Abstract
We establish in this paper an exact formula which links the dimension of the harmonic measure, the asymptotic entropy and the rate of escape for a random walk on a discrete subgroup of the isometry group of a Gromov hyperbolic space. This completes a result obtained by the author in a previous paper, where only an upper bound for the dimension was proved.
Highlights
Let Γ be a discrete subgroup of the isometry group of a Gromov hyperbolic space X and μ a probability measure on Γ
The distribution ν of x∞ is called the harmonic measure associated with the random walk
The aim of this paper is to established the following formula, which links the dimension of ν with respect to da, the asymptotic entropy h(μ) and the rate of escape l(μ) of the random walk: dim ν
Summary
Let Γ be a discrete subgroup of the isometry group of a Gromov hyperbolic space X and μ a probability measure on Γ. The aim of this paper is to established the following formula, which links the dimension of ν with respect to da, the asymptotic entropy h(μ) and the rate of escape l(μ) of the random walk (see Theorem 3.1 for a precise statement): dim ν. This result completes the result obtained in [11], where only the bound from above in (0.1) was proved. In our context of a hyperbolic space, and under the assumptions that μ is symmetric and finitely supported, the following result on the pointwise dimension is established in [2]: lim r→0 log νBa(ξ, log r r).
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