A horospherical ratio ergodic theorem for actions of free groups

Bowen, Lewis; Nevo, Amos

doi:10.4171/ggd/228

Cited by 4 publications

(3 citation statements)

References 20 publications

(26 reference statements)

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“…This phenomenon has been recently observed in certain algebraic settings involving "large" groups acting on infinite measure spaces, see e.g. the introduction of [3]. 2 However, our negative results exclude this as well; in the proofs we construct actions for which the ratios diverge.…”

Section: Introductionsupporting

confidence: 61%

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On the ratio ergodic theorem for group actions

Hochman

2013

Journal of the London Mathematical Society

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We study the ratio ergodic theorem (RET) of Hopf for group actions. Under a certain technical condition, if a sequence of sets {F n } in a group satisfy the RET, then there is a finite set E such that {EF n } satisfies the Besicovitch covering property. Consequently for the abelian group G = ⊕ ∞ n=1 Z there is no sequence F n ⊆ G along which the RET holds, and in many finitely generated groups, including the discrete Heisenberg group and the free group on ≥ 2 generators, there is no (sub)sequence of balls, in the standard generators, along which the RET holds.On the other hand, in groups with polynomial growth (including the Heisenberg group, to which our negative results apply) there always exists a sequence of balls along which the RET holds if convergence is understood as a.e. convergence in density (i.e. omitting a sequence of density zero).

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Section: Introductionsupporting

confidence: 61%

“…On h∈H I n+1,h set ψ ≡ 0 and ϕ ≡ v. There are no conflicts with previous definitions because of (3).…”

Section: Necessitymentioning

confidence: 99%

On the ratio ergodic theorem for group actions

Hochman

2013

Journal of the London Mathematical Society

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“…In the special case of hyperbolic groups, a short and very elegant proof of this theorem, using the method of Calegari and Fujiwara [15], was later given by Pollicott and Sharp [31]. Using the method of amenable equivalence relations, Bowen and Nevo [4], [5], [6], [7] established ergodic theorems for "spherical shells" in Gromov hyperbolic groups. The latter do not require any mixing assumptions.…”

Section: Introductionmentioning

confidence: 99%

Mean convergence of Markovian spherical averages for measure-preserving actions of the free group

Bowen¹,

Bufetov²,

Paris-Romaskevich³

2015

Geom Dedicata

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14 pages. Comments welcome!International audienceMean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space. We endow the set of generators with a generalized Markov chain and establish the mean convergence of resulting spherical averages in this case under mild nondegeneracy assumptions on the stochastic matrix $\Pi$ defining our Markov chain. Equivalently, we establish the triviality of the tail sigma-algebra of the corresponding Markov operator. This convergence was previously known only for symmetric Markov chains, while the conditions ensuring convergence in our paper are inequalities rather than equalities, so mean convergence of spherical averages is established for a much larger class of Markov chains

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