Date:

Tue, 10/11/201514:00-15:00

Location:

Manchester building, Hebrew University of Jerusalem, (Room 209)

Title: Self-affine measures with equal Hausdorff and Lyapunov dimensions

Abstract:

Let μ be the stationary measure on ℝd which corresponds to a self-affine iterated function system Φ and a probability vector p. Denote by A⊂Gl(d,ℝ) the linear parts of Φ. Assuming the members of A contract by more than 12, it follows from a result by Jordan, Pollicott and Simon, that if the translations of Φ are drawn according to the Lebesgue measure, then dimHμ=min{D,d} almost surely. Here D is the Lyapunov dimension, which is an explicit constant defined in terms of A and p.

I will present a new result which provides general conditions for μ to be exact dimensional with dimμ=D, whenever Φ satisfies strong separation. These conditions involve a lower bound on the dimension of the Furstenberg measure corresponding to A and p. The proof uses random matrix theory, and upper bounds on the dimension of exceptional sets of sections and projections of measures.

By using this I will present new explicit examples of self-affine measures whose dimension can be computed. These examples rely on new results by Hochman and Solomyak, Bourgain, and Benoist and Quint, regarding the Furstenberg measure.

Abstract:

Let μ be the stationary measure on ℝd which corresponds to a self-affine iterated function system Φ and a probability vector p. Denote by A⊂Gl(d,ℝ) the linear parts of Φ. Assuming the members of A contract by more than 12, it follows from a result by Jordan, Pollicott and Simon, that if the translations of Φ are drawn according to the Lebesgue measure, then dimHμ=min{D,d} almost surely. Here D is the Lyapunov dimension, which is an explicit constant defined in terms of A and p.

I will present a new result which provides general conditions for μ to be exact dimensional with dimμ=D, whenever Φ satisfies strong separation. These conditions involve a lower bound on the dimension of the Furstenberg measure corresponding to A and p. The proof uses random matrix theory, and upper bounds on the dimension of exceptional sets of sections and projections of measures.

By using this I will present new explicit examples of self-affine measures whose dimension can be computed. These examples rely on new results by Hochman and Solomyak, Bourgain, and Benoist and Quint, regarding the Furstenberg measure.