Yuri Lima

An Abramov formula for stationary spaces of discrete groups

Let (G, μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ-random 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 induced 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 Furstenberg-Poisson 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.

A combinatorial proof of Marstrand’s theorem for products of regular Cantor sets

In 1954 Marstrand proved that if K is a subset of R^2 with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^{1+a}, a>0, for which the sum of their Hausdorff dimension is greater than 1.

Yet another proof of Marstrand’s theorem

In a paper from 1954 Marstrand proved that if K ⊂ ℝ2 is a Borel set with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem, extending the techniques developed in our previous paper [9].

Graph-based Pólya’s urn: Completion of the linear case

Given a finite connected graph

d-actions with prescribed topological and ergodic properties

G

Annihilation and coalescence on binary trees

, place a bin at each vertex. Two bins are called a pair if they share an edge of

Law of large numbers for certain cylinder flows

G

A Marstrand Theorem for Subsets of Integers

. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. Previous works proved that when

Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems

G

Symbolic dynamics for one dimensional maps with non-uniform expansion

is not balanced bipartite, the proportion of balls in the bins converges to a point