Balázs Bárány

On the Ledrappier–Young formula for self-affine measures

Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier-Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some applications, namely, we give another proof for Hueter-Lalleys theorem and we consider self-affine measures and sets generated by lower triangular matrices.

On the dimension of the graph of the classical Weierstrass function

Abstract This paper examines dimension of the graph of the famous Weierstrass non-differentiable function W λ , b ( x ) = ∑ n = 0 ∞ λ n cos ⁡ ( 2 π b n x ) for an integer b ≥ 2 and 1 / b λ 1 . We prove that for every b there exists (explicitly given) λ b ∈ ( 1 / b , 1 ) such that the Hausdorff dimension of the graph of W λ , b is equal to D = 2 + log ⁡ λ log ⁡ b for every λ ∈ ( λ b , 1 ) . We also show that the dimension is equal to D for almost every λ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function f ( x ) = ∑ n = 0 ∞ λ n ϕ ( b n x ) for an integer b ≥ 2 and 1 / b λ 1 is equal to D for a typical Z -periodic C 3 function ϕ.

Slicing the Sierpinski gasket

We investigate the dimension of intersections of the Sierpinski gasket with lines. Our first main result describes a countable, dense set of angles that are exceptional for Marstrands theorem. We then provide a multifractal analysis for the set of points in the projection for which the associated slice has a prescribed dimension.

Dimension of slices of Sierpiński-like carpets

We investigate the dimension of intersections of the Sierpinski-like carpets with lines. We show a sufficient condition that for a fixed rational slope the dimension of almost every intersection w.r.t the natural measure is strictly greater than s− 1, and almost every intersection w.r.t the Lebesgue measure is strictly less than s − 1, where s is the Hausdorff dimension of the carpet. Moreover, we give partial multifractal spectra for the Hausdorff and packing dimension of slices. Mathematics Subject Classification (2010). 28A80; 28A78.

Shrinking targets on Bedford–McMullen carpets

We describe the shrinking target set for the Bedford-McMullen carpets, with targets being either cylinders or geometric balls.