Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus

By Jean Bourgain, A. Furman, Elon Lindenstrauss, Shahar Mozes. 2011

121 citations recorded by the source corpus.

Let Γ be a semigroup of d × d nonsingular integer matrices, and consider the action of Γ on the torus T. We assume throughout that the action is strongly irreducible: there is no subtorus invariant under a finite index subsemigroup of Γ. The strong irreducibility assumption in particular implies that Γ acts ergodically on T (equipped with the Lebesgue measure m). Therefore the Γ-orbit of Lebesgue almost every x ∈ T is dense and in an appropriate sense even becomes equidistributed. However, when Γ is cyclic, there is a set of full Hausdorff dimension of exceptional points x for which Γ.x fails to be dense. When Γ is bigger, the distribution of individual Γ-orbits can be expected to be much more restrictive. An important result in this direction is due to Furstenberg, who showed for d = 1 (in which case Γ 1 an abelian semigroup never satisfies condition (Γ-1); indeed, the group generated by a semigroup satisfying (Γ-1) is nonamenable. Assumption (Γ-2) is a technical condition which is in particular satisfied when Γ is a Zariski dense semigroup of SLd(Z) [16].

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5d3ecdda4d0a430911975a641b62fed426965929

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Citations in the corpus

Stationary measures and invariant subsets of homogeneous spaces (II) (2013, background): “4 was previously only known under the additional assumption that Γμ acts irreducibly on R: this is due to [12] in the proximal case and to [4] in general.”
Measure rigidity for random dynamics on surfaces and related skew products (2015, result): “Rn then every ν-stationary probability measure on Tn is either ﬁnitely supported or is Haar; in particular every ν-stationary probability measure is SL(n,Z)-invariant. Similar results was obtained in [BFLM] through completely different methods. In [BQ1] the authors obtain similar stiffness results f”
Random Walks on Reductive Groups (2016, methodology): “This theory has had recently nice applications to the study of subgroups of Lie groups (as in [58], [26] or [27, Section 12]).”
Random walks on random coset spaces with applications to Furstenberg entropy (2010, background): “There is significant interest in understanding the structure of stationary actions and their connections with random walks [13–16], rigidity theory [24–28] and classification of invariant measures [5– 8].”
Central limit theorem for linear groups (2016, background): “5 is similar to our proof in [7] of Guivarc’h theorem, which is inspired by [13].”
Stationary measures and invariant subsets of homogeneous spaces (III) (2013, background): “There exist a lower semicontinuous function u : X → [0,∞] which is bounded on K and 0 ≤ a < 1 and C > 0 with Pu ≤ au + C, such that, for any M > 0, the set XM = u −1([0,M ]) is compact.”