On the dimension of points which escape to infinity at given rate - PowerPoint PPT Presentation

On the dimension of points which escape to infinity at given rate under exponential iteration Krzysztof Barański University of Warsaw On geometric complexity of Julia sets II, 25 August 2020

This is a joint work with Bogusława Karpińska (Warsaw University of Technology)

Escaping set and Julia set Let f : C → C be a transcendental entire map. The escaping set is defined as I ( f ) = { z ∈ C : | f n ( z ) | → ∞ as n → ∞} , while the Julia set J ( f ) is J ( f ) = { z ∈ C : { f n } ∞ n = 1 is not a normal family in any nbhd of z } . • J ( f ) = ∂ I ( f ) (Eremenko 1989) • J ( f ) = I ( f ) for f ∈ B (Eremenko and Lyubich 1992) B = { maps with bounded set of critical and asymptotic values }

Dimension of escaping set and Julia set The exponential map is defined as E λ ( z ) = λ e z , z ∈ C , λ ∈ C \ { 0 } . • The Julia sets of exponential maps have Hausdorff dimension 2 (Mcmullen 1987) • Since then, many results on the dimension of J ( f ) , I ( f ) and their dynamically defined subsets (Bergweiler, Bishop, Karpińska, Kotus, Mayer, Osborne, Pawelec, Peter, Rempe-Gillen, Rippon, Rottenfußer, Rückert, Schleicher, Schubert, Sixsmith, Stallard, Urbański, Waterman, Zdunik, Zheng, Zimmer,...)

Various kinds of escaping • Fast escaping set (Bergweiler and Hinkkanen 1999) A ( f ) = { z ∈ I ( f ) : | f n + l ( z ) | ≥ M n f ( R ) , n ∈ N , for some l ≥ 0 } for M f ( r ) = max | z | = r | f ( z ) | . • Moderately slow escaping set (Rippon and Stallard 2011) 1 n log log | f n ( z ) | < ∞} M ( f ) = { z ∈ I ( f ) : lim sup n →∞ • Slow escaping set (Rippon and Stallard 2011) 1 n log | f n ( z ) | < ∞} L ( f ) = { z ∈ I ( f ) : lim sup n →∞ Theorem (Bergweiler, Karpińska, Stallard 2009, Rippon and Stallard 2014) Fast escaping set has Hausdorff dimension 2 for f ∈ B of finite order or ‘not too large’ infinite order.

Sets with prescribed escape rate For sequences a = ( a n ) ∞ n = 1 , b = ( b n ) ∞ n = 1 with 0 < a n ≤ b n let I a ( f ) = { z : | f n ( z ) | ≥ a n for large n ∈ N } , I b ( f ) = { z : | f n ( z ) | ≤ b n for large n ∈ N } , I b a ( f ) = { z : a n ≤ | f n ( z ) | ≤ b n for large n ∈ N } . Remark To guarantee that the sets are consideration are not empty, one usually assumes that the sequence a is admissible , which roughly means a n + 1 < M f ( a n ) .

Some results on I b a ( f ) • I b a ( E λ ) � = ∅ for every admissible sequence a = ( a n ) ∞ n = 1 with a n → ∞ and b n = ca n , c > 1 (Rempe 2006) • The same holds for arbitrary transcendental entire (or meromorphic) maps f (Rippon and Stallard 2011) • dim H ( I ( f ) ∩ I b ( f )) ≥ 1 for every transcendental entire map f in the class B and b n → ∞ (Bergweiler and Peter 2013) Remark The Julia sets of exponential maps contain hairs (Devaney and Krych 1984, Devaney and Tangerman 1986, Schleicher and Zimmer 2003). For exponential maps with an attracting fixed point the Julia set is the union of hairs together with their endpoints (Aarts and Oversteegen 1993). The hairs without endpoints are contained in the fast escaping set (Rempe, Rippon and Stallard 2010).

Results on I b a ( E λ ) In 2016 Sixsmith proved that dim H I b a ( E λ ) ≤ 1 for admissible sequences a = ( a n ) ∞ n = 1 with a n → ∞ and b n = ca n for c > 1. Moreover, he showed dim H I b a ( E λ ) = 1 in the following cases: (a) a n = c 1 R n and b n = c 2 R n , c 1 , c 2 > 0, R > 1 (b) a n = n (log + ) p ( n ) and b n = R n , where (log + ) p denotes the p -th iterate of log + , for p ∈ N , R > 1, (c) a n = e n (log + ) p ( n ) and b n = e e pn for p ∈ N , log a n + 1 (d) log( a 1 ··· a n ) = 0, b n = ca n for large c > 1. Remark In the cases (a)–(b) the sets I b a ( E λ ) are contained in the slow escaping set, while in the cases (c)–(d) they are in the moderately slow escaping set.

Remarks Points with bounded and unbounded trajectories Let K ( f ) = { z ∈ J ( f ) : { f n ( z ) } ∞ n = 1 is bounded } . • dim H ( K ( E λ )) > 1 (Karpińska 1999) • dim H ( J ( E λ ) \ I ( E λ )) ∈ ( 1 , 2 ) for hyperbolic exponential maps (Urbański and Zdunik 2003) • dim H ( K ( f )) > 1 for f ∈ B (B, Karpińska and Zdunik 2009) • dim H ( J ( f ) \ ( I ( f ) ∪ K ( f )) > 1 for f ∈ B (Osborne and Sixsmith 2016) Symbolic itineraries In 2006 Karpińska and Urbański computed the Hausdorff dimension of subsets of A ( E λ ) consisting of points whose symbolic itineraries (describing the imaginary part of f n ( z ) ) grow to infinity at a given rate. Possible values of dimension cover [ 1 , 2 ] .

Setup We consider non-autonomous exponential iteration E λ = ( E λ n ◦ · · · ◦ E λ 1 ) ∞ n = 1 for λ = ( λ n ) ∞ n = 1 ⊂ Λ N , where Λ ⊂ C \ { 0 } . We assume that Λ is a compact set in C \ { 0 } and set λ ∈ Λ | λ | , | λ | . λ min = inf λ max = sup λ ∈ Λ For a = ( a n ) ∞ n = 1 , b = ( b n ) ∞ n = 1 with 0 < a n ≤ b n we consider I b a ( E λ ) = { z : a n ≤ | E λ n ◦ · · · ◦ E λ 1 ( z ) | ≤ b n for large n ∈ N } . Remark The sequences a and b need not tend to infinity and need not be 1 n → ∞ . increasing. We only assume ( a 1 · · · a n )

Definition A sequence ( a n ) ∞ n = 1 is admissible , if a n > 100 λ max and a n + 1 ≤ | λ n + 1 | e qa n for large n and a constant q < 1. If a n → ∞ , then the condition reduces to a n + 1 ≤ e qa n , q < 1. We study the Hausdorff ( dim H ) and packing ( dim P ) dimension of the sets I b a ( E λ ) . Remark We have dim H ≤ dim P . Moreover, dim B ( I b a ( E λ ) ∩ D ( 0 , r )) = dim P I b a ( E λ ) for every large r > 0, where dim B denotes the upper box dimension (Rippon and Stallard 2005).

Theorem (B and Karpińska 2020) � 1 log b n + 1 � n a n + 1 If a n > 100 λ max for large n and lim inf = 0 , then a 1 · · · a n n →∞ dim H I b a ( E λ ) ≤ 1 . In particular, this holds provided log log b n + 1 n →∞ ( a 1 · · · a n ) 1 / n = ∞ a n + 1 lim and lim inf log( a 1 · · · a n ) < 1 n →∞ or log log b n + 1 n →∞ ( a 1 · · · a n ) 1 / n = ∞ a n + 1 lim sup and lim sup log( a 1 · · · a n ) < 1 . n →∞ Remark In Theorem 8 we can allow λ min = 0.

Theorem (B and Karpińska 2020) n →∞ ( a 1 · · · a n ) 1 / n = ∞ and b n ≥ ca n for If ( a n ) ∞ n = 1 is admissible, lim c > 1 , then n →∞ φ n ( x ) ≤ dim H I b 1 + inf x lim inf a ( E λ ) ≤ 1 + sup x lim inf n →∞ φ n ( x ) , n →∞ ψ n ( x ) ≤ dim P I b 1 + inf x lim sup a ( E λ ) ≤ 1 + sup x lim sup n →∞ ψ n ( x ) , where x = ( x 1 , x 2 , . . . ) ∈ [ a 1 , b 1 ] × [ a 2 , b 2 ] × · · · and min(log b 2 � a 2 , x 1 ) · · · min(log b n � log a n , x n − 1 ) φ n ( x ) = , log( x 1 · · · x n ) − log min(log b n + 1 a n + 1 , x n ) min(log b 2 a 2 , x 1 ) · · · min(log b n + 1 � � log a n + 1 , x n ) ψ n ( x ) = . log( x 1 · · · x n )

Corollary n →∞ ( a 1 · · · a n ) 1 / n = ∞ and b n ≥ ca n for If ( a n ) ∞ n = 1 is admissible, lim c > 1 , then log b 1 a 1 · · · log b n � � log a n 1 ≤ dim H I b a ( E λ ) ≤ 1 + lim inf , log( a 1 · · · a n − 1 ) + log + a n n →∞ log( b n + 1 / a n + 1 ) a 1 · · · log b n + 1 � log b 1 � log a n + 1 1 ≤ dim P I b a ( E λ ) ≤ 1 + lim sup . log( a 1 · · · a n ) n →∞ If, additionally, log b n + 1 a n + 1 ≤ Ca n for C > 1 ( e.g. if b n ≤ a C n ) for large n , then dim H I b a ( E λ ) = 1 , log b 1 a 1 · · · log b n + 1 � � log a n + 1 dim P I b a ( E λ ) ≥ 1 + lim sup . log( b 1 · · · b n ) n →∞

Corollary n →∞ ( a 1 · · · a n ) 1 / n = ∞ and Suppose ( a n ) ∞ n = 1 is admissible, lim b n ≥ ca n for c > 1 . bn + 1 log log = 0 , then dim H I b a ( E λ ) = dim P I b an + 1 (a) If lim a ( E λ ) = 1 . log a n n →∞ bn + 1 log log < 1 , then dim H I b an + 1 (b) If lim inf a ( E λ ) = 1 . log a n n →∞ bn + 1 log log ≥ 1 , then dim P I b an + 1 (c) If lim inf a ( E λ ) = 2 . log b n n →∞ bn + 1 log log > 1 , then dim H I b a ( E λ ) = dim P I b an + 1 (d) If lim inf a ( E λ ) = 2 . log b n n →∞ Remark The assertions (b)–(c) imply that if dim P I b a ( E λ ) < 2, then dim H I b a ( E λ ) = 1.

Moderately slowly escaping points Corollary 1 n = ∞ and (a) If a n > 100 λ max for large n , lim n →∞ ( a 1 · · · a n ) n →∞ (log b n ) 1 / n < ∞ , then dim H I b lim inf a ( E λ ) ≤ 1 . (b) If, additionally, ( a n ) ∞ n = 1 is admissible and b n ≥ ca n for c > 1 , then dim H I b a ( E λ ) = 1 . In particular, if ( a n ) ∞ n = 1 is admissible, b n ≥ ca n for c > 1 and I b a ( E λ ) is contained in the moderately slow escaping set 1 M ( E λ ) = { z ∈ I ( E λ ) : lim sup n log log | E λ n ◦ · · · ◦ E λ 1 ( z ) | < ∞} , n →∞ then dim H I b a ( E λ ) = 1 .

Points with exact growth rate Definition We say that the iterations of a point z ∈ C under E λ have growth n = 1 , if z ∈ I ca rate a = ( a n ) ∞ a / c ( E λ ) for some constant c > 1, i.e. a n c ≤ | E λ n ◦ · · · ◦ E λ 1 ( z ) | ≤ ca n for large n . Corollary n →∞ ( a 1 · · · a n ) 1 / n = ∞ , then (a) If a = ( a n ) ∞ n = 1 is admissible and lim the set of points with growth rate a has Hausdorff dimension 1 . (b) If a = ( a n ) ∞ n →∞ a n = ∞ , then the set of n = 1 is admissible and lim points with growth rate a has Hausdorff and packing dimension 1 .