Seminars: K. Kh. Rakhimov, Semicontiunity of small Lyapunov exponent for holomorphic endomorphisms of $\mathbb{P}^k$

SEMINARS


Seminar on Modern Problems of Complex Analysis (Sadullaev Seminar) January 21, 2021 12:00, Tashkent, National University of Uzbekistan, Room A304 (Department of Mathematics)

Semicontiunity of small Lyapunov exponent for holomorphic endomorphisms of $\mathbb{P}^k$ K. Kh. Rakhimov^ab ^a Centre National de la Recherche Scientifique, Paris ^b Université de Lille
Abstract: After a short introduction to dynamics of holomorphic endomorphisms of $\mathbb{P}^k$ we talk about dynamical stability of a holomorphic family of endomorphisms $\{f_\lambda\},$ $\lambda\in M$ (a complex manifold). We show that, if $\{f_\lambda\}$ is a stable family, and for a fixed $\lambda_0$ if $\nu_{\lambda_0}$ is an invariant probability measure on $J_{\lambda_0}$, the small Julia set for $f_{\lambda_0}$, then there exists invariant measures $\nu_\lambda$ on $J_{\lambda}$ which vary continuously on $\lambda$. In the second part of the talk we study Lyapunov exponents of $\nu_\lambda$ and we show that the small Lyapunov exponent is lower semicontinuous. Website: https://us02web.zoom.us/j/84574793587