Fix f ∈ F . Since the immediate attraction basin B = B ( z 0 ) of an attracting fixed point z 0 is simply connected, there exists a bounded simply connected domain D ⊂ ℂ , such that its closure D ¯ ⊂ B and boundary ∂ D is an analytic Jordan curve. Moreover, Sing ( f − 1 ) ¯ ⊂ D and f ( D ) ¯ ⊂ D , for more details see ( [8] , Lemma 3.1). Following [8] , the pre-images of ℂ \ D ¯ by f consists of countably many unbounded connected components called tracts of f . We denote the collection of all these tracts by R .

Since the closure of each tract is simply connected, there exists an open simple arc α : ( 0, ∞ ) → ℂ \ D ¯ , which is disjoint from the union of the closures of all tracts and such that α ( t ) tends to a point of ∂ D as t tends to 0 + , and α ( t ) tends to ∞ as t tends to + ∞ . We use this curve to define the fundamental domains on each tract as follows: since for every T ∈ R the map f | T is a cover of ℂ \ D ¯ , we have T \ f − 1 ( α ) is the union of infinitely many disjoint simply connected domains S such that the function

f | S : S → ℂ \ ( D ¯ ∪ α )

is bijective. Given T ∈ R , we denote by S T the collection of connected components of T \ f − 1 ( α ) . The elements of

S : = ∪ T ∈ R     S T (1)

are called fundamental domains.

For each S ∈ S , we have that the restriction f | S is univalent, so we denote its inverse branch by g S : = ( f | S ) − 1 : ℂ \ ( D ¯ ∪ α ) → S . For n ≥ 1 and each j ∈ { 0,1, ⋯ , n } denote by S j an element of S and put g S 0 S 1 ⋯ S n =   g S 0 ∘ ⋯ ∘ g S n . Then,

g S 0 ⋯ S n ( ℂ \ ( D ¯ ∪ α ) ) = { z ∈ ℂ : f j ( z ) ∈ S j , for   every   j = 0, ⋯ , n } . (2)

For each sequence S _ = ( S 0 S 1 ⋯ ) ∈ S ℕ , let K S _ : = ∩ n = 0 ∞   g S 0 S 1 ⋯ S n ( ℂ \ ( D ¯ ∪   α ) ) . Then, the Julia set of f is given by the disjoint union of K S _ , that is

J ( f ) = ⊔ S _ ∈ S ℕ     K S _ .

Since f has finite order and of disjoint-type, following [18] , the Julia set J ( f ) is a Cantor bouquet, that is a union of uncountably many pairwise disjoint curves tending to infinity (hair) and each curve is attached to the unique point accessible from the immediate basin B, called the endpoint of the hair. More precisely, either K S _ is empty or there is a homeomorphism h S _ : [ 0, + ∞ ) → K S _ such that lim t → + ∞ h S _ ( t ) =   ∞ , and such that for every t > 0 we have lim n → + ∞ f n ( h S _ ( t ) ) = ∞ . In the latter case z S _ : = h S _ ( 0 ) is the only point of K S _ accessible1 from the immediate basin B. See also [7] , which generalizes previous results for the exponential map having an attracting fixed point of [19] .

Following as in [10] , let us consider ρ f be the order of f and α 1 , α 2 > 0 be the corresponding constants of the rapid derivative growth condition of f . Fix τ ∈ ( 0, α 2 ) and let γ : ℂ \ { 0 } → ℝ ∪ { ∞ } defined by γ ( z ) = 1 | z | τ . Let θ be the Riemaniann metric on ℂ \ { 0 } defined by

d θ ( z ) = γ ( z ) | d z | ,

and we derive f with respect to θ instead of the Euclidean metric. So, for each z ∈ ℂ \ { 0 } we have

| f ′ ( z ) | θ = | f ′ ( z ) | γ ∘ f ( z ) γ ( z ) = | f ′ ( z ) | | z | τ | f ( z ) | τ . (3)

We consider C , the set of functions ψ from ∪ S ∈ S     S to ℝ + that are bounded from above and are constant on each element of S . That is, we write

C : = { ψ : ∪ S ∈ S     S → ℝ + : ψ   isboundedfromaboveandconstantovereach   S ∈ S } . (4)

We define the following class of potentials for f :

P f = { φ ψ , t ( z ) = log ψ ( z ) − t log | f ′ ( z ) | θ , ψ ∈ C , t > ρ f α 1 + τ } .

Observe that this class contains potentials − t log | f ′ | θ , which from (3) are cohomologous to − t log | f ′ | .

For each f ∈ F we denote by T f the class of tame potentials defined in [10] , that is

T f : = { φ = h − t log | f ′ | θ ; h   isboundedweaklyH o ¨ lderfunction ,   t > ρ ′ α 1 + α 2 }

Note that the class F does not include most of the functions considered in [10] . However, the class of potentials P f determined from f ∈ F intersects the class of tame potentials, in which the difference is non-empty. So our first result is the following

Theorem 1. There exists f ∈ F such that P f ∩ T f ≠ ∅ and P f \ T f ≠ ∅ .

Let Σ ℤ : = { s _ = ( s 0 s 1 ⋯ ) : s j ∈ ℤ , for   all   j ≥ 0 } be the full shift space, and the shift metric is defined as follows, for some θ ∈ ( 0,1 ) ,

d ( s _ , t _ ) = θ inf { k : s k ≠ t k } ∪ { ∞ } . (5)

For every n ≥ 1 , we denote a finite word s 0 ⋯ s n − 1 in ℤ n simply by s * , so we follows the following notation for cylinders

[ s * ] = { w _ ∈ Σ ℤ : w i = s i ,0 ≤ i ≤ n − 1 }

and for s ∈ ℤ , we simply denote [ s ] = { w _ ∈ Σ ℤ : w 0 = s } . Let σ : Σ ℤ → Σ ℤ be the left-sided shift map, given by σ ( s 0 s 1 ⋯ ) = ( s 1 s 2 ⋯ ) .

Observe that by definition the set S given in (1) is countably infinite, so we identify S with ℤ . Put

X : = { S _ ∈ S ℕ : K S _ ≠ ∅ } ⊆ Σ ℤ . (6)

Let

Z = ∪ S _ ∈ X     K S _ .

From (2), we have for each S _ ∈ S ℕ , f ( K S _ ) = K σ ( S _ ) , then the function f on the Julia set J ( f ) is semi-conjugate to σ on X, however f on the set

E P : = { z S _ = h S _ ( 0 ) : S _ ∈ X } ,

is conjugate to σ | X . Hence the set X is completely σ -invariant.

The set E P defined above, is the set of endpoints of hairs K S _ and it satisfies the following properties, it is the set of accessible points from the immediate attraction basin B. It is totally disconnected, however E P ∪ { ∞ } is connected, see [14] . Moreover, following [20] , the Hausdorff dimension of this set is equal to two, generalizing previous results of Karpińska [3] for the exponential map f λ ( z ) = λ e z with parameters λ ∈ ( 0, 1 / e ) . This exponential map is probably the best-known example in the family F , its Julia set is a Cantor bouquet and the set of endpoints is modeled by the symbolic space of all allowable sequences, see [19] and [21] .

In the following, we state some properties concerning the dynamics ( X , σ | X ) , endowed with a metric inherited from the euclidean metric on J ( f ) . It does not necessarily generate the topology induced by the cylinder sets.

Let H : X × [ 0, + ∞ ) → J ( f ) , and H | X × { 0 } : X × { 0 } → E P defined by H ( s _ ,0 ) = h s _ ( 0 ) , we have that H induces a metric ρ on X,

ρ ( s _ , w _ ) : = | h s _ ( 0 ) − h w _ ( 0 ) | .

The shift map σ : X → X is continuous with respect to ρ .

Given s _ ∈ Σ ℤ and w * ∈   ℤ n let us write w * s _ = ( w 0 ⋯ w n − 1 s 0 s 1 ⋯ ) . For a set A ⊂ Σ ℤ write w * A = { w * s _ : s _ ∈ A } . For s _ ∈ X and δ > 0 we define the following sets with respect to the metric ρ .

B ( s _ , δ ) : = { w _ ∈ X : ρ ( s _ , w _ ) < δ } = { w _ ∈ X : | h s _ ( 0 ) − h w _ ( 0 ) | < δ } ,

B ¯ ( 0 _ , δ ) : = { s _ ∈ X : ρ ( s _ , 0 _ ) ≤ δ } .

B 0 ( s _ , δ ) : = { w _ ∈ X : ρ ( s _ , w _ ) < δ & b 0   =   a 0 } .

For every n ≥ 1 and s _ ∈ X define

B n ( s _ , δ ) : = { w _ ∈ X : σ j ( w _ ) ∈ B 0 ( σ j ( s _ ) , δ ) , for   all   j = 0,1, ⋯ , n } .

The set X endowed with the metric ρ is non-compact, however, it can be approximated by an increasing sequence of compact and invariant subsets. Indeed, for all N ≥ 1 , define

Σ N : = { s _ = ( s 0 s 1 ⋯ ) ∈ X : for   j ≥ 0, s j ∈ { − N , ⋯ , N } } ,

so, the following holds

Theorem 2.

1) For all N ≥ 1 , Σ N ⊂ X , Σ N is compact with respect to ρ and invariant by σ . Moreover, for each compact subset Λ of X with respect to the metric ρ , so that σ ( Λ ) ⊂ Λ , we have, there exists N 0 ≥ 1 , such that Λ ⊂ Σ N 0 .

2) There exists δ 0 such that the following condition holds:

There exist C > 0 and λ > 1 such that for every n ∈ ℕ and s _ , t _ ∈ X and u * ∈ ℤ n , if ρ ( s _ , t _ ) < δ 0 then we have

ρ ( u * s _ , u * t _ ) ≤ C λ − n ρ ( s _ , t _ ) .

3) For every R > 0 there exists n ≥ 1 such that for every s _ ∈ B ( 0 _ , R ) , we have σ n ( B ( s _ , δ 0 ) ) ⊃ B ( 0 _ , R ) . Thus ( X , σ ) is topologically mixing.

4) The set ∪ N ≥ 1     Σ N is dense in X.

We recall what a conformal measure means; consider a measurable endomorphism T : Y → Y on a measurable space ( Y , B ) and a measurable non-negative function g on Y. A measure m on ( Y , B ) is called g-conformal for T on g if for all measurable set A which T ( A ) is measurable and T | A is invertible we have

m ( T ( A ) ) = ∫ A     g d m . (7)

Observe that (7) implies that m ∘ T is absolutely continuous with respect to m on the σ-algebra B ∩ A , for every set A ∈ B such that T : A →   T ( A ) is a measurable isomorphism.

Corollary 1. Let f ∈ F . Then for every potential ϕ ∈ P f we have the following properties.

1) The topological pressure P ( ϕ ) : = lim n → ∞ 1 n log ∑ z ∈ f − n ( w ) exp ( ∑ j = 0 n − 1 ϕ ∘ f j ( z ) ) . exists and is independent of w ∈ J ( f ) .

2) There exists a unique e P ( ϕ ) − ϕ -conformal measure ν ϕ of f .

3) There exists a unique probability Gibbs state μ ϕ . That is, μ ϕ is f-invariant and equivalent to ν ϕ . Moreover, both measures are ergodic and supported on the radial Julia set J r ( f ) , where

J r ( f ) = { z ∈ J ( f ) : lim n → ∞ f n ( z ) < ∞ } .

4) The density ρ ϕ = d μ ϕ / d ν ϕ is a nowhere vanishing continuous and bounded function on the Julia set J ( f ) .

Consider the exponential family { f λ ( z ) = λ e z , λ ∈ ( 0, 1 / e ) } . Each f λ belongs to F , because it has order equal to 1, satisfies the rapid derivative growth condition with α 1 = 0 and α 2 = 1 , and since 0 is the only singular value of f λ . Thus this map is hyperbolic. Moreover the potentials − t log | z | = − t log | f ′ λ ( z ) | + log γ 1 − log γ 1 ∘ f λ , where γ 1 = | z | − t , are tame potentials and also belong to the class P f λ .

On the other hand, let D be the open unit disk in ℂ , then

f λ ( { z : Re z < ln ( 1 λ ) } ) = D \ { 0 } ,

and since 1 < ln ( 1 λ ) we have f λ ( D ) ¯ ⊂ D . Moreover, since the immediate basin B of the attracting fixed point is the only Fatou component of f λ we have D ¯ ⊂ B .

Since f λ − 1 ( ℂ \ D ¯ ) = { z : Re z > ln ( 1 λ ) } , the only tract of f λ is the half plane T = { z : Re z > ln ( 1 λ ) } . Let us consider the ray α : ( 0, ∞ ) → ℂ \ D ¯ defined by α ( t ) = − ( 1 + t ) , then

f λ − 1 ( α ( 0 , ∞ ) ) = ∪ k ∈ ℤ { x + ( 2 k − 1 ) π i : x > ln ( 1 λ ) } ,

and for each k ∈ ℤ , put S k : = { z : Re z > ln ( 1 λ ) , ( 2 k − 1 ) π < Im z < ( 2 k + 1 ) π } . Then T \ f λ − 1 ( α ( t ) ) is the disjoint union of the fundamental domains S k .

Following [6] , let c : J ( f λ ) → ℝ + be a function such that for each k ∈ ℤ , this function is constant on J ( f λ ) ∩   ( S − k ∪   S k ) and we denote by c k its value on this set. Furthermore we assume that the sequence ( c k ) k ∈ ℤ of positive numbers satisfies

lim k → ∞ log c k log k = − ∞ . (8)

Define φ ( z ) : = log ( c ( z ) | z | − t ) , where t > 0 , c ( z ) = c k if z ∈ S − k ∪ S k and the sequence ( c k ) k ∈ ℤ satisfies (8). Observe that any potential as above φ ( z ) = log ( c ( z ) | z | − t ) satisfies lim k → + ∞ c k → 0 , so, φ is not a tame potential, however, this belongs to the class P f λ since the function c is bounded on each S k and | f ′ λ ( z ) | θ = | z | .

1) The classical Denjoy-Carleman-Ahlfors Theorem [22] implies that transcendental entire functions of finite order have only a finite number of tracts. We will assume for simplicity that for f ∈ F there is only one tract T, and there is no complication in the generalization to a finite number of tracts.

Let N ≥ 1 and denote by S N the union of 2 N + 1 fundamental domains in T, that is

S N = ∪ k = − N N     S k

and define

K N : = { z ∈ S N : for   every   j ≥ 0, f j ( z ) ∈ S N } .

Let ( s _ m ) m ≥ 1 be a sequence in Σ N . Taking a subsequence if it is necessary, one can assume that for some R large enough, the subsequence ( s _ m ) m ≥ 1 is contained in Σ N ∩ B ( 0 _ , R ) . So for every m ≥ 1 there is there is an endpoint z s _ m = h s _ m ( 0 ) ∈ K N ∩ B ( z 0 _ , R ) . Since K N ∩ B ( z 0 _ , R ) is bounded and K N is closed in J ( f ) we have there is a subsequence converging to some point z ∈ K N . Let s _ be the itinerary associated to z, then s _ ∈ Σ + N and ρ ( s _ m j , s _ ) = | z s _ m j − z | → 0 , j → ∞ .

On the other hand, let Λ be a compact subset of X with respect to the metric ρ with σ ( Λ ) ⊂ Λ . Let s _ ∈ Λ and let z ∈ H ( Λ ) with itinerary s _ , then since the compact subset H ( Λ ) intersects only a finite numbers of tracts (see ( [8] , Lemma 3.2)), there exists N 0 ≥ 1 such that for every j ≥ 0 we have | s j | ≤ N 0 . Therefore Λ ⊂ Σ N 0 .

2) Follows from the derivative grown condition and the uniformly expanding property of f , see ( [10] , Proposition 4.4).

3) This a standard fact described in ( [10] , Lemma 4.2). However, we include a short proof. That is, for each R > 0 there exists n ≥ 1 such that for every w ∈ B ( z 0 _ , R ) , f n ( B ( w , δ 0 ) ) ⊃ B ( z 0 _ , R ) , then the property follows. Let s _ , t _ ∈ X and U = B ( s _ , δ 1 ) , V = B ( t _ , δ 2 ) , for δ 1 , δ 2 > 0 . By expanding property in Part 2, follows there is n 1 > 0 such σ n 1 ( U ) ⊇ B ( σ n 1 s _ , δ 0 ) . Let R be large enough such that σ n 1 s _ , t _ ∈ B ( 0 _ , R ) and there is N 1 such that

σ N 1 ( B ( σ n 1 s _ , δ 0 ) ) ⊃ B ( 0 _ , R ) .

So, σ N 1 + n 1 ( U ) ⊃ σ N 1 B ( σ n 1 s _ , δ 0 ) ⊃ B ( 0 _ , R ) . Taking m > N 1 + n 1 , we have for every k ≥ m , σ k ( U ) ∩ V ≠ ∅ .

4) This property can be inferred from the general property of the density of periodic sources in the Julia set, as referred in ( [23] , Theorem 4). However, we include here a short proof: Let s _ = ( s 0 s 1 ⋯ ) ∈ X and ε > 0 . Then, there exists n 1 > 0 such that

B ( σ n 1 s _ , δ 0 ) ⊆ σ n 1 B ( s _ , ε ) .

It is enough to take n 1 > 0 such that C λ − n 1 δ 0 < ε . So, we have

s 0 s 1 ⋯ s n 1 − 1 B ( σ n 1 s _ , δ 0 ) ⊆ B ( s _ , C λ − n 1 δ 0 ) ⊆ B ( s _ , ε ) .

Hence, B ( σ n 1 s _ , δ 0 ) ⊆ σ n 1 B ( s _ , ε ) . Since X = ∪ R > 0     B ( 0 _ , R ) , then for some R > 0 we have σ n 1 s _ ∈ B ( 0 _ , R ) . Moreover from Part 3, there is n 2 > 0 such that B ( 0 _ , R ) ⊂ σ n 2 B ( σ n 1 s _ , δ 0 ) . Therefore, for n = n 1 + n 2 we follow that B ( 0 _ , R ) ⊂ σ n B ( s _ , ε ) . Hence, the set σ n B ( s _ , ε ) contains the sequence 0 _ = 00 ⋯ . Let w * = w 0 ⋯ w n − 1 ∈ ℤ n such that w * B ( 0 _ , R ) ⊂ B ( s _ , ε ) , then w 0 ⋯ w n − 1 0 _ ∈ B ( s _ , ε ) . Then, taking N : = max { | w 0 | , ⋯ , | w n − 1 | } we conclude w 0 w 1 ⋯ w n 0 _ ∈ Σ N .

Let CB ( J ( f ) , ℝ ) be denote the Banach space of bounded continuous functions on J ( f ) . For each potential φ ∈ P f , the transfer operator associated to φ and denoted by L φ acts continuously on CB ( J ( f ) , ℝ ) . So for each ψ ∈ CB ( J ( f ) , ℝ ) ,

L φ ψ ( z ) = ∑ f ( w ) = z     ψ ( w ) exp ( φ ( w ) ) .

To prove Corollary 1 one can adapt with minor modifications the approach given in [10] on the thermodynamic formalism for a large class of hyperbolic meromorphic functions f :   ℂ → ℂ ¯ of finite order ρ ′ satisfying a rapid growth condition and for a class of tame potentials, to the family of transcendental entire maps and potentials under study. Therefore, following as in [10] , we have that the following proposition remains valid for potentials ϕ c with c being only bounded from above.

Proposition 1. Given f ∈ F , we have each potential φ c , t ∈ P f satisfies the following properties.

1) sup w _ ∈ X L φ c , t ∘ H ( 1 l ) ( w _ ) < ∞

2) lim R → ∞ ∑ s ∈ ℤ exp ( sup w _ ∈ [ s ] ∩ ( X \ B ( 0 _ , R ) ) φ c , t ∘ H ( w _ ) ) = 0

3) lim z w _ → ∞ L φ c , t ∘ H 1 l ( w _ ) = 0

Proof. Let φ c , t = log c − t log | f ′ | θ be a potential in P f and ψ ∈ CB ( J ( f ) , ℝ ) ,

L φ c , t ∘ H ψ ∘ H ( w _ ) = ∑ σ ( s _ ) = w _     ψ ( z s _ ) c ( z s _ ) | f ′ ( z s _ ) | θ − t = ∑ σ ( s _ ) = w _     ψ ( z s _ ) c ( z s _ ) | f ′ ( z s _ ) | − t | z s _ | − τ t | f ( z s _ ) | τ t = | z w _ | τ t ∑ σ ( s _ ) = w _     ψ ( z s _ ) c ( z s _ ) | f ′ ( z s _ ) | − t | z s _ | − τ t .

Since f satisfies the derivative growth condition, we have

L φ c , t ∘ H ( 1 l ) ( w _ ) ≤ κ t | z w _ | τ t ∑ σ ( s _ ) = w _     c ( z s _ ) | z s _ | − α 1 t | f ( z s _ ) | − α 2 t | z s _ | − τ t = κ t | z w _ | τ t ∑ σ ( s _ ) = w _     c ( z s _ ) | z s _ | − α 1 t | z w _ | − α 2 t | z s _ | − τ t ≤ κ t | z w _ | t ( α 2 − τ ) ∑ σ ( s _ ) = w _     c ( z s _ ) | z s _ | − ( τ + α 1 ) t ≤ κ t | z w _ | t ( α 2 − τ ) sup s _ ∈ S ℕ c ( z s _ ) ∑ σ ( s _ ) = w _ | z s _ | − ( τ + α 1 ) t .

Since f is a transcendental entire function of finite order ρ and t > ρ / ( τ + α 1 ) , then the Borel-Picard Theorem (see ( [10] , Theorem 3.5)) states that the series has the exponent of convergence equal to ρ . So the last sum is finite. Following ( [10] , Proposition 3.6), there exists M t > 0 such for all w _ ∈ X we have

L φ c , t ∘ H ( 1 l ) ( w _ ) ≤ M t | z w _ | t ( α 2 − τ ) sup s _ ∈ S ℕ c ( z s _ ) . (9)

So, the Equation (9) implies lim R → ∞ ∑ s ∈ ℤ exp ( sup w _ ∈ [ s ] ∩ ( X \ B ( 0 _ , R ) ) φ c , t ∘ H ( w _ ) ) = 0 and

lim z w _ → ∞ L φ c , t ∘ H 1 l ( w _ ) = 0 . □