Dynamics of generic endomorphisms of Oka-Stein manifolds
aa r X i v : . [ m a t h . C V ] F e b DYNAMICS OF GENERIC ENDOMORPHISMSOF OKA-STEIN MANIFOLDS
LEANDRO AROSIO AND FINNUR L ´ARUSSON
Abstract.
We study the dynamics of a generic endomorphism f of an Oka-Steinmanifold X . Such manifolds include all connected linear algebraic groups and, moregenerally, all Stein homogeneous spaces of complex Lie groups. We give several de-scriptions of the Fatou set and the Julia set of f . In particular, we show that the Juliaset is the derived set of the set of attracting periodic points of f and that it is alsothe closure of the set of repelling periodic points of f . Among other results, we provethat f is chaotic on the Julia set and that every periodic point of f is hyperbolic.We also give an explicit description of the “Conley decomposition” of X induced by f into chain-recurrence classes and basins of attractors. For X = C , we prove thatevery Fatou component is a disc and that every point in the Fatou set is attracted toan attracting cycle or lies in a dynamically bounded wandering domain (whether suchdomains exist is an open question). Introduction and main theorem
This paper continues a line of research begun with our previous papers [3] and [4]. Here,we investigate the dynamics of a generic endomorphism of an Oka-Stein manifold. The setting is very general and includes all Stein homogeneous spaces of complex Liegroups, in particular all connected linear algebraic groups. Even in the much-studiedcase of complex affine space C n , n ≥
1, most of our results are new. The classes ofOka manifolds and Stein manifolds are fundamental in modern complex analysis andgeometry. Manifolds in their intersection have the right degree of flexibility to possessrich dynamics.The genericity assumption rules out exceptional or pathological behaviour, andstrong, clean theorems emerge. We give several descriptions of the Fatou set and theJulia set of a generic endomorphism f of an Oka-Stein manifold X . In particular, we Date : 1 February 2021.
Key words and phrases.
Dynamics, Stein manifold, Oka manifold, linear algebraic group, Fatou set,Julia set, periodic point, non-wandering point, chain-recurrent point.2020
Mathematics Subject Classification.
Primary 32H50. Secondary 14R10, 32M17, 32Q28, 32Q56,37F80.L. Arosio was supported by SIR grant “NEWHOLITE – New methods in holomorphic itera-tion”, no. RBSI14CFME, and partially supported by the MIUR Excellence Department Projectawarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.F. L´arusson was partially supported by Australian Research Council grant DP150103442. Part of thiswork was done when F. L´arusson visited Rome in February 2020. He thanks the University of RomeTor Vergata for financial support and hospitality. By an Oka-Stein manifold we simply mean a complex manifold that is both Oka and Stein. how that the Julia set is the derived set of the set of attracting periodic points of f and that it is also the closure of the set of repelling periodic points of f . Among otherresults, we prove that f is chaotic on the Julia set and that every periodic point of f ishyperbolic (this is one half of the Kupka-Smale theorem in our setting). We also givean explicit description of the “Conley decomposition” of X induced by f into chain-recurrence classes and basins of attractors. For X = C , we prove that every Fatoucomponent is a disc and that every point in the Fatou set is attracted to an attractingcycle or lies in a dynamically bounded wandering domain (whether such domains existis an open question).Before stating our first result, we need to clarify some definitions and establish somenotation. We denote by End X the monoid of endomorphisms of a complex manifold X ,that is, holomorphic maps X → X , with the compact-open topology, which is separableand defined by a complete metric, so End X is a Polish space. • hyp( f ) is the set of hyperbolic periodic points of an endomorphism f of X . Aperiodic point p of f of period n is hyperbolic if the derivative D p f n of the n th iterate f n at p has no eigenvalue of absolute value 1. • att( f ) is the set of attracting periodic points of f . The periodic point p isattracting if all the eigenvalues of D p f n have absolute value less than 1. • The periodic point p is super-attracting if D p f n = 0 (we consider a super-attracting point to be attracting). • rep( f ) is the set of repelling periodic points of f . The periodic point p is repellingif all the eigenvalues of D p f n have absolute value greater than 1. • sad( f ) is the set of saddle periodic points of f . The periodic point p is a saddlepoint if it is hyperbolic and some of the eigenvalues of D p f n have absolute valueless than 1 and some have absolute value greater than 1. • p ∈ X is an escaping point of f if f n ( p ) converges to the point at infinity as n → ∞ . • clo( f ) is the set of points p ∈ X whose f -orbit can be closed in the weaksense, meaning that for every neighbourhood U of p in X and V of f in End X ,there is an endomorphism in V with a periodic point in U . Note that clo( f )is a closed subset of X . The f -orbit of p can be closed in the strong senseif for every neighbourhood V of f in End X , there is an endomorphism in V for which p itself is periodic. If X is homogeneous in the weak sense that theautomorphism group of X acts transitively on X , as is the case for all knownOka-Stein manifolds, then the two notions coincide. • Ω f is the non-wandering set of f , that is, the set of points p ∈ X such that forevery neighbourhood U of p , there is k ≥ U ∩ f k ( U ) = ∅ . Note thatΩ f is a closed subset of X . • F f is the Fatou set of f , the open set of normality of the iterates of f . Moreexplicitly, F f is the set of points in X with a neighbourhood U such that everysubsequence of the sequence of iterates of f has a subsequence that convergeslocally uniformly on U to a holomorphic map into X or to the point at infinity.Clearly, F f is backward invariant, that is, f − ( F f ) ⊂ F f . J f is the Julia set of f , defined as the complement of F f in X . Since F f isbackward invariant, J f is forward invariant, that is, f ( J f ) ⊂ J f . • rne( f ) is the open set of points p ∈ X at which f is robustly non-expelling,meaning that there are neighbourhoods U of p in X and V of f in End X anda compact subset K of X such that g j ( U ) ⊂ K for all g ∈ V and j ≥
0. If X isStein, by Montel’s theorem, rne( f ) ⊂ F f .The following results from our previous paper [4], building on the groundbreaking workof Fornæss and Sibony [8], will be important here. If f is an endomorphism of anOka-Stein manifold X , then rne( f ) ∩ Ω f = att( f )and Ω f ⊂ clo( f ) (closing lemma) . Also, for a generic endomorphism f of X , that is, an endomorphism in a suitableresidual subset of End X ,hyp( f ) = Ω f (general density theorem) . Our main theorem is as follows. It is proved in Section 2.
Theorem 1.
A generic endomorphism f of an Oka-Stein manifold X has the followingproperties. (a) Every periodic point of f is hyperbolic. (b) The Fatou set is completely invariant, that is, f − ( F f ) = F f , and its connectedcomponents are Runge (in particular Stein). Moreover, F f = rne( f ) . When X is homogeneous, F f consists of those points of X whose orbit underevery sufficiently small perturbation of f is relatively compact. (c) The Julia set J f is completely invariant, perfect, not compact, and has no inte-rior, so J f and F f are not empty. Also, J f is the boundary of the set of pointsin X with relatively compact f -orbit. Moreover, J f = rep( f ) = att( f ) \ att( f ) , and when dim X ≥ , J f = sad( f ) . When X is homogeneous, J f consists of those points of X that are escaping withrespect to arbitrarily small perturbations of f . (d) clo( f ) = Ω f = att( f ) = J f ∪ att( f ) . (e) f is chaotic on J f . If U is a neighbourhood of a point in J f , then S n ≥ f n ( U ) is dense in X . Every basin of attraction of an attracting cycle has J f as itsboundary. (f) The iterates of f have the same Fatou set and Julia set as f itself. emark 1. (a) Perhaps the most novel part of the theorem is the inclusion J f ⊂ att( f )for a generic endomorphism f . As far as we know, this result is new even for X = C n .It implies, for example, that any neighbourhood of a point in J f intersects infinitelymany basins of attraction. It follows that f has infinitely many Fatou components. Wecan think of att( f ) as an infinite discrete metric space on which f acts in such a waythat every point has a finite orbit. This dynamical system determines J f and the actionof f on it.(b) If dim X ≥ ≤ k ≤ dim X −
1, then J f is the closure of the set ofsaddle periodic points of f at which the derivative of the appropriate iterate of f has k eigenvalues λ with | λ | < X − k eigenvalues λ with | λ | >
1. This is evidentfrom the proof of Theorem 3.(c) The most important classes of Oka-Stein manifolds are connected linear algebraicgroups and, more generally, Stein homogeneous spaces of complex Lie groups. Steinmanifolds with the density property or the volume density property are Oka. The veryspecial examples C m × C ∗ n , m, n ≥ m + n ≥
1, are of course of great interest. Thereare not many known ways to produce new Oka-Stein manifolds from old. The twoproperties do not go well together in this respect. All we can say is that products,retracts, and covering spaces of Oka-Stein manifolds are Oka-Stein. (The definitivereference on Oka theory is [10]. See also the survey [11].)(d) In the proof of part (a) of the theorem, we use a dominating spray on X , givenby the fact that for a Stein manifold, the Oka property and Gromov’s ellipticity areequivalent. In the proofs of parts (b)–(f), we directly apply the Oka property, morespecifically the basic Oka property with approximation and jet interpolation. It is oneof the nontrivially equivalent formulations of the Oka property and says the followingfor a complex manifold X .Let S be a reduced Stein space, K be a holomorphically convex compact subset of S , and T be a closed analytic subvariety of S . Every continuous map S → X that isholomorphic on a neighbourhood of K ∪ T can be deformed to a holomorphic map bya deformation that is arbitrarily small on K and fixed to any finite order along T .When X is Oka and Stein, we can take S = X and it is immediately clear that themonoid of endomorphisms of X is very large. For example, we can prescribe the valuesof an endomorphism of X on any discrete subset of X .(e) A residual set intersected with an open set is residual in the open set, so resultsabout generic endomorphisms of an Oka-Stein manifold X also apply to open sets ofendomorphisms. When X is affine algebraic, as the most important examples are, andtherefore strongly deformation-retracts onto a compact subset of itself [13, Theorem 1.1],every homotopy class of endomorphisms is open. So are unions of homotopy classes, ofcourse, such as the set of all endomorphisms that induce a given endomorphism of aparticular homotopy, homology, or cohomology group.(f) Non-hyperbolicity is generic.
While there is no standard definition of hyperbolicdynamics on a noncompact set, having both repelling periodic points and saddle pe-riodic points dense in the Julia set rules out a generic endomorphism of an Oka-Stein anifold of dimension at least 2 being hyperbolic on its Julia set in any reasonablesense.In the 1-dimensional case, Rempe-Gillen and Sixsmith have proposed a definitionof hyperbolicity for transcendental entire functions [17, Theorem and Definition 1.3].Their hyperbolicity property implies that the set of critical values is bounded. For both C and C ∗ , using the persistence of isolated critical points, an argument similar to theproof of Proposition 9(a) shows that the set of critical values of a generic endomorphismis not relatively compact.The remainder of the paper is organised as follows. In the next section we proveTheorem 1. In Section 3, we present two results about Fatou components of genericendomorphisms, the first for the complex plane C and the second for an arbitrary Oka-Stein manifold. In the final Section 4, we consider Conley’s fundamental theorems ongeneral dynamical systems in the setting of an Oka-Stein manifold.2. Proof of the main theorem
We start by proving part (a) of Theorem 1. Let X be an Oka-Stein manifold. Let K be a compact subset of X and m ≥ T ( m, K ) ⊂ End X to bethe open set of endomorphisms of X such that every periodic point p ∈ K of minimalperiod at most m is transverse. (A periodic point p of an endomorphism f of minimalperiod n is transverse if 1 is not an eigenvalue of the derivative D p f n .) Note that wedo not require the whole cycle of p to be contained in K . Define H ( m, K ) ⊂ End X tobe the open set of endomorphisms of X such that every periodic point in K of minimalperiod at most m is hyperbolic.Here is the idea of the proof of the theorem. We will show that for all compactsubsets K of X and all m ≥ H ( m, K ) is dense in End X . The sets T ( m, K ) aremore natural with respect to transversality theory and will help us prove the density of H ( m, K ). Then we consider an exhaustion ( K m ) of X and define G = \ m ≥ H ( m, K m ) . The set G is residual in End X and every periodic point of an endomorphism in G ishyperbolic.The density of H ( m, K ) is proved by induction. The steps are the following. Proposition 1.
Let K be a compact subset of an Oka-Stein manifold X . Then T (1 , K ) is dense in End X .Proof. This is an immediate consequence of Thom’s transversality theorem for holo-morphic maps from a Stein manifold to an Oka manifold [10, Corollary 8.8.7]. (cid:3)
Proposition 2.
Let K be a compact subset of an Oka-Stein manifold X . Then H ( m, K ) is dense in T ( m, K ) for all m ≥ . roof. Let f ∈ T ( m, K ). Let Λ be the set of periodic points of f of minimal period atmost m in K . It is finite, of cardinality, say, N . By persistence of transverse periodicpoints, there is a neighbourhood U of Λ in X and a neighbourhood N of f in End X such that every g ∈ N has exactly N periodic points of minimal period at most m in U . Arguing as in [4, Lemma 1], we obtain a sequence ( g n ) in End X convergingto f , such that for all n , every point in Λ is a hyperbolic periodic point of minimalperiod at most m for g n . Hence for n large enough, all the periodic points of g n ofminimal period at most m in U are hyperbolic. The result now follows from the factthat the non-existence of periodic points of minimal period at most m in K \ U is anopen condition. (cid:3) Proposition 3.
Let K be a compact subset of an Oka-Stein manifold X . Then, for all m ≥ , T ( m, K ) ∩ H ( m − , K ) is dense in H ( m − , K ) .Proof. Let f ∈ H ( m − , K ). Recall that H ( m − , K ) is open. We will obtain aperturbation of f , that is, a continuous map f : P → End X , where the parameterspace P is a neighbourhood of the origin 0 in some C k , with f (0) = f , such that thereare arbitrarily small t ∈ P with f t = f ( t ) ∈ T ( m, K ). Equivalently, we will show thatthere are arbitrarily small t ∈ P such that the map ρ m ( f t ) : X → X × X, x ( x, f mt ( x )) , is transverse to the diagonal ∆ ⊂ X × X on K . Indeed, considering the derivative d a ρ m ( f t ) : T a X → T a X ⊕ T a X, v ( v, d a f mt ) , we see that d a ρ m ( f t )( T a X ) ∩ ∆ = { } if and only if 1 is not an eigenvalue of d a f mt .By the parametric transversality theorem [14, Theorem 3.2.7], this is the case if theassociated map(*) F : X × P → X × X, F ( x, t ) = ( x, f mt ( x )) , is C and transverse to the diagonal ∆ on K × P .Consider the finite set Λ of periodic points of f of minimal period at most m − K . Let U be a neighbourhood of Λ and let N be a neighbourhood of f in H ( m − , K ).By persistence of hyperbolic periodic points, we can choose U and N small enough thatno g ∈ N has periodic points of minimal period m in U ∩ K . Since N ⊂ H ( m − , K ),it follows that for all g ∈ N , the map ρ m ( g ) : x ( x, g m ( x )) is transverse to ∆ on U ∩ K .Take x ∈ K \ U and let V ⊂ X be a neighbourhood of x . Since x Λ, the points x, f ( x ) , . . . , f m − ( x ) are distinct. Hence, by choosing V small enough, we can ensurethat for j = 1 , . . . , m −
1, the set f j ( V ) is contained in a compact set L j such that • the sets V , L , . . . , L m − are mutually disjoint, and • the union V ∪ L ∪ · · · ∪ L m − is O ( X )-convex.Let b be a holomorphic function on X , close to 1 on V and close to 0 on L ∪ · · · ∪ L m − .Since K \ U is compact, there is a finite cover { V , . . . , V ℓ } of K \ U and associated olomorphic functions b , . . . , b ℓ . Let s : X × C r → X be a dominating spray. Considerthe perturbations ϕ V j t ( x ) = s ( x, b j ( x ) t ) , j = 1 , . . . , ℓ. Define the perturbation f : X × ( C r ) ℓ → X, f t ( x ) = f ( x, t , . . . , t ℓ ) = ϕ V t ◦ · · · ◦ ϕ V ℓ t ℓ ◦ f ( x ) . To complete the proof we just need to verify that for k = 1 , . . . , ℓ , the associated map F : X × C rℓ → X × X, F ( x, t ) = ( x, f mt ( x )) , is transverse to ∆ on V k × { } . Fix k and let a ∈ V k have f m ( a ) = a . Set t j = 0 for j = k . Fix x = a and let t k vary. It suffices to show that the mapΘ : C r → X, t k ( ϕ V k t k ◦ f ) m ( a ) , is a submersion at the origin. Denoting by ˜ s : X × C r → X the modified spray with˜ s ( x, t ) = s ( x, b k ( x ) t ), we see that the derivative d Θ : C r → T a X is given by d Θ = ∂ ˜ s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( a, + d f m − ( a ) f ◦ ∂ ˜ s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( f m − ( a ) , + · · · + d f ( a ) f m − ◦ ∂ ˜ s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( f ( a ) , . Choose a hermitian metric h on X . Let C > j = 1 , . . . , m − x ∈ f j ( V k ), we have k d x f m − j k ≤ C (using the operator norm associated to h ).Since s is dominating, there is ǫ > x ∈ V k , the ball B (cid:18) ∂s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( x, , ǫ (cid:19) in the space of linear maps from C k to T x X is contained in the open subset of maps ofmaximal rank. We can choose b k close enough to 1 on V k that for all x ∈ V k , ∂ ˜ s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( x, ∈ B ∂s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( x, , ǫ ! , and close enough to 0 on f ( V k ) ∪· · ·∪ f m − ( V k ) that for all y ∈ f ( V k ) ∪· · ·∪ f m − ( V k ), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ ˜ s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( y, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ǫ C ( m − . It follows that d Θ belongs to the ball B (cid:18) ∂s∂t (cid:12)(cid:12)(cid:12)(cid:12) ( x, , ǫ (cid:19) and thus has maximal rank. (cid:3) Remark 2.
Let f ∈ End X , let K ⊂ X be compact, and fix m ≥
1. One couldtry to avoid the induction process and find a perturbation f : P → End X such thatthe associated map F in (*) above is C and transverse to the diagonal ∆ on K × P .A strategy to obtain such a perturbation could be to show that if a ∈ K satisfies f m ( a ) = a , then the linear map A = ∂f mt ( x ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) ( a, : C r → T a X s surjective. There is, however, a simple obstruction to the surjectivity of A . Assumethat the minimal period m of a is smaller that m and let d = m/m . Assume moreoverthat B = d a f m admits as an eigenvalue a d -th root ζ = 1 of unity. Then A is equal to A = ( B d − + B d − + · · · + I ) ◦ T, where T : C r → T a X is a linear map. The map B d − + B d − + · · · + I : T a X → T a X isnot surjective since ζ is an eigenvalue of B , so A cannot be surjective. This is why, inthe induction process, we take care of points with period smaller than m first.A key ingredient in the proof of the remainder of Theorem 1, parts (b)–(f), is thefollowing result, which builds on the proofs of the closing lemma and general densitytheorem in our previous paper [4, Theorem 2(c) and Corollary 1(b)]. Theorem 2.
For a generic endomorphism f of an Oka-Stein manifold X , the set X \ rne( f ) lies in the closure of att( f ) , equals the closure of rep( f ) , and, if dim X ≥ ,equals the closure of sad( f ) . To prove the theorem we use the following closing lemma. The term non-degeneratein its proof refers to an endomorphism that has maximal rank at some point or, in otherwords, is a local biholomorphism outside a proper subvariety.
Theorem 3.
Let f be an endomorphism of an Oka-Stein manifold X , let K ⊂ X becompact, and let p ∈ X \ rne( f ) . Let W be a neighbourhood of f in End X and V be aneighbourhood of p in X . Then: (a) There are h ∈ W and q ∈ V such that q is a super-attracting periodic point of h and the h -orbit of q leaves K . (b) There are h ∈ W and q ∈ V such that q is a repelling periodic point of h andthe h -orbit of q leaves K . (c) Suppose that dim X ≥ . There are h ∈ W and q ∈ V such that q is a saddleperiodic point of h and the h -orbit of q leaves K .Proof. Take a holomorphically convex compact subset L of X containing K . Since p ∈ X \ rne( f ), there are g ∈ W and q ∈ V such that the g -orbit of q is not containedin L . Say g k ( q ) ∈ L for 0 ≤ k < m and g m ( q ) / ∈ L . By Proposition 4 below, we mayassume that g is non-degenerate, and by Lemma 1 below, we may assume that g m hasmaximal rank at q .Let φ : X → X be continuous, equal to g on a neighbourhood of L , and holomorphicon a neighbourhood of g m ( q ) with φ ( g m ( q )) = q and an arbitrarily prescribed derivativeat g m ( q ). Since X is Stein and Oka, φ can be deformed to h ∈ End X , arbitrarily close to g on L , such that the 1-jet of h coincides with that of g at the points q, g ( q ) , . . . , g m − ( q ),and such that h takes g m ( q ) to q with the prescribed derivative at g m ( q ). Then h isarbitrarily close to g on L with q as a periodic point and h m ( q ) / ∈ K . Now( h m +1 ) ′ ( q ) = h ′ ( g m ( q ))( g m ) ′ ( q ) , o we can get q to be super-attracting by taking h ′ ( g m ( q )) = 0. Since ( g m ) ′ ( q ) is non-singular, we can get q to be a repelling periodic point or, when dim X ≥
2, a saddleperiodic point by a suitable choice of h ′ ( g m ( q )). (cid:3) Lemma 1.
Let g be a non-degenerate endomorphism of a complex manifold X . Let m ≥ . Then there is a dense open subset V of X such that the m th iterate g m hasmaximal rank at every point of V .Proof. Let U be an open dense subset of X . Then the open set g − ( U ) is also dense.Indeed, suppose that there exists a nonempty open subset W such that g ( W ) ∩ U = ∅ .Since g has maximal rank on a dense set, there are points in W at which g has maximalrank, so g ( W ) has nonempty interior, which contradicts U being dense.Now let U be the set of points where g has maximal rank. Then V = U ∩ g − ( U ) ∩· · · ∩ g − m ( U ) is open and dense and g m has maximal rank at every point of V . (cid:3) Proposition 4.
Let X be an Oka-Stein manifold and p ∈ X . A generic endomorphismof X is a local biholomorphism at p . Hence, a generic endomorphism of X is non-degenerate.Proof. The set of endomorphisms of maximal rank at p is open. We need to show that itis dense. Embed X in C m with a tubular neighbourhood that retracts holomorphicallyonto X . Take p to be the origin in C m and suppose that f ∈ End X is singular at0 with f (0) = 0. Add to f the map ǫχλ , where λ is a linear automorphism of C m , ǫ >
0, and the continuous function χ : X → [0 ,
1] is 1 on a large compact subset of X and zero near infinity. If ǫ is small enough, the image of the new map lies in thetubular neighbourhood, and by a suitable generic choice of λ , the new map composedby the retraction has maximal rank at 0. Finally apply the basic Oka property withapproximation and jet interpolation and obtain an endomorphism close to f and ofmaximal rank at 0. (cid:3) Proof of Theorem 2.
Let { U n : n ≥ } be a countable basis for the topology of X . Let S n be the open set of all f ∈ End X such that f has an attracting cycle intersecting U n .Then G = T End X \ ∂S n is a residual subset of End X . We will show that if f ∈ G and p ∈ X \ rne( f ), then p lies in the closure of att( f ).Suppose that p ∈ U n . It suffices to show that att( f ) ∩ U n = ∅ . By definitionof G , we have f / ∈ ∂S n . Hence either f ∈ S n , in which case att( f ) ∩ U n = ∅ isimmediate, or f / ∈ S n . The latter case is ruled out by Theorem 3. Indeed, by thetheorem, if p ∈ U n \ rne( f ), then there is an endomorphism arbitrarily close to f witha super-attracting periodic point in U n .The proofs for rep( f ) and sad( f ) are analogous. It easily follows from Cauchyestimates, as in [3, Claim 3], that rep( f ) and sad( f ) are subsets of X \ rne( f ). (cid:3) Lemma 2.
Let f be an endomorphism of a complex manifold X . (a) The basin of attraction of an attracting cycle of f is contained in rne( f ) . . . (b) . . . and is a union of Fatou components. c) att( f ) is a closed subset of F f and hence, if X is Stein, of rne( f ) . Let us review the notion of the basin of attraction of an attracting cycle. It is notquite as straightforward as the special case of an attracting fixed point. Consider anattracting cycle A of f of length m . Every point of A is an attracting fixed point of f m . There are at least four ways to define the basin of attraction of A .Say x ∈ B if ( f mn ( x )) converges to a point in A as n → ∞ . Say x ∈ B ′ if( f mn ) converges uniformly to a point in A on a neighbourhood of x as n → ∞ . Then B = B ′ is the union of the basins of attraction of the points of A viewed as fixedpoints of f m . Say x ∈ B if ( f n ( x )) converges to A as n → ∞ , meaning that for everyneighbourhood V of A , f n ( x ) ∈ V for all sufficiently large n . Finally say x ∈ B ′ if ( f n )converges uniformly to A on a neighbourhood U of x as n → ∞ , meaning that for everyneighbourhood V of A , f n ( U ) ⊂ V for all sufficiently large n . Clearly, B ′ ⊂ B ⊂ B .It is now a basic observation that f n → A uniformly on a neighbourhood of A . Inother words, A ⊂ B ′ . It follows that B ⊂ B ′ , so B = B ′ = B = B ′ . Thus there is asingle natural notion of the basin B of A . The basin is open and completely invariant,that is, f − ( B ) = B . Also, the basin is a subset of the Fatou set F f . Indeed, if x ∈ B ,given any sequence ( f n k ) of iterates of f , there are infinitely many k for which n k liesin the same congruence class modulo m , so we can extract a subsequence of ( f n k ) thatconverges uniformly in a neighbourhood of x to a point in the cycle. Proof. (a) Another basic observation is that there is a compact neighbourhood K of A such that f ( K ) ⊂ K ◦ (in fact an arbitrarily small one, but this is not needed here).Take p ∈ B and find a compact neighbourhood L of p with f n ( L ) ⊂ K ◦ for some n ≥
0. Then, for all g ∈ End X sufficiently close to f , g ( K ) ⊂ K ◦ and g n ( L ) ⊂ K ◦ , so g j ( L ) ⊂ K for all j ≥ n . It follows that p ∈ rne( f ).(b) Let us prove that if a Fatou component U intersects B , then U ⊂ B . Take p ∈ U ∩ B . There is a neighbourhood V of p such that f mn converges uniformly on V to a point q in the cycle as n → ∞ . Since U is a Fatou component, and no subsequenceof ( f mn ) converges to infinity locally uniformly on U , every subsequence of ( f mn ) has asubsequence that converges locally uniformly on U to a holomorphic map g : U → X .Since g is constantly equal to q on V , it is constantly equal to q on all of U . Hence, f mn converges to q locally uniformly on U .(c) Let ( x n ) be a sequence in att( f ) converging to a point y in F f . The Fatoucomponent containing y can contain at most one point of att( f ). Hence the points x n are eventually equal to y . (cid:3) Proposition 5.
For a generic endomorphism f of an Oka-Stein manifold X , F f = rne( f ) . Moreover, F f is the set of points for which there is a neighbourhood V and a compactsubset K of X with f n ( V ) ⊂ K for all n ≥ .Proof. Take p ∈ F f \ rne( f ) and let U be the component of F f containing p . By Lemma2(a), U is not a component of a basin of attraction; yet, by Theorem 2, there are ttracting periodic points arbitrarily close to p and hence in U , which is absurd byLemma 2(b).Let E be the set of points for which there is a neighbourhood V and a compactsubset K of X with f n ( V ) ⊂ K for all n ≥
1. Then rne( f ) ⊂ E ⊂ F f and the secondclaim follows. (cid:3) The following lemma is Fornæss and Sibony’s [9, Proposition 2.5]. They proved itfor affine space, but the proof easily extends to Stein manifolds.
Lemma 3.
Let f be an endomorphism of a Stein manifold and let U be a Fatou com-ponent of f . If ( f n ) is locally bounded on U , then U is Runge (in particular Stein). If,moreover, f is non-degenerate, then f ( U ) ⊂ F f . Proposition 6.
The following hold for a generic endomorphism f of an Oka-Steinmanifold X . (a) The Fatou set F f is forward invariant and the components of F f are Runge. (b) Both F f and J f are completely invariant.Proof. (a) follows from Propositions 4 and 5 and Lemma 3.(c) follows immediately from (a) and the easy observation that F f is backwardinvariant. (cid:3) Proposition 7.
The following hold for a generic endomorphism f of an Oka-Steinmanifold X . (a) J f = rep( f ) and, if dim X ≥ , J f = sad( f ) . (b) For every m ≥ , J f m = J f and F f m = F f . Proof. (a) follows from Theorem 2 and Proposition 5.(b) Clearly, F f ⊂ F f m , so J f m ⊂ J f . Since J f is the closure of rep( f ), which iscontained in J f m , the reverse inclusions hold. (cid:3) From Theorem 2, Lemma 2(c), and Proposition 5, we deduce the following result.
Proposition 8.
For a generic endomorphism f of an Oka-Stein manifold X , clo( f ) = Ω f = att( f ) = J f ∪ att( f ) . Also, J f = X \ rne( f ) has no interior.Proof. Since att( f ) is closed in F f and J f lies in the closure of att( f ), we have att( f ) = J f ∪ att( f ). Since also att( f ) ∩ J f = ∅ , the interior of J f is empty.Now the non-wandering set Ω f is closed and att( f ) ⊂ Ω f , so att( f ) ⊂ Ω f . Theclosing lemma [4, Theorem 2(c)] says that Ω f ⊂ clo( f ). The proof of the generaldensity theorem [4, Corollary 1] shows that hyp( f ) is dense in clo( f ) for a genericendomorphism f , so clo( f ) = hyp( f ) = att( f ) ∪ J f . Thus the four sets att( f ) ⊂ Ω f ⊂ clo( f ) = J f ∪ att( f ) are equal. (cid:3) roposition 9. The following hold for a generic endomorphism f of an Oka-Steinmanifold X . (a) For every compact subset K of X , f has an attracting fixed point, a saddle fixedpoint, and a repelling fixed point outside K . (b) J f is not compact. (c) J f is perfect.Proof. (a) Let K ⊂ X be compact. We will show that the open set of endomorphisms of X with an attracting fixed point outside K is dense in End X , and similarly for saddlefixed points and repelling fixed points.Let f ∈ End X , let L be a holomorphically convex compact subset of X containing K , and let p ∈ X \ L . Let φ : X → X be continuous, equal to f on a neighbourhood of L , and holomorphic on a neighbourhood of p with φ ( p ) = p and an arbitrarily prescribedderivative at p . Since X is Stein and Oka, φ can be deformed to an endomorphism of X , arbitrarily close to f on L , whose 1-jet at p coincides with that of φ .(b) follows from (a).(c) Suppose that p is an isolated point of J f . Let U be a coordinate ball centredat p such that U \ { p } ⊂ F f . By Proposition 8, p ∈ att( f ), so there are two points inatt( f ) ∩ U that lie in distinct cycles. Since U \ { p } is connected, the two points mustlie in the same Fatou component, which is absurd. (cid:3) Proposition 10.
A generic endomorphism of an Oka-Stein manifold is chaotic on itsJulia set.
The usual definition of chaos is that periodic points are dense and there is a denseorbit. Instead, we verify Touhey’s characterisation of chaos [19], which says that forevery two nonempty open subsets, there is a cycle that visits both of them.
Proof.
Let Λ ⊂ End X be the dense open set of non-degenerate endomorphisms. Takea countable basis { U n : n ≥ } for the topology of X . Let S m,n be the open subset ofEnd X of endomorphisms with a repelling cycle through U m and U n . Then G = Λ ∩ \ m,n End X \ ∂S m,n is residual. We claim that each f ∈ G is chaotic on J f .Take U m and U n both intersecting J f . By definition of G , we have f ∈ S m,n or f / ∈ S m,n . In the former case, f has a repelling cycle through U m and U n . Such acycle lies in J f and we are done. We rule out the latter case by showing that f can beapproximated by endomorphisms in S m,n .Let K be a holomorphically convex compact subset of X . There are p ∈ U m and q ∈ U n such that f j ( p ) and f k ( q ) lie outside K for some j, k ≥
1. Since f ∈ Λ, by Lemma1, we can assume that f j has maximal rank at p and that f k has maximal rank at q .Let φ : X → X be continuous, equal to f on a neighbourhood of K and at the points p, f ( p ) , . . . , f j − ( p ) and q, f ( q ) , . . . , f k − ( q ), holomorphic on neighbourhoods of f j ( p ) nd f k ( q ), with φ ( f j ( p )) = q and φ ( f k ( q )) = p and an arbitrarily prescribed derivativeof maximal rank at those points. Deform φ to an endomorphism g of X , arbitrarilyclose to f on K , equal to f at the points p, f ( p ) , . . . , f j − ( p ) and q, f ( q ) , . . . , f k − ( q ),and with the prescribed 1-jet at f j ( p ), f k ( q ). By taking the derivatives φ ′ ( f j ( p )) and φ ′ ( f k ( q )) to be sufficiently large, we get g ∈ S m,n . (cid:3) Proposition 11.
The following hold for a generic endomorphism f of an Oka-Steinmanifold X . (a) The set of points in J f whose f -orbit is relatively compact is dense in J f . (b) The set of points in J f whose f -orbit is not relatively compact is dense in J f . (c) J f is the boundary of the set of points in X whose f -orbit is relatively compact.Proof. (a) By Proposition 10, points with finite orbit are dense in J f .(b) By Proposition 10, points with dense orbit are dense in J f , so by Proposition9(b), points whose orbit is not relatively compact are dense in J f .(c) follows from (b) and Proposition 5. (cid:3) Proposition 12.
The following hold for a generic endomorphism f of an Oka-Steinmanifold X . (a) If U is a neighbourhood of a point in J f , then S n ≥ f n ( U ) is dense in X . (b) J f is the boundary of each basin of attraction.Proof. (a) Let { U n : n ≥ } be a countable basis for the topology of X . Let S m,n bethe open set of all f ∈ End X such that f k ( U m ) intersects U n for some k ≥
0. Let G = T End X \ ∂S m,n . Then G is a residual subset of End X . We prove (a) by showingthat for every f ∈ G and every m, n ≥ U m intersects J f , there is k ≥ f k ( U m ) intersects U n . If f ∈ S m,n , this is obvious. By the definition of G , if f / ∈ S m,n , then f / ∈ S m,n , but this can be ruled out as follows.Let K be a holomorphically convex compact subset of X . Since U m intersects J f ,there is p ∈ U m such that f k ( p ) / ∈ K for some k ≥
1. Let φ : X → X be continuous,equal to f on a neighbourhood of K and at the points p, f ( p ) , . . . , f k − ( p ), and with φ ( f k ( p )) ∈ U n . Deform φ to an endomorphism g of X , arbitrarily close to f on K andequal to φ at the points p, f ( p ) , . . . , f k ( p ). Then g ∈ S m,n approximates f as well asdesired.(b) Let B be the basin of attraction of an attracting cycle of f and let U be aneighbourhood of a point in J f . By (a), f n ( U ) intersects B for some n ≥
0, so U itselfintersects B . Hence, J f ⊂ ∂B . The opposite inclusion is evident. (cid:3) Proposition 13.
Let f be an endomorphism of an Oka-Stein manifold X and let p ∈ X \ rne( f ) . Take a neighbourhood U of p in X and V of f in End X . Thenthere is an endomorphism in V with an escaping point in U .Proof. Choose for convenience an embedding of X in some affine space and let k·k bethe sup-norm there. Take a holomorphically convex compact subset K of X and ǫ > uch that { g ∈ End X : k f − g k K ≤ ǫ } ⊂ V . Since p ∈ X \ rne( f ), there is q ∈ U and h ∈ End X such that k f − h k K < ǫ and h m ( q ) K for some m ≥
0. Let q j = h j ( q ), j = 0 , . . . , m .Exhaust X by holomorphically convex compact subsets K n , n ≥
0, such that K = K . We may assume that q m ∈ K \ K . Applying the basic Oka property withapproximation and interpolation, we obtain h ∈ End X such that k h − h k K < ǫ/ h ( q j ) = q j +1 for j = 0 , . . . , m −
1, and h ( q m ) ∈ K \ K . Continuing in this way, weobtain a sequence ( h n ) in End X such that for all n ≥ k h n +1 − h n k K n < ǫ n +1 ,h jn ( q ) = h jn − ( q ) , for j = 0 , . . . , m + n − , and h m + nn ( q ) ∈ K n +1 \ K n . Hence, ( h n ) converges to an endomorphism in V for which q is escaping. (cid:3) The final claims in parts (b) and (c) of Theorem 1 now follow from Propositions 5and 13 and the equivalence of transitivity and micro-transitivity for a continuous actionof a Polish group on a Polish space (theorem of Effros; see the introduction to [4] andthe references there).The proof of Theorem 1 is complete.3.
Fatou components of generic endomorphisms
Let us first review a few more definitions. A Fatou component U of an endomorphism f of a complex manifold is wandering if the sets f n ( U ), n ≥
0, are mutually disjoint.The Fatou component U is recurrent if for some point p in U , a subsequence of ( f n ( p ))converges to a point q in U (so q is an ω -limit point of p ). Let us call U pre-recurrent (this is not standard terminology) if some point in U has an ω -limit point q in the Fatouset of f . Being an ω -limit point, q is non-wandering. Clearly, a Fatou component inthe basin of attraction of an attracting cycle is pre-recurrent.The Fatou component U is non-wandering if there are m < n such that f m ( U ) and f n ( U ) intersect. Write n = m + k , k ≥
1. If the manifold is Oka-Stein and f is generic,then the Fatou set of f is forward invariant by Theorem 1; assume this. Then f m ( U )lies in a Fatou component V , and V and f k ( V ) intersect, so f k ( V ) ⊂ V . Thus V isperiodic and U is preperiodic. (In our usage, a periodic component is preperiodic anda recurrent component is pre-recurrent.) Remark 3.
As mentioned above, if f is an endomorphism of an Oka-Stein manifold,then rne( f ) ∩ Ω f = att( f ), and by Theorem 1(b), if f is generic, then rne( f ) = F f .Hence, by Lemma 2(b), every pre-recurrent Fatou component of a generic endomor-phism lies in the basin of attraction of an attracting cycle.In the case of the complex plane C , our results show that a generic endomorphismdoes not have parabolic cycles, Siegel discs, or Baker domains. It only has Fatou omponents of one type – or perhaps two, if dynamically bounded wandering domainsexist, which is an open question. (For dynamics of entire functions, see the survey [18].) Theorem 4.
A point in the Fatou set of a generic endomorphism f of C is attractedto an attracting cycle or lies in a dynamically bounded wandering domain. Every Fatoucomponent of f is a disc. Hoping that it may be of interest to the reader, we give a proof that does not usethe classification of Fatou components for entire functions.
Proof.
First note that transcendental functions are generic among all entire functions.On a multiply-connected Fatou component of a transcendental function, the iterates ofthe function converge to infinity (this is noted by Baker at the beginning of the proofof Theorem 1 in [5]; see also [18, Theorem 2.5]). By Theorem 1, points in the Fatouset of a generic endomorphism f of C have relatively compact orbits and the Fatou setis not all of C . Hence, the Fatou components of f are discs. Also, a wandering domainof f cannot be escaping or oscillating: it must be dynamically bounded.Now let U be a periodic component of F f = rne( f ) of period m . If U is recurrent,then U lies in the basin of attraction of an attracting cycle by Remark 3. Next assumethat U is not recurrent and take a sequence of iterates of f m converging locally uniformlyon U to a limit function g with image in J f . Since J f has no interior by Theorem 1, g is constant, and its value p is a fixed point of f m . By Theorem 1 once again, p ishyperbolic and hence repelling, which is absurd. (cid:3) Motivated by Theorem 4, we pose the following open question.
Question.
Are all the Fatou components of a generic endomorphism f of an Oka-Steinmanifold pre-recurrent? Equivalently (by Remark 3), is the Fatou set of f the union ofthe basins of attraction of the attracting cycles of f ?We will not hazard a conjecture as to whether the answer is affirmative or negative.Theorem 1(a) shows that arbitrarily small perturbations of an endomorphism can de-stroy all non-hyperbolic periodic points at once. An affirmative answer would have asimilar but apparently much stronger consequence, namely that all non-pre-recurrentFatou components can be destroyed at once by arbitrarily small perturbations.We begin to shed some light on the question with the following result. In the nextsection, we reformulate the question in several different ways using the notion of chain-recurrence. Theorem 5.
Let X be an Oka-Stein manifold. The set of endomorphisms with a non-pre-recurrent Fatou component is dense in End X . It follows that the property of the entire Fatou set consisting of basins of attractioncan be destroyed by an arbitrarily small perturbation.To produce non-pre-recurrent Fatou components, we use a theorem of Hakim [12,Theorem 1.6] (see also [1] and [2]). She proved that for every endomorphism f of aneighbourhood of the origin 0 in C n , fixing 0 with a suitable 2-jet there, for example ( z , . . . , z n ) = ( z + z , . . . , z n + z n ), there is a bounded connected open set U in C n with 0 ∈ ∂U and f ( U ) ⊂ U , such that f k ( z ) → k → ∞ for every z ∈ U , butthe family of iterates of f is not normal on any neighbourhood of 0. It follows thatan endomorphism of a complex manifold with a suitable 2-jet at a fixed point p has anon-pre-recurrent Fatou component with p in its boundary. Proof.
Let f ∈ End X . By Hakim’s theorem, it suffices to show that by an arbitrarilysmall perturbation of f we can obtain an endomorphism g with a fixed point at which g has a prescribed 2-jet.Let K be a holomorphically convex compact subset of X . Take p ∈ X \ K . Let φ : X → X be continuous, equal to f on a neighbourhood of K , holomorphic on aneighbourhood of p , and with any prescribed 2-jet at p . Since X is Stein and Oka, thebasic Oka property with approximation and jet interpolation holds for maps X → X ,so we can deform φ to an endomorphism g of X , arbitrarily close to f on K , and withthe prescribed 2-jet at p . (cid:3) Holomorphic Conley theory
In the final section, we bring together our results on generic dynamics on Oka-Stein man-ifolds and Conley’s fundamental theorems on general dynamical systems [7] as adaptedto the noncompact setting by Hurley [15]. We start by summarising the necessarydefinitions and stating the two fundamental theorems.Let X be a locally compact second countable metric space and let f : X → X be continuous. Choose a metric d on X compatible with the topology of X . Let ǫ : X → (0 , ∞ ) be continuous. A finite sequence x , x , . . . , x n , n ≥
1, of points in X isan ǫ -chain or ǫ -pseudo-orbit of length n if d ( f ( x j ) , x j +1 ) < ǫ ( f ( x j )) for j = 0 , . . . , n − p in X is chain-recurrent for f if for every function ǫ , there is an ǫ -chain thatbegins and ends at p . We denote by C f the set of chain-recurrent points of f . Clearly,a non-wandering point is chain-recurrent.A nonempty open subset U of X is absorbing for f if f ( U ) ⊂ U . The closed set A = T n ≥ f n ( U ) is the attractor determined by U (this is a decreasing intersection). Thebasin of A relative to U is the open set B ( A, U ) = S n ≥ f − n ( U ) ⊃ A . Finally, the basinof A is the open set B ( A ) = S B ( A, U ), where U ranges over the absorbing sets thatdetermine A .The first fundamental theorem [15, Theorem 1] states that X \ C f = [ A B ( A ) \ A, where the union is taken over all the attractors A of f . The right-hand side of theequation is independent of the choice of the metric d and invariant under topologicalconjugation, so C f is as well. The equation also shows that C f is closed. n equivalence relation is defined on C f by declaring points p and q equivalent if forevery continuous ǫ : X → (0 , ∞ ), there is an ǫ -chain from p to q and an ǫ -chain from q to p . The equivalence classes are called chain-recurrence classes.The second fundamental theorem [15, Theorem 2] states that there is a completeLyapunov function L : X → R for f . It has the following properties.(1) L is continuous.(2) L ( f ( x )) ≤ L ( x ) for all x ∈ X , with equality if and only if x ∈ C f .(3) L is constant on each chain-recurrence class and takes different values on differ-ent classes.(4) If C and C ′ are distinct classes such that for each ǫ there is an ǫ -chain from C to C ′ , then L ( C ) > L ( C ′ ).(5) L ( C f ) is nowhere dense in R .We can think of L as denoting “height”, so that points in X \ C f move down with time,but points in C f stay at the same height.We now specialise to the case of X being an Oka-Stein manifold and f : X → X being holomorphic. Theorem 6.
A generic endomorphism f of an Oka-Stein manifold X has the followingproperties. (a) The Julia set J f lies in a single chain-recurrence class. (b) Each attracting cycle is an attractor and a chain-recurrence class. (c)
The basin of attraction of an attracting cycle contains no chain-recurrent pointsexcept the attracting cycle itself. (d)
Every point in a non-pre-recurrent Fatou component is chain-recurrent and liesin the same chain-recurrence class as the Julia set. (e)
The union of the non-pre-recurrent Fatou components equals the set of wanderingchain-recurrent points.Proof. (a) follows from the Julia set containing a dense orbit by Theorem 1(e).(b) A sufficiently small neighbourhood of an attracting cycle is absorbing. Theattracting cycle is the attractor determined by any such neighbourhood. It is easilyseen that one cannot escape from an attracting cycle along an ǫ -pseudo-orbit if ǫ issmall enough.(c) If V is the basin of attraction of an attracting cycle A , then clearly V ⊂ S f − n ( U ), where U is a neighbourhood of A as in the proof of (b), so V ⊂ B ( A )and V \ A ⊂ B ( A ) \ A ⊂ X \ C f .(d) Take a point p in a non-pre-recurrent Fatou component. The orbit of p isrelatively compact by Theorem 1(b), so p has an ω -limit point q ∈ J f . By Theorem1(e), arbitrarily close to q is a point r whose orbit comes arbitrarily close to p . Let ustravel from p along its orbit, jump to q when we get close to it, jump to r , travel alongits orbit, and jump to p when we get close to it. This makes an ǫ -pseudo-orbit for an rbitrarily small ǫ , beginning and ending at p , and visiting J f along the way, so (d) isproved.(e) follows from the previous parts and Ω f = J f ∪ att( f ) (Theorem 1(d)). (cid:3) The theorem shows that the “Conley decomposition” of an Oka-Stein manifold X ,induced by a generic endomorphism f of X , is as follows. • C f is partitioned into the following chain-recurrence classes. – The union of the Julia set and the non-pre-recurrent Fatou components. – Each attracting cycle is a chain-recurrence class. • X \ C f is the union of the basins of attraction of the attracting cycles with thecycles themselves removed.Thinking of the Lyapunov function L as denoting height, we can picture the union ofthe Julia set and the non-pre-recurrent Fatou components at constant height at the“top” of X . The attracting cycles are at the “bottom” of X , each at a different height,and each point in a basin of attraction moves down with time towards the cycle thatattracts it.The following corollary of Theorem 6 is immediate. Corollary 1.
For a generic endomorphism of an Oka-Stein manifold X , the followingare equivalent. (i) Every Fatou component is pre-recurrent. (ii)
The Fatou set is the union of the basins of attraction of the attracting cycles. (iii)
The robustly non-expelling set is the union of the basins of attraction of theattracting cycles. (iv)
Every chain-recurrent point is non-wandering. (v)
The periodic points are dense in the chain-recurrent set. (vi)
The Julia set is a chain-recurrence class. (vii)
The chain-recurrent set has no interior.If endomorphisms of X satisfy the weak closing lemma for chain-recurrent points, thena generic endomorphism of X has these properties. The weak closing lemma for chain-recurrent points would imply that periodic pointsare dense in the chain-recurrent set of a generic endomorphism, and thus give an affir-mative answer to the open question posed in Section 3, just as the weak closing lemmafor non-wandering points implies that periodic points are dense in the non-wandering setof a generic endomorphism [4, Corollary 1(b)]. The closing lemma for chain-recurrentpoints of diffeomorphisms of compact smooth manifolds, or rather the stronger con-necting lemma for pseudo-orbits, is a major result in modern smooth dynamics [6,Th´eor`eme 1.2]. This lemma states that if p is a chain-recurrent point of f ∈ End X , V is a neighbourhood of p in X , and W is a neighbourhood of f in End X , then there is an endomorphism in W with a periodicpoint in V . e conclude this section by determining the topological structure of each of the sixsets of endomorphisms defined by the properties (i)–(vii) in the corollary. According tothe corollary, the symmetric difference of any two of them is meagre (meaning that thecomplement is residual, that is, contains a dense G δ ). A subset E of a topological space Y is said to be nearly open or have the Baire property if there is an open subset U of Y such that the symmetric difference E △ U is meagre. A set is nearly open if and only ifit is the union of a G δ set and a meagre set or, equivalently, the complement of a meagreset in an F σ set. The nearly open subsets of Y form a σ -algebra, the smallest σ -algebracontaining all open subsets and all meagre subsets of Y . (See [16, Section 8.F].) Theorem 7.
Let X be an Oka-Stein manifold. The subsets of End X defined by theproperties (i)–(vii) above and their complements are nearly open.Proof. Consider the set U of pairs ( p, f ) in X × End X such that p ∈ rne( f ). Let V bethe subset of pairs ( p, f ) such that p lies in the basin of attraction of some attractingcycle of f . Clearly, U is open. We claim that V is also open. Let ( p, f ) ∈ V and takea small neighbourhood W of the attracting cycle of f to which p is attracted. Then f m ( p ) ∈ W for some m ≥
0. By persistence of attracting cycles, every endomorphismin a sufficiently small neighbourhood Z of f has an attracting cycle in W with W in itsbasin of attraction. The pairs ( q, g ) ∈ X × Z such that g m ( q ) ∈ W lie in V and form aneighbourhood of ( p, f ).We conclude that U \ V is G δ in End X . Let π : X × End X → End X be theprojection. Then π ( U \ V ) is an analytic subset of the Polish space End X [16, Section14.A]. It is the set of endomorphisms f for which the basins of attraction do not coverall of rne( f ), that is, the complement of the set defined by property (iii). By a theoremof Lusin and Sierpi´nski [16, Theorem 21.6], an analytic set in a Polish space is nearlyopen. (cid:3) By Theorem 5, the set of endomorphisms of X with a non-pre-recurrent Fatoucomponent is dense in End X . It is the complement of the subset of End X defined byproperty (i). By Theorem 7, it is the union of a G δ set and a meagre set. If the meagreset could be taken to be empty, then we could conclude that the negations of properties(i)–(vii) are generic and consequently that the weak closing lemma for chain-recurrentpoints fails in our setting. References [1] M. Abate.
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Dipartimento Di Matematica, Universit`a di Roma “Tor Vergata”, Via Della RicercaScientifica 1, 00133 Roma, Italy
Email address : [email protected] School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Aus-tralia
Email address : [email protected]@adelaide.edu.au