This work is motivated by the paper Annular itineraries for entire - PowerPoint PPT Presentation

Annular Itineraries for C ∗ David Martí-Pete Dept. of Mathematics and Statistics The Open University — supervised by Phil Rippon and Gwyneth Stallard — Winter School on Kleinian Groups and Transcendental Dynamics Universität Bremen April 10, 2014

This work is motivated by the paper Annular itineraries for entire functions by Phil Rippon and Gwyneth Stallard (January 2013). P. Rippon and G. Stallard, Annular itineraries for entire functions , to appear in Trans. Amer. Math. Soc.; arXiv:1301.1328.

Sketch of the talk 1. Introduction to holomorphic self-maps of C ∗ 2. The escaping set 3. Symbolic dynamics and annuli covering results 4. Types of orbits 5. The fast escaping set

Holomorphic self-maps of C ∗ Let f : S ⊆ � C → S be holomorphic. By Montel’s thm., if J ( f ) � = ∅ then: ◮ S = � C = C ∪ {∞} , the Riemann sphere (rational functions); ◮ S = C , the complex plane (transcendental entire functions); ◮ S = C ∗ = C \ { 0 } , the punctured plane . Holomorphic self-maps of C ∗ were first studied in 1953 by H. Rådström. H. Rådström, On the iteration of analytic functions . Math. Scand. 1 (1953), 85–92. P. Bhattacharyya, Iteration of analytic functions . PhD Thesis (1969), University of London, 1969.

Holomorphic self-maps of C ∗ Let f : S ⊆ � C → S be holomorphic. By Montel’s thm., if J ( f ) � = ∅ then: ◮ S = � C = C ∪ {∞} , the Riemann sphere (rational functions); ◮ S = C , the complex plane (transcendental entire functions); ◮ S = C ∗ = C \ { 0 } , the punctured plane . Holomorphic self-maps of C ∗ were first studied in 1953 by H. Rådström. Theorem (Bhattacharyya 1969) Every transcendental holomorphic function f : C ∗ → C ∗ is of the form � � f ( z ) = z n exp g ( z ) + h ( 1 / z ) for some n ∈ Z and g , h non-constant entire functions. H. Rådström, On the iteration of analytic functions . Math. Scand. 1 (1953), 85–92. P. Bhattacharyya, Iteration of analytic functions . PhD Thesis (1969), University of London, 1969.

Holomorphic self-maps of C ∗ Let f : S ⊆ � C → S be holomorphic. By Montel’s thm., if J ( f ) � = ∅ then: ◮ S = � C = C ∪ {∞} , the Riemann sphere (rational functions); ◮ S = C , the complex plane (transcendental entire functions); ◮ S = C ∗ = C \ { 0 } , the punctured plane . Holomorphic self-maps of C ∗ were first studied in 1953 by H. Rådström. Theorem (Bhattacharyya 1969) Every transcendental holomorphic function f : C ∗ → C ∗ is of the form � � f ( z ) = z n exp g ( z ) + h ( 1 / z ) for some n ∈ Z and g , h non-constant entire functions. Later results are due to: L. Keen, J. Kotus, P. M. Makienko, A. N. Mukha- medshin, L. Fang, W. Bergweiler, I. N. Baker, P. Domínguez Soto, etc. H. Rådström, On the iteration of analytic functions . Math. Scand. 1 (1953), 85–92. P. Bhattacharyya, Iteration of analytic functions . PhD Thesis (1969), University of London, 1969.

Motivation: complexification of circle maps The complexification of the Arnol’d standard family F αβ ( θ ) = θ + α + β sin ( θ ) ( mod 2 π ) , θ ∈ R are transcendental hol. self-maps of C ∗ : � F αβ ( w ) = we i α e β ( w − 1 / w ) / 2 . α = 3 . 1 , β = 0 . 8 . α = 3 . 1 , β = 5. N. Fagella, Dynamics of the complex standard family . J. Math. Anal. Appl. 229 (1999), no. 1, 1–31.

Summary of results about holomorphic self-maps of C ∗ Relation with the lift F : C → C (i.e. exp ◦ F ≡ f ◦ exp): ◮ exp − 1 J ( f ) = J ( F ) . [Ber95] About the Julia set : ◮ J ( f ) has no compact component; [BD98] ◮ J ( f ) has either one or infinitely many components. [BD98] About the Fatou set : if f is not a Möbius transformation, ◮ the Fatou components are simply or doubly-connected and there is at most one doubly connected component (if f ( z ) �≡ kz n ); [Kee88] ◮ if A is doubly-connected comp. which is relatively compact in C ∗ , then A is either a Herman ring, pre-periodic, or wandering. [BD98] I. N. Baker and P. Domínguez, Analytic self-maps of the punctured plane . Complex Variables Theory Appl. 37 (1998), no. 1-4, 67–91. W. Bergweiler, On the Julia set of analytic self-maps of the punctured plane . Analysis 15 (1995), no. 3, 251–256. L. Keen, Dynamics of holomorphic self-maps of C ∗ . Holomorphic functions and moduli, Vol. I, Math. Sci. Res. Inst. Publ., vol. 10, Springer, 1988, 9–30.

The escaping set The escaping set was firstly studied for transcendental entire functions by A. Eremenko in 1989, I ( f ) := { z ∈ C : | f n ( z ) | → ∞} . Theorem (Eremenko 1989, Eremenko & Lyubich 1992) Let f be a transcendental entire function. Then, ◮ I ( f ) ∩ J ( f ) � = ∅ ; ◮ J ( f ) = ∂ I ( f ) ; ◮ all the components of I ( f ) are unbounded; ◮ if f ∈ B , then I ( f ) ⊆ J ( f ) . A. Eremenko, On the iteration of entire functions , Dynamical Systems and Ergodic Theory, Banach Center Publ. 23 (1989), 339-345. A. Eremenko, and M. Lyubich Dynamical properties of some classes of entire functions , Ann. Inst. Fourier (Grenoble) 42 (1992), 989–1020.

The escaping set in C ∗ If f : C ∗ → C ∗ with 0 , ∞ essential singularities, � � I ( f ) := { z ∈ C ∗ : � log | f n ( z ) | � → ∞} which contains I 0 ( f ) := { z ∈ C ∗ : | f n ( z ) | → 0 } , I ∞ ( f ) := { z ∈ C ∗ : | f n ( z ) | → ∞} and also for ( s n ) ∈ {− 1 , + 1 } N � � | f n + ℓ ( z ) | > 1 if s n = + 1 , I s ( f ) := z ∈ I ( f ) : ∃ ℓ ∈ N , . | f n + ℓ ( z ) | < 1 if s n = − 1 Theorem (Fang 1998, Baker, Domínguez & Herring 2001) Let f be a transcendental self-map of C ∗ , then ◮ J ( f ) ∩ I 0 ( f ) � = ∅ , J ( f ) ∩ I ∞ ( f ) � = ∅ ; ◮ J ( f ) = ∂ I 0 ( f ) = ∂ I ∞ ( f ) . I.N. Baker, P. Domínguez and M.E. Herring, Dynamics of functions meromorphic outside a small set , Ergodic Theory and Dynamical Systems 21 (2001), no. 3, 647–672. Fang L., On the Iteration of Holomorphic Self-Maps of C ∗ , Acta Mathematica Sinica, New Series 14 (1998), no. 1, 139–144.

Different rates of escape f ( z ) = z + 1 + e − z : Cantor bouquet (blue) & Baker domain (red).

The maximum and minimum modulus Let f be a holomorphic self-map of C ∗ and R > 0, the functions M ( R ) = max | z | = R | f ( z ) | < + ∞ , m ( R ) = min | z | = R | f ( z ) | > 0 are continuous and unimodal.

The maximum and minimum modulus Let f be a holomorphic self-map of C ∗ and R > 0, the functions M ( R ) = max | z | = R | f ( z ) | < + ∞ , m ( R ) = min | z | = R | f ( z ) | > 0 are continuous and unimodal. Main properties : log M ( R ) i) → ∞ as R → ∞ ; log R ii) log M ( R ) is a convex function of log R ; iii) ∃ R 0 > 0 such that M ( R k ) � M ( R ) k for every R � R 0 and k > 1; iv) if k > 1, M ( kR ) M ( R ) → ∞ as R → ∞ .

The maximum and minimum modulus Let f be a holomorphic self-map of C ∗ and R > 0, the functions M ( R ) = max | z | = R | f ( z ) | < + ∞ , m ( R ) = min | z | = R | f ( z ) | > 0 are continuous and unimodal. Main properties : log M ( R ) i) → ∞ as R → ∞ ; log R ii) log M ( R ) is a convex function of log R ; iii) ∃ R 0 > 0 such that M ( R k ) � M ( R ) k for every R � R 0 and k > 1; iv) if k > 1, M ( kR ) M ( R ) → ∞ as R → ∞ . The analogous properties hold at 0, and similarly for m ( R ) .

Defining an annular partition Let R + be big enough such that for all R > R + , M ( R ) > R , and let R − be small enough such that for all R < R − , m ( R ) < R . Define A 0 := A ( R − , R + ) = { z ∈ C ∗ : R − < | z | < R + } and, for every n > 0, A n := A ( M n − 1 ( R + ) , M n ( R + )) , A − n := A ( m n ( R − ) , m n − 1 ( R − )) .

Itineraries The annular itinerary of z ∈ C ∗ is a 0 a 1 a 2 . . . , with a n ∈ Z , and f n ( z ) ∈ A a n , for all n � 0 .

Itineraries The annular itinerary of z ∈ C ∗ is a 0 a 1 a 2 . . . , with a n ∈ Z , and f n ( z ) ∈ A a n , for all n � 0 . What kind of sequence can be realized as an itinerary? By construction, a n + 1 � a n + 1 if a n > 0 , a n + 1 � a n − 1 if a n < 0 . because of the Maximum Modulus Principle.

Constructing orbits with a prescribed itinerary We will use the following well-known result: Lemma Let C m , m � 0, be compact sets in C and f continuous such that f ( C m ) ⊇ C m + 1 , for m � 0 . Then ∃ ξ such that f m ( ξ ) ∈ C m , for m � 0. We are going to construct a sequence of annuli B n ⊂ A n such that ◮ f ( B n ) ⊇ B n + 1 if n > 0; ◮ f ( B n ) ⊇ B n − 1 if n < 0.

Annuli covering lemma Theorem (Bergweiler, Rippon & Stallard 2011) There exists an absolute constant δ > 0 such that if f : A ( R , R ′ ) → C ∗ is analytic, where R ′ > R, then for all z 1 , z 2 ∈ A ( R , R ′ ) such that ρ A ( R , R ′ ) ( z 1 , z 2 ) < δ and | f ( z 2 ) | � 2 | f ( z 1 ) | , we have f ( A ( R , R ′ )) ⊃ A ( | f ( z 1 ) | , | f ( z 2 ) | ) . W. Bergweiler, P. Rippon and G. Stallard, Multiply connected wandering domains of entire functions , to appear in Proc. London Math. Soc.; arXiv:1109.1794

Construction of the annuli B n (for n > 0) Let µ ( R ) := ε M ( R ) for ε > 0. For large enough values of R , µ n ( R ) � M n ( ε R ) for n ∈ N . P. Rippon and G. Stallard, Fast escaping points of entire functions , Proc. London Math. Soc. (3) 105 (2012) 787–820.