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François Bacher

Publications and source records attributed to François Bacher.

6 recordsLinked to original sources
arXiv2026

Hyperbolic foliated entropy of suspensions

We study the hyperbolic entropies of foliations obtained by suspensions of a representation, in the sense of Dinh, Nguyên and Sibony (topological and measure-theoretic). We establish a link between this type of entropy and an adapted version of an entropy defined by Ghys, Langevin and Walczak for pseudo-groups of homeomorphisms. Such a link has various consequences. Among them, it implies that the hyperbolic entropy of foliations is not invariant by diffeomorphisms, and that a minimal entropy suspension admits an invariant measure. Finally, this allows us to study thoroughly the simple case in which the image of the representation is isomorphic to~$\mathbb{Z}$. In that case, we give the first exact estimate of the hyperbolic entropy, and prove a Brin--Katok type theorem and a variational principle, relying strongly on the standard ones for the entropy of maps.

math.DS↗
arXiv2025

Hénon maps with biholomorphic Kato surfaces

Let $F$ and $G$ be generalized Hénon maps. We show that the Kato surfaces associated to the germs of $F$ and $G$ near infinity are biholomorphic if and only if $F$ and $G$ are conjugate in $\text{Aut}(\mathbb{C}^2)$. This answers a question raised by Favre in Xénelkis de Hénon's survey. Moreover, we describe all the Kato surfaces associated to the germ of a generalized Hénon map near infinity. We also show that two generalized Hénon maps are conjugate near infinity, if and only if they are affinely conjugate.

math.CV↗
arXiv2024

Hyperbolic entropy for harmonic measures on singular holomorphic foliations

Let $\mathscr{F}=(M,\mathscr{L},E)$ be a Brody-hyperbolic singular holomorphic foliation on a compact complex manifold $M$. Suppose that $\mathscr{F}$ has isolated singularities and that its Poincaré metric is complete. This is the case for a very large class of singularities, namely, non-degenerate and saddle-nodes in dimension $2$. Let $μ$ be an ergodic harmonic measure on $\mathscr{F}$. We show that the upper and lower local hyperbolic entropies of $μ$ are leafwise constant almost everywhere. Moreover, we show that the entropy of $μ$ is at least $2$.

math.DS↗