Markov random walks on homogeneous spaces and Diophantine approximation on fractals

Prohaska, Roland; Sert, Cagri

doi:10.1090/tran/8181

Cited by 5 publications

(23 citation statements)

References 25 publications

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“…If Γ µ has Zariski dense image in Ad(H), then it is proved in [8] that the orbit closure Γ + µ x is a homogeneous subspace of X inside which the random walk equidistributes. Our next result is a generalization of this and other rigidity results for the random trajectory of points proved in [8,75,89].…”

Section: Orbit Closures and Equidistributionsupporting

confidence: 57%

“…For unipotent random walks, recurrence is not always guaranteed [17, §10.2.1]. On the other hand, in the particular case of measures on parabolic subgroups, slightly weaker expansion properties were first used in the work of Simmons-Weiss [89] and subsequently in [75] to prove measure rigidity and equidistribution results in a setting corresponding to the case H = G in our framework. See also Remark 1.3. We next introduce the terminology necessary to state our main results.…”

Section: Introductionmentioning

confidence: 99%

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Expanding measures: Random walks and rigidity on homogeneous spaces

Prohaska¹,

Sert²,

Shi³

2021

Preprint

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Let G be a real Lie group, Λ < G a lattice and H G a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures µ on H and, applying recent work of Eskin-Lindenstrauss, prove that µ-stationary probability measures on G/Λ are homogeneous. Transferring a construction by Benoist-Quint and drawing on ideas of Eskin-Mirzakhani-Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on G/Λ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in G/Λ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons-Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a non-conformal and weighted setting. Contents1. Introduction 1 2. H-expansion: Definition and basic properties 11 3. Examples of H-expanding measures 14 4. Measure rigidity 22 5. Countability of homogeneous subspaces 28 6. Height functions with contraction properties 31 7. Recurrence, equidistribution, topology of homogeneous measures 41 8. Birkhoff genericity 45 9. Connections to Diophantine approximation on fractals 50 Appendix A. Epimorphic subgroups and subalgebras 55 References 57

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Section: Orbit Closures and Equidistributionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

Expanding measures: Random walks and rigidity on homogeneous spaces

Prohaska¹,

Sert²,

Shi³

2021

Preprint

Self Cite

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“…The following is a consequence of the above proof regarding positivity of the constant Λ based on the Markov property (see e.g. [35,59] for similar uses of this idea in close contexts). The hypotheses of this proposition are a little awkward but are satisfied in many cases: • (Furstenberg [32]) H = SL d (R) and the ρ(Γ) is a strongly irreducible and nonrelatively compact, • (Guivarc'h [44]) the image of Γ is non-amenable and has at most exponential | • |growth (see more precisely Kaimanovich-Kapovich-Schupp [47, Proposition 2.5]), • (Maher-Tiozzo [53]) the group Γ acts non-elementarily on a geodesic Gromov-hyperbolic space.…”

Section: Law Of Large Numbers For Subadditive Functionsmentioning

confidence: 92%

“…Now fix an edge (v 0 , v 1 ) in the maximal component B and consider the induced (renewal) measure ν on Γ obtained by return times the vertices v 0 and v 1 consecutively (see e.g. [59,Definition 3.4]). Then, by the Markov property, the induced law on Γ N of this Markovian random walk along the return times to (v 0 , v 1 ) is the Bernoulli law ν N on Γ N (see e.g.…”

Section: Law Of Large Numbers For Subadditive Functionsmentioning

confidence: 99%

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Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Cantrell¹,

Sert²

2022

Preprint

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Given a Gromov-hyperbolic group G endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of G. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry-Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates provide a positive answer to a question of Kaimanovich-Kapovich-Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson-Sullivan measures on the boundary of the group.Contents 42 9. On a question of Kaimanovich-Kapovich-Schupp 47 References 49

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Stationary probability measures on projective spaces 1: block-Lyapunov dominated systems

Aoun

Sert

2023

Math. Ann.

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Given a finite-dimensional real vector space V, a probability measure $$\mu $$ μ on $${\text {PGL}}(V)$$ PGL ( V ) and a $$\mu $$ μ -invariant subspace W, under a block-Lyapunov contraction assumption, we prove existence and uniqueness of lifts to $$P(V){\setminus } P(W)$$ P ( V ) \ P ( W ) of stationary probability measures on the quotient P(V/W). In the other direction, i.e. block-Lyapunov expansion, we prove that stationary measures on P(V/W) have lifts if any only if the group generated by the support of $$\mu $$ μ stabilizes a subspace $$W'$$ W ′ not contained in W and exhibiting a faster growth than on $$W \cap W'$$ W ∩ W ′ . These refine the description of stationary probability measures on projective spaces as given by Furstenberg, Kifer and Hennion, and under the same assumptions, extend corresponding results by Aoun, Benoist, Bruère, Guivarc’h, and others.

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Markov random walks on homogeneous spaces and Diophantine approximation on fractals

Cited by 5 publications

References 25 publications

Expanding measures: Random walks and rigidity on homogeneous spaces

Expanding measures: Random walks and rigidity on homogeneous spaces

Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Stationary probability measures on projective spaces 1: block-Lyapunov dominated systems

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