Dynamical Systems seminar: Dimension of SL(2,C) measures for Furstenberg Products

Speaker: Haojie Ren (Technion – Israel Institute of Technology)

Abstract: Let (\mathcal{A} = {A_i}_{i \in \Lambda}) be a finite subset of (\mathrm{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 $\dim\mu=\min\left{2, H(p)/(2\chi(\theta))\right}$. Here, $H(p)$ is the entropy of $p$, 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.

Time: 16:00, Thursday, December 19, 2024

Location: Room 1210, SIMIS, Block A, No. 657 Songhu Road, Yangpu District, Shanghai

Zoom Meeting No.: 894 9814 1985 (Passcode: 675743)

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