arXiv subjects

Miri Son

Publications and source records attributed to Miri Son.

3 recordsLinked to original sources

arXiv2026

Rigidity phenomena for time changes of products of Anosov flows

Our main results establish two rigidity phenomena in the class of time changes of a fixed product of Anosov flows. Our first result shows that two time changes having the same stabilizers for all periodic orbits are conjugate up to automorphism. The second rigidity result proves that if the stabilizers of periodic orbits can be simultaneously diagonalized, then the time change is conjugate to a product of flows up to automorphism. We allow our time changes to be H\"older continuous, which by structural stability implies that our results hold for $C^1$ perturbations of products of Anosov flows. We apply our main results to $C^1$ time changes of products of Anosov flows on $3$-dimensional manifolds. For such actions, we show that being totally Anosov, being conjugate to a product of flows and having the kernels of Lyapunov functionals not depend on the periodic orbits are all equivalent properties. We also build counterexamples to the Katok-Spatzier conjecture as time changes of products of any transitive Anosov flows, extending a result of Vinhage beyond the continuously accessible case.

math.DS↗

arXiv2025

Real analytic $\mathrm{SL}(n,\mathbb{R})$-actions on closed manifolds

We classify real-analytic $\mathrm{SL}(n,\mathbb{R})$-actions on closed manifolds of dimension m for $3\leq n\leq m\leq2n-3$, which extends Fisher--Melnick's work for $\mathrm{SL}(n,\mathbb{R})$-actions on closed n-manifolds. Additionally, we classify smooth $\mathrm{SL}(n,\mathbb{R})$-actions on closed m-manifolds that are fixed-point free. As a corollary, we obtain the density or non-density of structural stability of fixed-point free $\mathrm{SL}(n,\mathbb{R})$-actions.

math.DS↗

arXiv2023

${\rm SL}_2$ quantum trace in quantum Teichmüller theory via writhe

Quantization of the Teichmüller space of a punctured Riemann surface $S$ is an approach to $3$-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop $γ$ in $S$ gives rise to a natural trace-of-monodromy function $\mathbb{I}(γ)$ on the Teichmüller space. For any ideal triangulation $Δ$ of $S$, this function $\mathbb{I}(γ)$ is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of $Δ$. An important problem was to construct a quantization of this function $\mathbb{I}(γ)$, namely to replace it by a noncommutative Laurent polynomial in the quantum variables. This problem, which is closely related to the framed protected spin characters in physics, has been solved by Allegretti and Kim using Bonahon and Wong's ${\rm SL}_2$ quantum trace for skein algebras, and by Gabella using Gaiotto, Moore and Neitzke's Seiberg-Witten curves, spectral networks, and writhe of links. We show that these two solutions to the quantization problem coincide. We enhance Gabella's solution and show that it is a twist of the Bonahon-Wong quantum trace.

math.GT↗