On the ergodic theory of the real Rel foliation

Jon Chaika; Barak Weiss

doi:10.1017/fmp.2024.6

On the ergodic theory of the real Rel foliation

Jon Chaika, Barak Weiss

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

Let H be a stratum of translation surfaces with at least two singularities, let m_H denote the Masur-Veech measure on H, and let Z₀ be a flow on (H, m_H) obtained by integrating a Rel vector field. We prove that Z₀ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces (L, m_L), where L ⊂ H is an orbit-closure for the action of G = SL₂ (R) (i.e., an affine invariant subvariety) and m_L is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of Z₀ with respect to any of the measures m_L is zero.

Original language	American English
Article number	e7
Journal	Forum of Mathematics, Pi
Volume	12
DOIs	https://doi.org/10.1017/fmp.2024.6
State	Published - 2 Apr 2024

All Science Journal Classification (ASJC) codes

Analysis
Algebra and Number Theory
Statistics and Probability
Mathematical Physics
Geometry and Topology
Discrete Mathematics and Combinatorics

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AB - Let H be a stratum of translation surfaces with at least two singularities, let mH denote the Masur-Veech measure on H, and let Z0 be a flow on (H, mH) obtained by integrating a Rel vector field. We prove that Z0 is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces (L, mL), where L ⊂ H is an orbit-closure for the action of G = SL2 (R) (i.e., an affine invariant subvariety) and mL is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of Z0 with respect to any of the measures mL is zero.

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JF - Forum of Mathematics, Pi

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ER -

On the ergodic theory of the real Rel foliation

Abstract

All Science Journal Classification (ASJC) codes

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