Speaker: Virgile Tapiero (Institute for Mathematics and Fundamental Physics)
Time: 10:00-11:00, Nov 13, 2025
Venue: room 1124, Material Science Research Building (Section C)
Let f be an endomorphism of the complex projective plane P^2 of degree d>1. There exists a unique probability measure mu of maximal entropy (equal to log d^2) for f. It is ergodic and admits Lyapunov exponents L_1 \geq L_2. A fundamental result due to J.-Y. Briend and J. Duval (1999) states that the Lyapunov exponents of mu are bounded below by a constant depending only on the degree d of f, more precisely, they are bounded below by (log d)/2. The case of equality L_1 = L_2 = (log d)/2 was characterized by several authors, who showed that this happens if and only if mu is absolutely continuous with respect to the Lebesgue measure.
mu is also a Monge-Ampère mass: mu = T \wedge T the autointersection of the Green current T of f. R. Dujardin proved (2012) that mu << Trace(T) implies L_2 = (log d)/2. He also addressed the question of the reverse property. In this talk, we will focus on the case L_1 > L_2 = (log d)/2. I will explain how to prove that L_2 = (log d)/2 implies mu << Trace(T), answering Dujardin’s question. The techniques involved are based on pluripotential theory, ergodic theory, and the use of normal forms for the dynamics
