Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of CP(k)

François Berteloot, Fabrizio Bianchi, Christophe Dupont
2018 Annales Scientifiques de l'Ecole Normale Supérieure  
We introduce a notion of stability for equilibrium measures in holomorphic families of endomorphisms of P k and prove that it is equivalent to the stability of repelling cycles and equivalent to the existence of some measurable holomorphic motion of Julia sets which we call equilibrium lamination. We characterize the corresponding bifurcations by the strict subharmonicity of the sum of Lyapunov exponents or the instability of critical dynamics and analyze how repelling cycles may bifurcate. Our
more » ... methods deeply exploit the properties of Lyapunov exponents and are based on ergodic and pluripotential theory. Theorem 1.1 leads us to define the bifurcation current of a holomorphic family of endomorphisms of P k as the closed positive current dd c λ L, and the bifurcation locus as the support of this current. The family is stable if its bifurcation locus is empty, stability is clearly a local notion. This is coherent with the one-dimensional definition of the bifurcation current, due to DeMarco [dM]. We stress that Theorem 1.1 stays partially true for general families (see Theorem 1.6). Let us now specify the definitions. A central notion is the set J := γ : M → P k : γ is holomorphic and γ(λ) ∈ J λ for every λ ∈ M . The graph {(λ, γ(λ)) λ ∈ M } of any element γ ∈ J is denoted Γ γ . We endow J with the topology of local uniform convergence and note that f induces a continuous self-map F : J → J given by F · γ(λ) := f λ (γ(λ)). DYNAMICAL STABILITY AND LYAPUNOV EXPONENTS FOR HOLOMORPHIC ENDOMORPHISMS OF P k 3 Definition 1.2. For every λ ∈ M , a repelling J-cycle of f λ is a repelling cycle which belongs to J λ . We say that these cycles move holomorphically over M if, for every period n, there exists a finite subset {ρ n,j , 1 ≤ j ≤ N n } of J such that {ρ n,j (λ), 1 ≤ j ≤ N n } is precisely the set of n periodic repelling J-cycles of f λ for every λ ∈ M . The holomorphic motion of repelling J-cycles over M also means that for every repelling periodic point z 0 ∈ J λ0 of f λ0 there exists γ ∈ J such that γ(λ 0 ) = z 0 and γ(λ) is a periodic repelling point of f λ for every λ ∈ M . Our notions of equilibrium webs and laminations are as follows. Definition 1.3. An equilibrium web is a probability measure M on J such that (1) M is F -invariant and its support is a compact subset of J , (2) for every λ ∈ M the probability measure M λ := J δ γ(λ) dM(γ) is equal to µ λ . This notion is related to Dinh's theory of woven currents and somehow means that the measures (µ λ ) λ∈M are holomorphically glued together. In this article we shall also say that the (µ λ ) λ∈M move holomorphically when such a web exists. Definition 1.4. An equilibrium lamination is a relatively compact subset (3) Γ γ does not meet the grand orbit of the critical set of f for every γ ∈ L, (4) the map F : L → L is d k to 1. The existence of an equilibrium lamination corresponds to the property of structural stability for sets of full µ λ -measure. It is rather easy to show that the existence of an equilibrium lamination implies that of an equilibrium web. The converse is much more delicate, equilibrium laminations will be extracted from the support of equilibrium webs by using ergodic theory for the dynamical system (J , F , M). Equilibrium webs 2.1. Sufficient conditions for the existence of an equilibrium web. Let us consider the set O M, P k of holomorphic maps from M to P k , endowed with the metric space topology of local uniform convergence, and the closed subspace J := γ ∈ O M, P k : γ(λ) ∈ J λ for every λ ∈ M . For any probability measure M on O M, P k and every λ ∈ M we define the measure This is a probability measure on P k which is actually equal to p λ⋆ M, where the mapping p λ : O M, P k → P k is given by p λ (γ) := γ(λ). We shall sometimes say that the measures µ λ move holomorphically over M when f admits an equilibrium strucural web. Equilibrium webs will be obtained as limits of discrete measures on O M, P k . To this purpose we shall use the following simple tool which somehow plays the role of the classical λ-lemma. Lemma 2.1. Let f : M × P k → M × P k be a holomorphic family of endomorphisms of P k . Let (M n ) n≥1 be a sequence of Borel probability measures on O M, P k such that: 1) lim n (M n ) λ = µ λ for every λ ∈ M ,
doi:10.24033/asens.2355 fatcat:v74oj5cvdjfobohgznx47kdomy