finite length. We show that the degree determines a homotopy class of a continuous function on SG with values in the circle. This provides an analog of the Hopf degree theorem for continuous functions on SG. We then move on to analyze the HMs on SG. At the heart of our method lies an original construction of the covering spaces for the SG. After lifting continuous functions on the SG with values in the unit circle to continuous real-valued functions on the covering space, we use the harmonic extension algorithm to obtain a harmonic function on the covering space. The desired HM is obtained by restricting the domain of the resultant harmonic function to the fundamental domain and projecting the range to the circle. Each covering space is constructed separately for HMs of a given homotopy class, capturing its intrinsic topology. We show that with suitable modifications, the method applies to p.c.f. fractals, a large class of self-similar domains. We illustrate our method of constructing the HMs using numerical examples of HMs from the SG to the circle and discuss the construction of the covering spaces for several representative p.c.f. fractals, including the 3-level SG, the hexagasket, and the pentagasket. The results of this paper provide the foundation for the follow-up work, where we give a complete description of the attractors in the KM on graphs approximating p.c.f. fractals. Specifically, we show that all HMs identified in this paper are stable steady states of the KM.

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