Dynamics of transcendental Hénon maps III: Infinite entropy
Leandro Arosio, Anna Miriam Benini, John Erik Fornæss, Han Peters
FFebruary 11, 2021
DYNAMICS OF TRANSCENDENTAL H´ENON MAPS III: INFINITE ENTROPY
LEANDRO AROSIO † , ANNA MIRIAM BENINI ‡ , JOHN ERIK FORNÆSS, AND HAN PETERS Abstract.
Very little is currently known about the dynamics of non-polynomial entire maps in severalcomplex variables. The family of transcendental H´enon maps offers the potential of combining ideasfrom transcendental dynamics in one variable, and the dynamics of polynomial H´enon maps in two.Here we show that these maps all have infinite topological and measure theoretic entropy. The proofalso implies the existence of infinitely many periodic orbits of any order greater than two. Introduction A transcendental H´enon map is a holomorphic automorphism of C of the form F ( z, w ) = ( f ( z ) − δw, z ) , where δ ∈ C \ { } , and f is a transcendental entire function. Transcendental H´enon maps form a bridgebetween two distinct families of holomorphic maps whose dynamical behaviors have been studied inten-sively in recent years: the family of complex (polynomial) H´enon maps, and the family of transcendentalentire functions.In two previous papers [ABFP19, ABFP20] we studied the dynamics of these maps, demonstratingnon-trivial dynamical behavior. For example, the Julia set is always non-empty. Here we provide furtherevidence of non-trivial dynamics: Theorem 1.1.
Any transcendental H´enon map has infinite topological entropy.
As an immediate corollary we obtain an alternative proof that the Julia set is non-empty, and by theVariational Principle that the metric entropy is also infinite. The proof implies that a transcendentalH´enon map has infinitely many periodic cycles of any order greater than 2. This result gives a completedescription on the possible periodic cycles, since there exist transcendental H´enon maps without anyperiodic cycles of orders 1 and 2 [ABFP20]. We recall the analogy with one-dimensional transcendentalfunctions, which may not have any fixed points, but always have infinitely many periodic cycles of anyorder greater than 1.The topological entropy of holomorphic maps is a topic with an interesting history. It was shownby Gromov that the topological entropy of a rational function of degree d is log( d ), a result writtenin a preprint in 1977, but not published until 2003 [Gro03]. In the meantime the result was obtainedindependently by Lyubich [Lju83].Smillie [Smi90] proved in 1990 that a polynomial H´enon map of degree d has topological entropy log( d ).Preliminary results for transcendental H´enon maps were obtained by Dujardin [Duj04], who proved thatthe entropy of a H´enon-like map of degree d is log( d ) as well, and used this fact to construct examples oftranscendental H´enon maps with infinite topological entropy. † Supported by the SIR grant “NEWHOLITE - New methods in holomorphic iteration” no. RBSI14CFME. Partiallysupported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of RomeTor Vergata, CUP E83C18000100006. ‡ This project has been partially supported by the project ’Transcendental Dynamics 1.5’ inside the program FIL-QuotaIncentivante of the University of Parma and co-sponsored by Fondazione Cariparma, and by Indam via the research groupGNAMPA. a r X i v : . [ m a t h . D S ] F e b L. AROSIO, A.M. BENINI, J.E. FORNÆSS, AND H. PETERS
The fact that transcendental functions in one complex variables always have infinite entropy was provedin the paper [BFP19] by the three last authors. However, after completing our paper we learned thatthis result was obtained earlier by Markus Wendt [Wen02, Wen05b, Wen05a], who never published thiswork. The proof we present in this paper will closely follow ideas from the proof of Wendt.1.1.
Outline of the proof.
Following Wendt we give different proofs depending on whether the familyof rescaled maps f n ( z ) := f ( n · z ) /n is quasi-normal or not (see Definition 2.8). If this family is quasi-normal, Wendt showed that f acts as a polynomial-like map of arbitrarily large degree on larger and largerdomains, hence has infinite entropy. Similarly, we show that F acts as a H´enon-like map of arbitrarilylarge degree, hence by Dujardin’s result F also has infinite entropy.When the family ( f n ) is not quasi-normal, Wendt shows that one can find an arbitrarily large numberof disks with pairwise disjoint closures, such that each of these disks contains a univalent preimage of allbut at most 2 of the disks; a consequence of the Ahlfors Five Islands Theorem [Ber00]. In the H´enonsetting, we prove similarly that any suitable graph over each of these disks contains a preimage of asuitable graph over all but at most 2 of the other disks. In both the quasi-normal and the non quasi-normal setting we obtain completely invariant compact subsets on which the entropy is arbitrarily large.It follows that the topological entropy is infinite.In section 2 we recall background on topological entropy, including the definition of entropy on non-compact spaces that we will use. We also discuss the notion of quasi-normality, and recall AhlforsFive-Islands Theorem and some of its consequences. In section 3 we prove Theorem 1.1, first under theassumption that the family ( f n ) is quasi-normal, and then under the assumption that the family is notquasi-normal. In section 4 we prove the existence of periodic cycles of any period at least 3. In section5 we construct examples of transcendental H´enon maps with arbitrarily slow or fast growing entropy interms of the size of the compact sets. Acknowledgment.
The result obtained here answers a question asked to us by both Romain Dujardinand Nessim Sibony. We are grateful for their suggestion, which stimulated this research. The proof ofour result closely follows the ideas of Markus Wendt in unpublished work. We are grateful for WalterBergweiler for bringing this work to our attention, and for further discussion on this topic.2.
Preliminaries
Entropy.
For maps acting on compact spaces the concept of topological entropy has been introducedin [AKM65].
Definition 2.1 (Definition of topological entropy for compact sets) . Let f : X → X be a continuousself-map of a compact metric space ( X, d ). Let n ∈ N and δ >
0. A set E ⊂ X is called ( n, δ ) -separated if for any z (cid:54) = w ∈ E there exists k ≤ n − d ( f k ( z ) , f k ( w )) > δ . Let K ( n, δ ) be the maximalcardinality of an ( n, δ )-separated set. Then the topological entropy h top ( X, f ) is defined as h top ( X, f ) := sup δ> (cid:26) lim sup n →∞ n log K ( n, δ ) (cid:27) . In the literature there are several non-equivalent natural generalizations for the definition of topologicalentropy on non-compact spaces (see for example [Bow73b], [Bow71], [Bow73a], [Hof74], and more recently[HNP08]). We will use the definition introduced by [CR05] which is smaller than or equal to all the onesmentioned above.
Definition 2.2.
Let f : Y → Y be a continuous self-map of a metric space ( Y, d ). Then the topologicalentropy h top ( Y, f ) is defined as the supremum of h top ( X, f ) over all forward invariant compact subsets X ⊂ Y. If there is no forward invariant compact subset the topological entropy is defined to be 0.
YNAMICS OF TRANSCENDENTAL H´ENON MAPS III: INFINITE ENTROPY 3
Remark 2.3.
Notice that this definition does not depend on the metric inducing the topology on Y ,and is invariant by topological conjugacy, hence the name “topological entropy” is justified. Notice alsothat in [REF] the last three named authors used a slightly different definition of topological entropy, apriori larger than or equal to the above one. Lemma 2.4.
Assume that the map f : Y → Y is injective. Then h top ( Y, f ) is equal to as the supremumof h top ( X, f ) over all completely invariant compact subsets X ⊂ Y. Proof.
Let X be a compact forward invariant subset of Y . Consider the compact set Λ := (cid:84) n ≥ f n ( X ) . Since f is injective, it follows that the map f | Λ : Λ → Λis bijective, and in particular Λ is completely invariant by f . The following classical result yields thelemma. (cid:3) Theorem 2.5.
Let g : K → K be a continuous self-map of a compact metric space ( K, d ) and let Λ := (cid:84) n ≥ g n ( K ) . Then h top ( K, f ) = h top (Λ , f ) . For the proof, see e.g. Block and Coppel.2.2.
Ahlfors Theorem and quasinormality.
The following is a version of Ahlfors five islands Theoremwhich can be found in [Ber00], Theorem A.1. A more classical formulation of Ahlfor’s five islands theoremand Corollary 2.7 in terms of regularly exhaustible Riemann surfaces can be found in [Sch93], Chapter1.9.
Theorem 2.6 (Ahlfors five islands Theorem) . Let D , . . . , D be Jordan domains on the Riemann spherewith pairwise disjoint closures and let D ⊂ C be a domain. Then the family of all meromorphic functions f : D → ˆ C with the property that none of the D j has a univalent preimage in D is normal. As observed in [Ber00] after the statement of Theorem B.3, if the functions are holomorphic on D andthe domains D i are bounded the number 5 can be replaced by 3. Corollary 2.7.
Let D , . . . , D k with k ≥ be bounded Jordan domains on the Riemann sphere withpairwise disjoint closures and let D ⊂ C be a domain. Let F be family of holomorphic functions f : D → ˆ C which is not normal in D . Then for all but at most 2 values of j , D j has a univalent preimage in D . We recall the definition of quasi-normality from the Appendix in [Sch93].
Definition 2.8.
Let Ω ⊂ C be a domain. A family F of holomorphic functions on Ω is quasi-normal iffor every sequence ( f n ) of functions in Ω there exists a finite set Q ⊂ Ω and a subsequence ( f n k ) of ( f n )which converges uniformly on compact subsets of Ω \ Q .The rest of this subsection is devoted to the proof of the following Proposition 2.9, which in turn willbe used in the proof of the not quasi-normal case. Proposition 2.9.
Let Ω ⊂ C be a domain and let F be a not quasi-normal family of holomorphicfunctions Ω → C . Then there exists a sequence ( f n ) ⊂ F and an infinite subset Q = ( x j ) j ≥ ⊂ Ω suchthat no subsequence of ( f n ) converges uniformly in any neighborhood of any x j . Lemma 2.10.
Let Ω ⊂ C be a domain and let F be a not quasi-normal family of holomorphic functions Ω → C . Then there exist a sequence ( f n ) in F with the following property: for every subsequence ( f n k ) ,there exists an infinite set E ( f n k ) ⊂ Ω such that ( f n k ) is not normal in any neighborhood of a point in E ( f n k ) . L. AROSIO, A.M. BENINI, J.E. FORNÆSS, AND H. PETERS
Proof.
Assume F is not quasi-normal. Then there exists a sequence ( f n ) in F such that for any finiteset L ⊂ Ω and every subsequence ( f n k ) of ( f n ), ( f n k ) does not converge uniformly on compact subsets inΩ \ L . For every subsequence ( f n k ), define E ( f n k ) as the set of all points x in Ω such that the sequence( f n k ) is not normal in any neighborhood of x . We just need to prove that E ( f n k ) is not a finite set. Ifby contradiction E ( f n k ) is a finite set, then for all points y ∈ Ω \ E ( f n k ), the sequence ( f n k ) is locallynormal around y . Since normality is a local property, it follows that ( f n k ) is normal on Ω \ E ( f n k ), andthus we can extract a subsequence of ( f n k ) converging on Ω \ E ( f n k ), which is a contradiction. (cid:3) Lemma 2.11.
Let Ω ⊂ C be a domain and let x ∈ Ω . If a sequence of holomorphic functions ( f n : Ω → C ) is not normal in any neighborhood of x , then we can extract a subsequence ( f n k ) with the property thatno subsequence of ( f n k ) converges uniformly in any neighborhood of x .Proof. Recall that a sequence ( f n ) is normal if and only if it is equicontinuous with respect to the sphericalmetric on the Riemann sphere. Since ( f n ) is not normal on any neighborhood of x , it follows that ( f n )is not equicontinuous in x . This means that there exists a constant ε > j there exist | x j − x | < /j and an integer n j such that d ( f n j ( x j ) , f n j ( x )) ≥ ε. But then the sequence ( f n j ) cannot have a subsequence converging uniformly in any neighborhood of x . (cid:3) Proof of Proposition 2.9.
Let ( f n ) be the sequence given by Lemma 2.10, and E ( f n ) be the associatednon-normality infinite set. Choose x ∈ E ( f n ). By Lemma 2.11 there exists a subsequence ( f n ( h ) ) of( f n ) such that every subsequence of ( f n ( h ) ) does not converge in any neighborhood of x .Let now E ( f n ( h ) )) be the infinite set given by Lemma 2.10 for the subsequence ( f n ( h ) ). Choose x ∈ E (( f n ( h ) )) different from x . By Lemma 2.11 there exists a subsequence ( f n ( h ) ) such that everysubsequence of ( f n ( h ) ) does not converge uniformly in any neighborhood of the points x , x . By in-duction we obtain an infinite set Q := ( x j ) j ≥ and a family (( f n k ( h ) )) k ≥ of nested subsequences of ( f n )such that for all k ≥ f n k ( h ) ) converges uniformly in any neighborhood of the points x , . . . , x k . The diagonal subsequence ( g h := f n h ( h )) gives the result. (cid:3) Proof of Theorem 1.1
Let F ( z, w ) = ( f ( z ) − δw, z ) be a transcendental H´enon map. For n ∈ N and z ∈ C let us define f n ( z ) := f ( nz ) n . Observe that for each n, f and f n are topologically conjugate via the map z (cid:55)→ nz , so they have the sameentropy. Analogously, the maps F n ( z, w ) = ( f n ( z ) − δw, z ) are topologically conjugate to F and hencehave the same entropy as F . Example 3.1.
For f ( z ) = e z the functions f n diverge on the right half plane, and converge to 0 on theleft half plane, thus ( f n ) is not quasi-normal in any neighborhood of any point on the imaginary axis.Consider a sequence of complex numbers ( a (cid:96) ) with | a (cid:96) | → ∞ and | a (cid:96) +1 /a (cid:96) | → ∞ , and define f ( z ) = (cid:89) (cid:96) ≥ (1 − z/a (cid:96) ) . Since the infinite product converges for every z by choice of the a (cid:96) , and since it is not a polynomial, f is a transcendental entire function. Notice that f n (0) →
0, that the zeros of f are { a (cid:96) } (cid:96) ≥ , and that thezeros of f n are Z n := { a (cid:96) /n } (cid:96) ≥ . YNAMICS OF TRANSCENDENTAL H´ENON MAPS III: INFINITE ENTROPY 5
Given any sequence in ( f n ) we can find a subsequence ( f n j ) for which the sets of zeros Z n j = { a (cid:96) /n j } (cid:96) ≥ converge as n j → ∞ to the set Z ∞ , which is either { , ∞} or { , ∞ , q } for some q ∈ C \ { } , in terms ofthe Hausdorff metric on the Riemann sphere.Indeed, if a sequence of zeros a (cid:96) j /n j accumulates on a point q (cid:54) = 0 , ∞ , then up to passing to asubsequence we may assume that a (cid:96) j /n j → q as j → ∞ . Since | a j +1 /a j | → ∞ it follows that as j → ∞ we have that a i j /n j tends to 0 whenever i j < (cid:96) j , and converges to ∞ whenever i j > (cid:96) j .Let us work with the case Z ∞ = { , ∞ , q } . Write f n j ( z ) as a product of three terms as follows: f n j ( z ) = n j (cid:89) (cid:96)<(cid:96) j (cid:18) − zn j a (cid:96) (cid:19) (cid:18) − zn j a (cid:96) j (cid:19) (cid:89) (cid:96)>(cid:96) j (cid:18) − zn j a (cid:96) (cid:19) . Observe that on any compact subset of C \ { , q } the second of these terms converges uniformly to thenon-zero function 1 − z/q , while the third term converges uniformly to the constant function 1. Thefirst term diverges uniformly, proving quasi-normality. In the case Z ∞ = { , ∞} one writes f n j ( z ) as aproduct of two terms, similarly obtaining locally uniform divergence on C \ { } .The proof of Theorem 1.1 is divided into two cases, with different proofs, depending on whether F := ( f n ) is a quasi-normal family or not. As mentioned in the introduction, the outline of our prooffollows Wendt’s proof [Wen02, Wen05b, Wen05a] for the one-dimensional case.3.1. Quasinormal Case.
In this subsection we prove the following result:
Theorem 3.2.
Let F : ( z, w ) (cid:55)→ ( f ( z ) − δw, z ) be a transcendental H´enon map, and suppose that thetranscendental functions defined by f n ( z ) = f ( nz ) /n form a quasi-normal family. Then F has infiniteentropy. For any r ∈ R let us denote by D r the Euclidean disk of radius r centered at 0. Let f be entiretranscendental and let F be the family of rescalings f n ( z ) = f ( nz ) /n . Assume that F is quasi-normal.Then there is a subsequence ( f n k ) of ( f n ) and a finite set Q such that ( f n k ) converges uniformly oncompact sets of C \ Q . Lemma 3.3.
The set Q contains the origin, and there exists < s < such that f n k → ∞ uniformlyon compact subsets of D s \ { } .Proof. Observe first that for every r >
0, any subsequence of ( f n ) is unbounded in the circle ∂ D r . Indeed,for any n we have that f n ( D / √ n ) = f ( D √ n ) /n , and the maximum modulus of a transcendental functionon a disk of radius r grows faster than r .We claim that ( f n k ) does not converge uniformly in a neighborhood of 0, so in particular, 0 ∈ Q .Indeed, f n k (0) = f (0) /n k → n k → ∞ , while ( f n k ) is unbounded in any neighborhood of 0. Since Q is finite we can find s such that f n k → g uniformly on compact subsets of D s \ { } , with g : D s \ { } → C or g = ∞ . Since ( f n k ) is unbounded in any circle ∂ D r we obtain g = ∞ . (cid:3) Proposition 3.4.
Let s, ( f n k ) be as in Lemma 3.3. Let < r < s , and let R > and m ∈ N . Thenthere exists k ∈ N such that for k > k we have(1) | f n k ( z ) | > R for every z ∈ ∂ D r ,(2) the winding number of the curve f n k ( ∂ D r ) around the origin is larger than or equal to m .Proof. (1) is an immediate consequence of Lemma 3.3. We now prove (2). Let a ∈ D R be a non-exceptional point for f . Fix m ∈ N , and let ρ = ρ ( m ) such that a has at least m preimages in D ρ under f . Let M such that f ( D ρ ) (cid:98) D M . It follows that there is a connected component W of f − ( D M ) whichcontains D ρ , and hence contains at least m preimages of a under f . L. AROSIO, A.M. BENINI, J.E. FORNÆSS, AND H. PETERS
Let k be large enough such that for all k ≥ k we have M/n k < R , and such that (1) holds. Let k ≥ k . Denote by W/n k the set { z/n k : z ∈ W } . Then if z ∈ W/n k we have n k z ∈ W and hence | f n k ( z ) | < R. Thus
W/n k ⊂ f − n k ( D R ). Notice that 0 ∈ W/n k . It follows by (1) that W/n k ⊂ D r .We now claim that W/n k contains at least m preimages of a k := a/n k under f n k . Indeed W containsat least m preimages of a under f , and for any such preimage z we have that f n k ( zn k ) = f ( z ) n k = a k . Since a k ∈ D R , the result follows by the argument principle. (cid:3) Let ∆ = D r × D r be a bidisk, ∂ v ∆ , ∂ h ∆ denote its vertical and horizontal boundary respectively.The following definition of H´enon-like maps is Definition 2.1 in [Duj04]. Definition 3.5 (H´enon-like map) . An injective holomorphic map H defined in a neighborhood of ∆ iscalled H´enon-like if(1) H (∆) ∩ ∆ (cid:54) = ∅ ;(2) H ( ∂ v (∆)) ∩ ∆ = ∅ ;(3) H (∆) ∩ ∂ ∆ ⊂ ∂ v (∆).Let π z , π w : C → C denote the projection to the z and to the w axis respectively. Definition 3.6 (Degree of a H´enon-like map) . Let H be a H´enon-like map defined in a neighborhood of∆ = D r × D r and let L h be any horizontal line intersecting ∆. Consider the holomorphic function π z ◦ H : H − (∆) ∩ ∆ ∩ L h → D r . (3.1)Then by condition (3) of Definition 3.5 we have that if ( z, w ) ∈ ∂ ( H − (∆) ∩ ∆ ∩ L h ), then H ( z, w ) ∈ ∂ v ∆,which means that the function in (3.1) is proper, and thus a branched covering. By Proposition 2.3 in[Duj04], its degree is independent of the chosen horizontal line. This integer is the degree of the H´enon-likemap H .The following theorem is proved in [Duj04, Theorem 3.1]. Theorem 3.7.
Let H be a H´enon-like map of degree d . The topological entropy of H is log d . Lemma 3.8.
Let f be a holomorphic function defined in a neighborhood of D r , let δ (cid:54) = 0 , and supposethat | f ( z ) | > ( | δ | + 1) · r whenever | z | = r . Assume that the winding number of the curve f ( ∂ D r ) aroundthe origin is d ≥ . Then the map F : ( z, w ) (cid:55)→ ( f ( z ) − δw, z ) is a H´enon-like map of degree d on ∆ = D r × D r .Proof. We check the three properties in Definition 3.5. The estimate | f ( z ) | > ( | δ | + 1) · r gives that | f ( z ) − δw | > r for all ( z, w ) ∈ ∂ v ∆, which implies property (2). The formula for F therefore impliesthat F (∆) cannot intersect ∂ h ∆, giving property (3). Since f ( ∂ D r ) winds around 0 exactly d ≥ a ∈ D r . Hence F ( a,
0) = (0 , a ) ∈ ∆ which gives Property (1).We now show that F has degree d on ∆. By Definition 3.6 it is enough to show that 0 ∈ D r has d preimages counted with multiplicity in F − (∆) ∩ ∆ ∩ L under π z ◦ F , where L is the horizontal linepassing through 0. It is easy to see that these points coincide with the preimages in D r of the originunder the function f , and the result follows by the argument principle since the curve f ( ∂ D r ) winds d times around 0. (cid:3) Proof of Theorem 3.2.
Recall that F n ( z, w ) := ( f n ( z ) − δw, z ) , and that F n is topologically conjugate to F for all n ≥ m ∈ N . Let s, ( f n k ) be as in Lemma 3.3 and fix r < s, R > ( | δ | + 1) r . Let k be given byProposition 3.4. Then, if k ≥ k , it follows by Lemma 3.8 that F n k is H´enon-like of degree at least m on YNAMICS OF TRANSCENDENTAL H´ENON MAPS III: INFINITE ENTROPY 7 the bidisk D r × D r . By Theorem 3.7 we have that the entropy of F n k is larger than or equal to log m ,and by topological invariance the same holds for the map F . (cid:3) Non Quasinormal Case.
We will now prove the following:
Theorem 3.9.
Let F : ( z, w ) (cid:55)→ ( f ( z ) − δw, z ) be a transcendental H´enon map, and suppose that thetranscendental functions defined by f n ( z ) = f ( nz ) /n do not form a quasi-normal family. Then F hasinfinite entropy. Proof of Theorem 3.9.
Assume that the family ( f n ) is not quasi-normal. Let ( f n h ) be the subse-quence of ( f n ) given by Proposition 2.9 and let Q = ( x j ) j ≥ be the associated infinite set. Fix k ≥
1. Let
R > D R ( x j ), for j = 1 , . . . , k are pairwise disjoint. Next define0 < r < R such that | δ | r < R − r . Recall that no subsequence of ( f n h ) is normal in any of the k disks D r ( x j ) , j = 1 , . . . , k . Lemma 3.10.
For a given n h , and for i, (cid:96) ∈ { , . . . , k } let J ( i, (cid:96) ) := { j ∈ { , . . . , k } : D R ( x j + δx (cid:96) ) admits a biholomorphic preimage under f n h in D r ( x i ) } . Then there exists n h such that J ( i, (cid:96) )) ≥ k − for every i, (cid:96) ∈ { , . . . , k } .Proof. Assume by contradiction that this is not the case. Then for all n h there exist i, (cid:96) ∈ { , . . . , k } and3 distinct values j , j , j ∈ , . . . , k such that the disks D R ( x j + δx (cid:96) ) , D R ( x j + δx (cid:96) ) , D R ( x j + δx (cid:96) ) donot admit biholomorphic preimages via f n h in the disk D r ( x i ). It follows that we can find a subsequence( f m h ) with the following property: there exist i, (cid:96) ∈ , . . . , k and 3 distinct values j , j , j ∈ { , . . . , k } such that for all m h the disks D R ( x j + δx (cid:96) ) , D R ( x j + δx (cid:96) ) , D R ( x j + δx (cid:96) ) do not admit biholomorphicpreimages via f m h in the disk D r ( x i ). By Ahlfors five islands Theorem (see Corollary 2.7) ( f m h ) is normalin D r ( x i ), which gives a contradiction. (cid:3) In what follows we denote the map f n h given by the previous lemma simply as f n . We will considerthe dynamics of the H´enon map F n ( z, w ) := ( f n ( z ) − δw, z ), which is linearly conjugate to F . Definition 3.11.
Let i, (cid:96) both lie in { , . . . , k } . A holomorphic disk D is called a ( i, (cid:96) )-disk if • it is a holomorphic graph over D r ( x i ), that is D can be parametrized as ( z, w ( z )) with w ( z )holomorphic in D r ( x i ); • π w ( D ) ⊂ D r ( x (cid:96) ), where π w is the projection to the second coordinate. Lemma 3.12.
Let i, (cid:96) ∈ { , . . . , k } . Then for all j ∈ J ( i, (cid:96) ) and for any ( i, (cid:96) ) -disk D there exists aholomorphic disk V ⊂ D for which F n ( V ) is a ( j, i ) -disk.Proof. It is clear that the w -coordinates of F n ( V ) are contained in D r ( x i ), regardless of the choice of V ⊂ D . We therefore merely need to find a holomorphic disk V ⊂ D such that F n ( V ) is a graph overthe disk D r ( x j ) in the z -coordinate. Since j ∈ J ( i, (cid:96) ) there is a biholomorphic preimage W ⊂ D r ( x i ) of D R ( x j + δx (cid:96) ) under f n . It follows that the function f n − δx (cid:96) : W → D R ( x j ) is a biholomorphism as well. Let z (cid:55)→ ( z, w ( z )) be the graph parametrization of D . We claim that there exists an open subdomain ˜ W ⊂ W such that f n ( z ) − δw ( z ) : ˜ W → D r ( x j ) is a biholomorphism. Once this is proved, setting V := D ∩ ( ˜ W × C )yields the result. Notice that up to shrinking R we can assume that f n − δx (cid:96) : W → D R ( x j ) is ahomeomorphism. For all z ∈ ∂W we have | ( f n ( z ) − δw ( z )) − ( f n ( z ) − δx (cid:96) ) | = | δ || x (cid:96) − w ( z ) | ≤ | δ | r < R − r by assumption, hence by Rouch´e’s Theorem it follows that for every u ∈ D r ( x j ) there exists exactlyone point z ∈ W such that f n ( z ) − δw ( z ) = u . Setting ˜ W := ( f n − δw ) − ( D r ( x j )) we have that f n − δw : ˜ W → D r ( x j ) is a biholomorphism. (cid:3) L. AROSIO, A.M. BENINI, J.E. FORNÆSS, AND H. PETERS
CC D r ( x i ) D r ( x (cid:96) ) D r ( x j ) D r ( x i ) D r ( x (cid:96) ) D r ( x j + δx (cid:96) ) F n D F n ( V ) V f n ( z ) − δw ( z ) f n ( z ) − δw ( z ) + δx (cid:96) ˜ W Figure 1.
Illustration of the statement and proof of Lemma 3.12. The disks D r ( x i ) arecontained in larger disks D R ( x i ), which do not appear in this picture. We conclude the proof of non quasi-normal case by showing that Lemma 3.12 implies that the topo-logical entropy of F n is at least log( k − C H := (cid:91) ≤ i,(cid:96) ≤ k D r ( x i ) × D r ( x (cid:96) ) , L := (cid:92) m ≥ F − mn ( H ) . Clearly L is forward F n -invariant. We say that a sequence ( i , i , i , . . . ) ∈ { , . . . k } N is admissible if i m +1 ∈ J ( i m , i m − ) for every m ≥ i , i , i , . . . ), there exists a point P ∈ L for which F mn ( P ) lies in a ( i m +1 , i m )-disk for all m ≥
0. Moreover for all m ≥ k · ( k − m − admissible words of length m .Thus L contains at least ( k − m points with distinct symbolic representations, which are therefore( m, ε )-separated as soon as ε < min i,(cid:96) dist( D r ( x i ) , D r ( x (cid:96) )) . This proves the claim that F n : L → L has topological entropy at least log( k − Periodic cycles of arbitrary order
We continue to a consider transcendental H´enon map F of the form( z, w ) (cid:55)→ ( f ( z ) − δw, z ) . In the previous paper [ABFP20] we showed that when δ = − F may not have any fixed point orperiodic orbits of period 2, but if F has neither, then it must have periodic points of order 4. The proofof this fact relied upon algebraic manipulations of the equation F ( z, w ) = ( z, w ). Using the techniquespresented in the previous sections we can now obtain the following description. Theorem 4.1.
A transcendental H´enon map has infinitely many periodic cycles of any order N ≥ . YNAMICS OF TRANSCENDENTAL H´ENON MAPS III: INFINITE ENTROPY 9
Proof.
We consider again the family of rescaled transcendental functions ( f n ). We have shown that ifthis sequence is quasi-normal then appropriate restrictions of the H´enon map F act as H´enon-like mapsof larger and larger degrees. It was proved by Dujardin in [Duj04], Proposition 5.7, that a H´enon-likemap of degree d has exactly d N points which are fixed under F N , counted with multiplicity. It followsthat if the family ( f n ) is quasi-normal then F has infinitely periodic cycles of any period.Let us therefore assume that the family ( f n ) is not quasi-normal and fix N ≥
3. Let k > N −
1, andlet f n h be the function given by Lemma 3.10. Since the subsequence ( n h ) plays no further role in thisproof, we will just write n instead of n h , and write as before F n := ( f n ( z ) − δw, z ) . Consider the ( i, (cid:96) )-disksconstructed in Definition 3.11, for i, (cid:96) = 1 , . . . , k . Recall from Lemma 3.12 that for any i, (cid:96) = 1 , . . . , k there exists a subset J ( i, (cid:96) ) ⊂ { , . . . , k } with J ( i, (cid:96) )) ≥ k − j ∈ J ( i, (cid:96) ), any( i, (cid:96) )-disk D i,(cid:96) contains a holomorphic disk V which F n maps onto an ( j, i )-disk. We first claim that thenumber of N -tuples ( i , i , . . . , i N − ) with distinct entries satisfying i j +1 ∈ J ( i j , i j − ) , j = 0 , . . . , N − , (where the indices are taken modulo N ) tends to infinity as k → ∞ . Indeed, the number of N -tupleswhose entries are all distinct over k symbols is k · ( k − · . . . · ( k − N + 1); on the other hand byLemma 3.12, the number of such N -tuples which violate the condition i j +1 ∈ J ( i j , i j − ) in at least oneindex is at most 2 N k · ( k − · . . . · ( k − N + 2). Hence the number of admissible sequences is at least k · ( k − · . . . · ( k − N + 2)( k − N + 1) → ∞ as k → ∞ . Notice that this counting argument breaks downfor N = 2, in agreement with the fact that there exists transcendental H´enon maps without periodicpoints of period 2.We will now argue that corresponding to any sequence { ( i , i ) , . . . , ( i N − , i ) } of length N which isperiodic in the sense discussed above we can find a periodic cycle of minimal period N .Observe that in the proof of Lemma 3.12 the holomorphic disk V ⊂ D is of the form D ∩ ( ˜ W × C ), where˜ W ⊂ W depends on D , but W is independent of the chosen ( i, (cid:96) )-disk D . Indeed, it is by constructiona simply connected domain W ∈ D r ( x i ) that is mapped univalently onto D R ( x j + δx (cid:96) ) by the function f n , hence it depends only on the three indices i, j, (cid:96) of the domain, the ( i, (cid:96) )-disk, and the codomain, the( j, i )-disk.It follows that having chosen the domain W , the intersection of the bidisk W × D r ( x (cid:96) ) with thepreimage F − n ( D r ( x j ) × D r ( x i )) is connected; a union of straight horizontal disks V w ⊂ W × { w } for w ∈ D r ( x (cid:96) ).Let us now consider the periodic sequence ( i , i , . . . , i N − ) discussed earlier, where each i j +1 ∈ J ( i j , i j − ). For each triple ( i j − , i j , i j +1 ) we select a disk W j ⊂ D r ( x i j ) as above, for j ≥ N we de-fine these sets inductively by W j = W j − N , obtaining a periodic sequence. We will consider the nestedsets ( W j × D r ( x i j − )) ∩ F − n ( W j +1 × D r ( x i j )) ∩ · · · ∩ F − mn ( W j + m × D r ( x i j + m − )) , and show that the intersection for all m ∈ N is a unique holomorphic disk which is a holomorphic graph D r ( x i j ) (cid:51) z (cid:55)→ ( ϕ ( z ) , z ) ∈ W j × D r ( x i j − ) , and which is actually the local stable manifold of a saddle periodic point.Define the compact and forward invariant setΓ := (cid:91) j =1 ,...,N (cid:92) m ≥ F − mn ( W j + m × D r ( x i j + m − )) . Let D be the intersection of a ( i j , i j − )-disk with W j × D r ( x i j − ). We know that the image F n ( D )contains a holomorphic graph over the disk D R −| δ | r ( x i j +1 ) ⊃⊃ D r ( x i j +1 ) . So the modulus of the annulus D \ F − n ( W j +1 × D r ( x i j )) is bounded away from zero. Applying thisobservation repeatedly and using the Gr¨oztsch Inequality we have that D ∩ Γ consists of a single point.Applying this argument to the trivial foliation of W j × D r ( x i j − ) consisting of disks D of the form { w = c } we immediately get that Γ ∩ ( W j × D r ( x i j − )) is a graph z (cid:55)→ ( ϕ ( z ) , z ) for some function ϕ : D r ( x i j ) → W j .We claim that the function ϕ is actually holomorphic. Recall that in the proof of Lemma 3.12 we canchoose the ratio between the radii r and R as large as we wish. The function f n maps W j univalentlyonto D R ( x i j +1 + δx i j − ). By applying Cauchy estimates to f − n from D R ( x i j +1 + δx i j − ) into D r ( x i j ) itfollows that | f (cid:48) n ( z ) | can be made arbitrarily large on the subset of W j that is mapped by f n onto D r + | δ | r ( x i j +1 + δx i j − ) ⊂⊂ D R ( x i j +1 + δx i j − ) . It follows that we may assume that the derivative | f (cid:48) n | is arbitrarily large on ( W j × D r ( x i j − )) ∩ ( F − n ( W j +1 × D r ( x i j ))) for every j .Recall that DF n ( z, w ) = (cid:18) f (cid:48) n ( z ) − δ (cid:19) , hence when | f (cid:48) n ( z ) | is sufficiently large the horizontal cone field C h containing the tangent vectors ( v , v )with | v | ≤ | v | is forward invariant. Let C v be the vertical cone field, given by the pullback under dF n of the constant vertical cone field defined by | v | ≥ | v | . It follows that C v is backwards invariant for anypoint in F n ( W j × D r ( x i j − )), and moreover, any non-constant tangent vector in C v is contracted by someuniform factor, while vectors in C h are uniformly expanded. Thus Γ is a hyperbolic forward invariantset by the cone criterion, and through every point ( z, w ) ∈ Γ there exists a stable manifold W s ( z, w ). Itimmediately follows that Γ ∩ ( W j × D r ( x i j − )) has to coincide with a local stable manifold, and thus thefunction ϕ is actually holomorphic.By the forward invariance of Γ we know that the holomorphic disk Γ ∩ ( W j × D r ( x i j − )) is mappedinto itself by F Nn . The existence of a saddle periodic orbit of period N follows.Since the maps F n are all conjugate to F it follows that F has infinitely many periodic cycles of anyorder N ≥ (cid:3) For polynomial H´enon maps saddle periodic points form a dense subset of the Julia set J = J + ∩ J − .While the periodic points constructed above in the not quasi-normal setting are all saddle points, it isunclear to the authors whether there also exist (infinitely many) saddle points of any order N ≥ Arbitrary Growth of entropy
In [Duj04], Dujardin constructed transcendental H´enon maps with infinite entropy by letting f ( z )be an entire function which, on suitable disks D i , is well approximated by polynomials of some degree d i → ∞ , to deduce that the corresponding H´enon map is H´enon-like on the bidiscs D i × D i of the samedegree d i . It follows that the H´enon map has topological entropy at least log d i → ∞ .The rate of the growth of entropy is then given by the relation between d i and the radii of the disks D i .In this section we show that the entropy of lacunary power series, i.e. power series with mostlyvanishing coefficients, can grow at any prescribed rate. We will first prove the statement for entirefunctions in one variable: YNAMICS OF TRANSCENDENTAL H´ENON MAPS III: INFINITE ENTROPY 11
Theorem 5.1.
Let h ( R ) be a continuous positive increasing function h : [0 , ∞ ) → [0 , ∞ ) with h (0) = 0 and lim R →∞ h ( R ) = ∞ . Then there exists an entire function f ( z ) and a sequence of radii R j (cid:37) ∞ sothat the topological entropy of f on D R j equals h ( R j ) . Lemma 5.2.
Let P ( z ) := az n with a (cid:54) = 0 and n ≥ . Let r > , set R := | a | r n , and assume that R/ > r . Let g : D r → C be a holomorphic function such that | g ( z ) | < R/ n for all z ∈ D r . Then thefunction defined as f := P + g , f : D r ∩ f − ( D R ) → D R is a polynomial-like map of degree n .Proof. The function f satisfies f ( ∂ D r ) ∩ D R/ = ∅ and by Rouch´e’s Theorem the winding number ofthe curve f ( ∂ D r ) around the origin is n . It follows that f : D r ∩ f − ( D R/ ) → D R/ is a proper map ofdegree n , and by the maximum principle every connected component of its domain is simply connected.To prove that it is polynomial-like it suffices to show that D r ∩ f − ( D R/ ) is connected. Notice that | f | > | z | > r/
2, hence all preimages of 0 under f are contained in D r/ , and hence all connectedcomponents of f − ( D R/ ) have to intersect D r/ . On the other hand, D r/ ⊂ f − ( D R/ ), hence there isonly one connected component of f − ( D R/ ) in D R as claimed. (cid:3) Recall that the entropy of a polynomial-like map of degree n is log n . It follows from the fact thatsuch maps are topologically conjugate (in fact, hybrid conjugate) to a true polynomial of degree d byDouady-Hubbard Straightening Theorem [DH85] in a neighborhood of their Julia set, or one can proveit directly as for polynomials following for example [Lju83]. Proof of Theorem 5.1.
We construct f as a lacunary series (cid:80) ∞ i =1 a i z n i with ( a i ) positive real numbers.Define g j := (cid:80) i (cid:54) = j a i z n i . By choosing a i , r i , n i appropriately we will ensure that for each j the monomial a j z n j = f − g j is the leading term on the circle of radius r j , in the precise way needed to apply Lemma 5.2.We will construct the series inductively, along with a sequence of radii ( r j ) such that for all integer j ≥ h ( r j ) = log n j ; (5.1) | g j ( z ) | ≤ r j n j , ∀ z ∈ D r j ; (5.2) a j r n j j > r j ; (5.3) a j ≤ − ( j +1) j/ . (5.4)By (5.4) the series converges to an entire function f . By (5.2),(5.3), and Lemma 5.2 we immediatelyobtain that the topological entropy of f on D r j equals log n j , which by (5.1) is equal to h ( r j ).We start setting a = 1 / r > h ( r ) = log( n ) for some integer n ≥
2. We will choose a , r , n such that a r n ≤ a r n n +1 , (5.5)and a r n ≤ a r n n +1 . (5.6)Consider all possible radii r > r for which h ( r ) is of the form log( n ) for some integer n . Set a := a r n − n / n +1 , which satisfies (5.5). Substituting in (5.6) we obtain (cid:18) r r (cid:19) n − n ≥ n + n +2 , which is satisfied once r (and hence n ) is chosen large enough. Notice that a = 1 / , hence (5.4)is satisfied, and similarly if r (and hence n ) is chosen large enough (5.3) is satisfied. Iterating thisprocedure yields the desired series. (cid:3) Corollary 5.3.
Let h, f be as in Theorem 5.1. Then the topological entropy of F ( z, w ) = ( f ( z ) − δw, z ) on D r j × D r j equals h ( r j ) for all j sufficiently large.Proof. In the proof of Theorem 5.1 we obtained a sequence of disks D r j with r j (cid:37) ∞ such that | f ( z ) | > ( | δ | + 1) · r j for | z | = r j and j sufficiently large, and that f ( z ) winds n j times around the origin as z runsaround the circle ∂ D r j . It follows from Lemma 3.8 that the restriction of F to the bidisk D r j × D r j is aH´enon-like map of degree n j , which by Theorem 3.7 implies that the topological entropy on D r j × D r j equals h ( r j ) for all j sufficiently large. (cid:3) References [ABFP19] Leandro Arosio, Anna Miriam Benini, John Erik Fornæss, and Han Peters,
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L. Arosio: Dipartimento Di Matematica, Universit`a di Roma “Tor Vergata”, Italy
Email address : [email protected] A.M. Benini: Dipartimento di Matematica Fisica e Informatica, Universit´a di Parma, IT.,
Email address : [email protected] H. Peters: Korteweg de Vries Institute for Mathematics, University of Amsterdam, the Netherlands
Email address : [email protected] YNAMICS OF TRANSCENDENTAL H´ENON MAPS III: INFINITE ENTROPY 13
J.E. Fornaess: Department of Mathematical Sciences, NTNU Trondheim, Norway
Email address ::