The stable subset of a univalent self-map

Theorem 1.1 (A.–Bracci). Let X be Kobayashi hyperbolic and cocompact and let f : X → X be a univalent self-map. Then there exists an essentially unique model (Ω, h, ψ). More- over there exists a holomorphic retract Z of X, a surjective holomorphic submersion r: Ω→Z, and an automorphism τ : Z →Z with divergence rate c(τ) = c(f) = lim m→∞ sm(x) m , x ∈X, such that (Z, r ◦h, τ) is a semi-model for f. Moreover (Z, r ◦h, τ) satisfies the following universal property. If (Q, ℓ, ϕ) is another semi-model

Theorem 1.2 (Poggi-Corradini ). Let f : D →D be a holomorphic self-map, and let ζ ∈∂D be a boundary repelling fixed point with dilation 1 < λ < ∞. Then there exists a pre-model (H, t, ϑ) for f such that ϑ is the hyperbolic automorphism of H given by ϑ(z) = 1 λz, and the mapping t has non-tangential limit ζ at ∞. Poggi-Corradini also proves that the pre-model (H, t, ϑ) is essentially unique. The same strategy was used by Ostapyuk to generalize this result to the unit ball Bq ⊂Cq. She proved in [18

Theorem 1.3 (Ostapyuk). Let f : Bq →Bq a holomorphic self-map, and let ζ be a bound- ary repelling fixed point with dilation 1 < λ < ∞, which is isolated from other boundary repelling fixed points with dilation less or equal than λ. Then there exists a pre-model (H, t, ϑ) for f such that ϑ is the hyperbolic automorphism of H given by ϑ(z) = 1 λz, and the mapping t has non-tangential limit ζ at ∞.

Theorem 1.3 gives dynamical information on f only on the one-dimensional image t(H). This remark motivated the following open question [18, Question 6.2.1]. Recall that the

Theorem 1.5. Let X be Kobayashi hyperbolic and cocompact and let f : X →X be a univalent self-map. Then the stable subset S(f), if non-empty, is the disjoint union of completely invariant complex submanifolds Λ = G j∈J Σj, such that for all j ∈J there exists a holomorphic retract Zj and an injective holomorphic immersion gj : Zj →X satisfying gj(Zj) = Σj. For all j ∈J, there exists an automorphism τj : Zj →Zj with divergence rate c(τj) = lim m→∞ σm(x) m ,

Theorem 1.6. Let f : Bq →Bq be a univalent self-map and let ζ ∈∂Bq be a boundary repelling fixed point with dilation 1 < λ < ∞. Then the stable subset S(ζ) at ζ, if non- empty, is the disjoint union of completely invariant complex submanifolds Λ = G j∈J Σj. Fix j ∈J, let 1 ≤kj ≤q be the dimension of Σj and define µj by µj := lim m→∞e σm(x) m ≥λ, where x ∈Σj. Then µj does not depend on x ∈Σj and there exist an injective holomorphic immersion gj : Hkj →Bq with gj(Hkj) = Σj and

Theorem 1.5. The proof of Theorem 1.5 involves the study of the complex structure of a decreasing intersection. We recall some results for the dual problem, that is the study of the complex structure of a growing union, also called the union problem. Assume that we have a monotonically increasing sequence of domains of a complex manifold Ω: X0 ⊂X1 ⊂X2 ⊂. . . , and assume that Ω= S n≥0 Xn. Assume moreover that every Xj is biholomorphic to a given complex manifold X. One wants to understand the co

Theorem 1.7 (Fornæss–Sibony). If X is Kobayashi hyperbolic and cocompact, then there exists a holomorphic retract Z ⊂X and a surjective holomorphic submersion r: Ω→Z which satisfies the following universal property. If Q is a Kobayashi hyperbolic complex manifold and t: Ω→Q is a holomorphic mapping, then there exists a holomorphic mapping σ: Z →Q such that the following diagram commutes: Ω t / r  Q Z. σ ?⑦ ⑦

Theorem 1.8. Let X be Kobayashi hyperbolic and cocompact. Then the subset Λ := T n≥0 Xn, if non-empty, is the disjoint union of complex submanifolds Λ = G j∈J Σj, such that for all j ∈J there exists a holomorphic retract Zj and an injective holomorphic immersion gj : Zj →X satisfying gj(Zj) = Σj. For all j ∈J, the mapping gj satisfies the following universal property. If Q is a complex manifold and t: Q →X is a holomorphic mapping such that t(Q) ∩Σj ̸= ∅and t(Q) ⊂Λ, then there exists a holomorphi

Lemma 2.3. Let X, Z, Q be complex manifolds and let g: Z →X and f : Q →X be injec- tive holomorphic mappings. Assume that g extends t. Then the following are equivalent: (1) t extends g, (2) g(Z) ⊂t(Q), (3) η: Q →Z is surjective, (4) t and g are equivalent.

Lemma 2.8. Let Q be a complex manifold and let t: Q →X be a holomorphic mapping with t(Q) ⊂Λ. Let x, y ∈t(Q). Then the sequence (kX(f −1 n (x), f −1 n (y))) is bounded, that is x ∼y. Similarly, if x = t(z) and v = dzt(ζ) with ζ ∈TzQ, then the sequence (κX(f −1 n (x), dxf −1 n (v))) is bounded.

Lemma 2.9. We have that g(X) = [x0].

Lemma 2.11. The map α: X →X is a holomorphic retraction.

Lemma 2.13. We have that g(X) = g(Z) and that g|Z : Z →X is an injective holomorphic immersion.

Proposition 2.14. Let Q be a complex manifold and let t: Q →X be a holomorphic mapping such that t(Q) ∩Σ ̸= ∅and t(Q) ∈Λ. Then g: Z →X extends t.

Corollary 2.15. Let Q be a complex manifold and let t: Q →X be an injective holomor- phic mapping such that t(Q) = Σ. Then t: Q →X and g: Z →X are equivalent.

Proposition 2.19. For all x ∈Λ we have that Vx = TxΣ.

Proposition 2.22. Let z, w ∈Σ and let v ∈TzΣ. Then kXn(z, w) →kΣ(z, w), and κXn(z, v) →κΣ(z, v).

Lemma 3.4. If f is injective, then BO(f) = Λ.

Lemma 3.6. If f is injective, then Λ is completely invariant, and the mapping f|Λ : Λ →Λ is bijective.

Lemma 3.10. Let (Z, g, τ) be an injective pre-model for f and let (Q, t, ϑ) be a pre-model for f. Then there exists a morphism ˆη: (Q, t, ϑ) →(Z, g, τ) if and only if g: Z →X extends t: Q →X through the mapping η: Q →Z.

Corollary 3.11. Let (Z, g, τ) and (Q, t, ϑ) be two injective pre-models for f. Then (Q, t, ϑ) and (Z, g, τ) are isomorphic if and only if g and t are equivalent.

Lemma 3.19. Let Σ be a canonical submanifold and let x ∈Σ. Then (1) Σ is invariant if and only if σ1(x) < ∞, (2) Σ is q-periodic if and only if σk(x) < ∞for all k ∈qZ, (3) Σ is wandering if and only if σk(x) = ∞for all k ∈Z, k ̸= 0.

Proposition 3.21. Let Σ be an invariant canonical submanifold, and let g: Z →X be a holomorphic injective mapping such that g(Z) = Σ. Then there exists an automorphism τ

Proposition 3.23. Let Σ ⊂Λ be an invariant canonical submanifold. Let (Z, g, τ) be a canonical pre-model associated with Σ. Let x ∈Σ and let z ∈Z be such that g(z) = x. Then for all m ≥0, σm(x) = kZ(z, τ −m(z)) = kZ(z, τ m(z)).

Proposition 3.23 immediately yields the following corollary.

Corollary 3.25. Let Σ ⊂Λ be an invariant canonical submanifold. Let (Z, g, τ) be a canonical pre-model associated with Σ. Let x ∈Σ. Then c(τ) = lim m→∞ σm(x) m = inf m∈N σm(x) m . The proof of Theorem 1.5 follows easily from the results of this section. We now want to describe the dynamics on an invariant canonical submanifold. We first need to recall some definitions.

Theorem 3.27. Let Y be taut, and let h: Y →Y be a holomorphic self-map. Then the following are equivalent: (1) the sequence (hn) is not compactly divergent, (2) the subset {hn(y)} is relatively compact in Y for all y ∈Y , (3) there exists y ∈Y such that the subset {hn(y)} is relatively compact in Y . The next results show that it is possible to detect the type of an invariant canonical submanifold only by looking at the backward steps at any x ∈Σ. This, together with

Lemma 3.19, shows that every dynamical information concerning Σ is encoded in the sequence (σm(x)).

Proposition 3.28. Let Σ be an invariant canonical submanifold, and let x ∈Σ. Then (1) the type of Σ is elliptic if and only if the sequence (σm(x)) is bounded, (2) the type of Σ is parabolic if and only if the sequence (σm(x)) is unbounded and limm→∞ σm(x) m = 0, (3) the type of Σ is hyperbolic if and only if limm→∞ σm(x) m > 0 and in this case lim m→∞ σm(x) m

Proposition 3.30. Let Σ ⊂Λ be an invariant canonical submanifold. Let (Z, g, τ) be a canonical pre-model associated with Σ. Then c(τ) ≥c(f).

Corollary 3.31. Let f : X →X be a hyperbolic self-map. If Σ is an invariant canonical submanifold, then it is hyperbolic. For a proof of the following result, see, e.g., [1, Theorem 2.1.29].

Theorem 3.32. Let Y be a taut manifold, and let h: Y →Y be a holomorphic self-map such that the sequence (hn) is not compactly divergent. Then there exists a submanifold M of Y called the limit manifold and a holomorphic retraction ρ: Y →M which is a limit point of (hn) such that every holomorphic self-map k: Y →Y which is a limit point of (hn) is of the form k = γ ◦ρ, where γ is an automorphism of M. Moreover h(M) ⊂M and h|M is an automorphism of M. The next result characterizes the invariant c

Proposition 3.33. If f : X →X is elliptic then the limit manifold is an elliptic invariant canonical submanifold. Conversely, if an invariant canonical submanifold Σ ⊂X is elliptic, then f is elliptic and Σ is the limit manifold.

Theorem 4.3. Let f : Bq →Bq be holomorphic. Assume that f admits no fixed points in Bq. Then there exists a boundary regular fixed point p ∈∂Bq with dilation λ ≤1, called the Denjoy–Wolffpoint of f, such that (f n) converges uniformly on compact subsets to the constant map z 7→p. A proof of the following result is given in [1, Theorem 2.4.20] for the more general case of bounded convex domains.

Theorem 4.4. A holomorphic self-map f : Bq →Bq is elliptic if and only if it admits a fixed point z ∈Bq.

Proposition 4.7. Let f : Bq →Bq be univalent, and let ζ ∈∂Bq be a boundary regular fixed point. Then S(ζ), if non-empty, is the disjoint union of invariant canonical submanifolds, which are injectively immersed holomorphic balls Bk with 1 ≤k ≤q.

Proposition 4.11. Let f : Bq →Bq be univalent, and let ζ ∈∂Bq be a boundary repelling fixed point, with dilation 1 < λ < ∞. Let Σ be a canonical submanifold in the stable subset S(ζ), let 1 ≤k ≤q be the dimension of Σ, let x ∈Σ and µ := lim m→∞e σm(x) m ≥λ. Then Σ is hyperbolic, and there exists a (k −1) × (k −1) diagonal unitary matrix U, and an injective holomorphic immersion g: Hk →Bq with g(Hk) = Σ and K- lim z→∞g(z) = ζ, such that  Hk, g, τ : (z, w) 7→

Proposition 4.12. Let f : Bq →Bq be a univalent self-map. Let ζ ∈∂Bq be a boundary repelling fixed point, with dilation 1 < λ < ∞. Let Σ be an invariant canonical submanifold contained in S(ζ), and assume there exists x ∈Σ such that the backward orbit (f −n(x)) is special and restricted. Then µ = λ.