This is a brief survey of selected results obtained using the “transversality method”
developed for studying parametrized families of fractal sets and measures. We mostly focus …

Let {S i} i∈ Λ be a finite contracting affine iterated function system (IFS) on R d. Let (Σ, σ)
denote the two-sided full shift over the alphabet Λ, and let π: Σ→ R d be the coding map …

In this paper, we solve the long standing open problem on exact dimensionality of self-affine
measures on the plane. We show that every self-affine measure on the plane is exact …

In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of
self-affine sets, and the theory of random matrix products, it has often been found useful to …

In this paper, a sponge in R d R^d is the attractor of an iterated function system consisting of
finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable …

Using methods from ergodic theory along with properties of the Furstenberg measure we
obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box …

We completely describe the equilibrium states of a class of potentials over the full shift which
includes Falconer's singular value function for affine iterated function systems with invertible …

We consider one-parameter families of smooth uniformly contractive iterated function
systems {fj λ} on the real line. Given a family of parameter dependent measures {μ λ} on the …

A fundamental problem in the dimension theory of self‐affine sets is the construction of high‐
dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the …

Let (Σ T, σ) be a subshift of finite type with primitive adjacency matrix T, ψ: Σ T→ R a Hölder
continuous potential, and A: Σ T→ GL d (R) a 1-typical, one-step cocycle. For t∈ R consider …