Veech Surfaces and Expanding Twist Tori on Moduli Spaces of Abelian Differentials
2025/07/13 by Jon Chaika, Chaika, Jon, Osama Khalil +1
Mathematics · Computer Science · #advanced mathematical theories #Mathematical Dynamics and Fractals #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2507.09775
Abstract
Let (M,ω) be a translation surface such that every leaf of its horizontal foliation is either closed, or joins two zeros of ω. Then, M decomposes as a union of horizontal Euclidean cylinders. The twist torus of (M,ω), denoted \mathbbT(ω), consists of all translation surfaces obtained from (M,ω) by applying the horocycle flow independently to each of these cylinders. Let gt be the Teichmüller geodesic flow. We study the distribution of the expanding tori gt⋅ \mathbbT(ω) on moduli spaces of translation surfaces in cases where (M,ω) is a Veech surface. We provide sufficient criteria for these tori to become dense within the conjectured limiting locus M :=SL2(ℝ)⋅ \mathbbT(ω) as t→ ∞. We also provide criteria guaranteeing a uniform lower bound on the mass a given open set U\subsetM must receive with respect to any weak-∗ limit of the uniform measures on gt⋅ \mathbbT(ω) as t→∞. In particular, all such limits must be fully supported in M in such cases. Finally, we exhibit infinite families of well-known examples of Veech surfaces satisfying each of these results. A key feature of our results in comparison to previous work is that they do not require passage to subsequences.
Citations
- On Mirzakhani's twist torus conjecture
- Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow
- Horospherical dynamics in invariant subvarieties
- Effective equidistribution for some one parameter unipotent flows
- On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials
- Generalizations of the Eierlegende-Wollmilchsau
- Tremors and horocycle dynamics on the moduli space of translation surfaces
- Effective counting of simple closed geodesics on hyperbolic surfaces
- A tour through Mirzakhani's work on moduli spaces of Riemann surfaces
- Entropy, Lyapunov exponents, and rigidity of group actions
- Explicit bounds for separation between Oseledets subspaces
- Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group
- The algebraic hull of the Kontsevich-Zorich cocycle
- Horocycle dynamics: new invariants and eigenform loci in the stratum\n H(1,1)
- Isolation, equidistribution, and orbit closures for the SL(2,R) action\n on Moduli space
- Invariant and stationary measures for the SL(2,R) action on Moduli space
- Grid graphs and lattice surfaces
- An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum
- Teichmueller curves, triangle groups, and Lyapunov exponents
- An extraordinary origami curve
- Unipotent flows on the space of branched covers of Veech surfaces
- Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
- Billiards in rectangles with barriers
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