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- (with Chris Connell) Harmonicity of quasiconformal measures and
Poisson boundaries of hyperbolic spaces Submited to GAFA
Abstract:We consider a group $\group$ of isometries acting on a (not
necessarily geodesic) $\delta$-hyperbolic space $X$ and possessing a
radial limit set of full measure within its limit set. For any
continuous $\alpha$-quasiconformal measure $\nu$ supported on the
limit set, we produce a stationary measure $\mu$ on $\group$.
Moreover the limit set together with $\nu$ forms a $\mu$-boundary
and $\nu$ is harmonic with respect to the random walk induced by
$\mu$. In the case when $X$ is a CAT$(-\kappa)$ space and $\group$
acts cocompactly, for instance, we show that $\mu$ has finite first
moment. This implies that $(\pa X, \nu)$ is the unique Poisson
boundary for $\mu$. In the course of the proofs, we establish
sufficient conditions for a set of continuous functions to form a
positive basis, either in the $L^1$ or $L^\infty$ norm, for the
space of uniformly positive lower-semicontinuous functions on a
general metric measure space.
- A Note on stationarity of spherical measures accepted to Israel Journal of mathematics
Abstract: In this short note we show that any smooth probability measure on
the boundary $B(G)$ of a semisimple Lie group $G$ is stationary for
some probability measure on a lattice $\Ga$. This generalizes a
result of Furstenberg about Poisson boundaries for semisimple Lie
groups
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- Semigroup actions on T^n, to appear in Geometrae Dedicata
Abstract: Let S be a semigroup of non-singular nxn - matrices with
integer coefficients. There is a natural action of S on the n-dimensional
torus T^n. We give a complete characterization of S that satisfies the
following property(ID): The only infinite closed S-invariant subset of T^n
is T^n itself. We also prove that a semigroup of affine transformations,
whose linear parts satisfy property ID, also satisfies property ID.
This generalizes the results of Furstenberg for the circle and Berend for commutative semigroups. We also describe orbits for semigroups that are
not virtually cyclic and act strongly irreducibly on T^n.
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- Orbits of Zariski dense semigroups of SL(n, Z) ,
to appear in Ergodic Theory and Dyn. Sys.
Abstract: Let S$ be a semigroup of continuous automorphisms of
n-dimensional torus T^n. We prove that if S is Zariski dense in
SL(n, Z), then every closed S-invariant set of T^n is either finite
or T^n. This implies that the action of SL(n,Z) on T^n is taut. Similar
results for commutative nonlacunary semigroups were obtained by Furstenberg
and Berend.
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- (with Igor Pak) Percolation on Grigorchuk groups,
Comm. Algebra, vol. 29 (2001), 661-671.
Let p_c(G) be the critical probability of the site percolation
on the Cayley graph of group G. Benjamini and Schramm
conjectured that p_c<1, given the group is infinite and not a finite
extension of Z. The conjecture was proved earlier for groups of
polynomial and exponential growth and remains open for groups of
intermediate growth. In this note we prove the conjecture for a
special class of Grigorchuk groups, which contains all known
examples of groups of intermediate growth.
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- (with Igor Pak) On growth of Grigorchuk groups,
International Journal of Algebra and Computation,
vol. 11 (2001), 1-17.
We present an analytic technique for estimating the
growth for groups of intermediate growth.
We apply our technique to Grigorchuk groups, which
are the only known examples of such groups. Our estimates
generalize and improve various bounds by Grigorchuk, Bartholdi and
others.
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- (with Igor Pak) On tilings by ribbon
tetrominoes, J. Comb. Th., Ser. A, vol. 88, 1999, 188-193.
We resolve a problem posed in the previous paper by extending
the set of regions to all simply connected regions in the case
n=4. Conway-Lagarias type technique is employed.
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