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Isolation, equidistribution, and orbit closures for the SL2(R) action on moduli space
"... Abstract We prove results about orbit closures and equidistribution for the SL(2, R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of the first two authors and ..."
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Cited by 19 (2 self)
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Abstract We prove results about orbit closures and equidistribution for the SL(2, R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of the first two authors and a certain isolation property of closed SL(2, R) invariant manifolds developed in this paper.
Random conformal dynamical systems
, 2006
"... Abstract. We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversaly conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudogroup. We prove that either there exists a measure invar ..."
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Cited by 18 (6 self)
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Abstract. We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversaly conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudogroup. We prove that either there exists a measure invariant under all the elements of the group (or the pseudogroup), or almost surely a long composition of maps contracts exponentially a ball. We deduce some results about the unique ergodicity. 1.
Semisimplicity and rigidity of the KontsevichZorich cocycle ArXiv eprints
, 2013
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FURSTENBERG ENTROPY REALIZATIONS FOR VIRTUALLY FREE GROUPS AND LAMPLIGHTER GROUPS
"... Abstract. Let (G, µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G, µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have prop ..."
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Cited by 7 (6 self)
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Abstract. Let (G, µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G, µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have property (T). For these groups, we construct stationary actions with arbitrarily small, positive entropy. This construction involves building and lifting spaces of lamplighter groups. For some classical lamplighters, these spaces realize a dense set of entropies
An abramov formula for stationary spaces of discrete groups, arXiv preprint arXiv:1204.5414
, 2012
"... Abstract. Let (G, µ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A µrandom walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the rand ..."
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Cited by 6 (1 self)
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Abstract. Let (G, µ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A µrandom walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G, µ)stationary space, with respect to the action of (Γ, θ), is equal to the Furstenberg entropy with respect to the action of (G, µ), times the index of Γ in G. The index is shown to be equal to the expected return time to Γ. As a corollary, when applied to the FurstenbergPoisson boundary of (G, µ), we prove that the random walk entropy of (Γ, θ) is equal to the random walk entropy of (G, µ), times the index of Γ in G.