Invariant and stationary measures for the action on Moduli space

Eskin, Alex; Mirzakhani, Maryam

doi:10.1007/s10240-018-0099-2

Cited by 134 publications

(150 citation statements)

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“…As this introduction is quite long, we do not dwell on the comparison between the results here and similar classification results of stationary measures on homogeneous spaces. Nevertheless, this comparison is essential if one wishes to shape a reasonable set of expectations regarding stationary measures and closed invariant sets of semisimple groups acting on spaces such as X. Bourgain-Furman-Lindenstrauss-Mozes [BFLM11] and [EM13]. The recent preprint [EL18] is also highly relevant and as it provides an alternative proof for some of the results of Benoist and Quint, potentially, the techniques introduced there can simplify the analysis carried out in this paper.…”

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confidence: 99%

Dynamics on the space of 2-lattices in 3-space

Sargent

Shapira

2019

Geom. Funct. Anal.

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We study the dynamics of SL3(R) and its subgroups on the homogeneous space X consisting of homothety classes of rank-2 discrete subgroups of R 3 . We focus on the case where the acting group is Zariski dense in either SL3(R) or SO(2, 1)(R). Using techniques of Benoist and Quint we prove that for a compactly supported probability measure µ on SL3(R) whose support generates a group which is Zariski dense in SL3(R), there exists a unique µ-stationary probability measure on X. When the Zariski closure is SO(2, 1)(R) we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in X. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.proximally on R 3 , then P µ (Gr 2 (R 3 )) consists of a single element. We will refer to it as the Furstenberg measure of µ on Gr 2 (R 3 ) and denote it byν Gr 2 (R 3 ) .Remark 1.4. It will be important for us that the Furstenberg measure is atom free. This is ensured by the strong irreducibility assumption, since if there was an atom ofν Gr 2 (R 3 ) then the set of atoms with maximal weight is a finite Γ µ -invariant set.We fix {e 1 , e 2 , e 3 } the standard orthonormal basis of unit vectors in R 3 . For v ∈ R 3 and 1 ≤ i ≤ 3 we will write v i := v, e i . As before we consider the indefinite quadratic form Q :(1.1) Let H µ denote the Zariski closure of Γ µ . In what follows we will concentrate on two cases which will be referred to as Case I and Case II as follows:

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confidence: 99%

Dynamics on the space of 2-lattices in 3-space

Sargent

Shapira

2019

Geom. Funct. Anal.

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“…The essential new ingredient is the fact that any (L, µ) possesses a spectral gap (see §3 for the definition). This was proved by Avila,Gouëzel and Yoccoz [AGY06] for the case of strata, and by Avila and Gouëzel [AG13] for general loci (again, in an abstract framework, as [AG13] also preceded [EMi15]). Using the spectral gap it is possible to obtain an effective estimate of the difference |π L (Σ t )f (x) −´L f dµ|, in case f is a K-smooth function and t is large enough (depending on x and f ).…”

Section: Introductionmentioning

confidence: 81%

Effective counting on translation surfaces

Nevo

Rühr

Weiss

2020

Advances in Mathematics

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“…The only fixed points of 0 are corners and centers of sides of parallelograms given by parts 00, 01, 10 and 11 of , there are 12. Therefore, ( 0 , 0 ) ∈  ( 0 1 , 2 2 4 , 4 3 , (−1) 12 ) and the genus of 0 is 0 = 3 + 4 − 2. The involution 1 fixes all singular points of degree 0 and 2 and does not fix singularities of degree 1.…”

Section: F I G U R E 3 Surface ( ) = (λ )mentioning

confidence: 99%