Abstract: Let \mathcal{A} = \{A_i\}_{i \in \Lambda} be a finite subset of SL(2,\mathbb{C}), and let p = (p_i)_{i \in \Lambda} be a probability vector with positive entries. Set \theta := \sum_{i \in \Lambda} p_i \delta_{A_i}, and denote by \mathbf{S}_{\mathcal{A}} the semigroup generated by \mathcal{A}. Suppose that \mathbf{S}_{\mathcal{A}} is strongly irreducible and proximal, and let \mu be the Furstenberg measure on \mathbb{CP}^1 associated to \theta.
We are working towards establishing the following result: If \mathcal{A} is exponentially separated and no generalized circle C\subset \mathbb{C} is invariant under the action of \mathbf{S}_{\mathcal{A}} via Möbius transformations, then \dim\mu=\min\left\{2, h_{RW}(\theta)/(2\chi(\theta))\right\}. Here, h_{RW}(\theta) is the random walk entropy of a discrete measure \theta, and \chi(\theta) is the Lyapunov exponent associated to \theta.
Our approach relies on methods from additive combinatorics and involves an analysis of orthogonal projections of \mu (viewed as a measure on \mathbb{R}^2). This is joint work with Ariel Rapaport.
