I plan to talk on the Hausdorff dimension spectrum for Lyapunov exponents, in particular on the function $\alpha\mapsto HD\{x\in K: \lim_{n\to\infty} 1/n \log |(f^n)'(x)|=\alpha\}$, where either $K=J$ is Julia set for a rational map $f$ on the Riemann sphere, or $K$ is the maximal invariant set for a generalized multimodal map on a domain in $R$, provided $f|_K$ is topologically transitive of positive topological entropy. These results have been obtained jointly with Katrin Gelfert and Michal Rams and the real 1-dimensional background comes from a preprint by the speaker and Juan Rivera-Letelier.

If time allows I will sketch a proof that for $f$ rational with a critical point $c\in J$ being the only critical point with its forward trajectory accumulating in $J$, the lower Lyapunov exponent at the critical value, namely $\liminf_{n\to\infty} 1/n \log |(f^n)'(f(c))|$, is non-negative (the result joint with Genadi Levin and Weixiao Shen).