A study of a meromorphic perturbation of the sine family

Patricia Domínguez; Josué Vázquez

doi:10.1515/dema-2022-0002

Article Open Access

A study of a meromorphic perturbation of the sine family

Patricia Domínguez and Josué Vázquez

Published/Copyright: March 23, 2022

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

We study the dynamics of a meromorphic perturbation of the family λ sin z by adding a pole at zero and a parameter μ , that is, f λ , μ ( z ) = λ sin z + μ / z , where λ , μ ∈ C ⧹ { 0 } . We study some geometrical properties of f λ , μ and prove that the imaginary axis is invariant under f n and belongs to the Julia set when ∣ λ ∣ ≥ 1 . We give a set of parameters ( λ , μ ) , such that the Fatou set of f λ , μ has two super-attracting domains. If λ = 1 and μ ∈ ( 0 , 2 ) , the Fatou set of f 1 , μ has two attracting domains. Also, we give parameters λ , μ such that ± π / 2 are fixed points of f λ , μ and the Fatou set of f λ , μ contains attracting domains, parabolic domains, and Siegel discs, we present examples of these domains. This paper closes with an example of f λ , μ , where the Fatou set contains two types of domains, for λ , μ given.

Keywords: iteration; fixed points; Fatou set; Julia set

MSC 2010: 37F10; 30D05

1 Introduction

The iteration of transcendental meromorphic functions with at least one non-omitted pole was initially investigated by Baker et al. in [1,2, 3,4]; since then many mathematicians have investigated these functions. Transcendental meromorphic functions have ∞ as the only essential singularity; an example of these functions is f λ ( z ) = λ tan ( z ) , λ ∈ C ⧹ { 0 } , which has an infinite countable set of non-omitted poles; see [5] and [6] for a study of the family.

The family f λ , μ ( z ) = λ sin z + μ / z can be obtained by adding a non-omitted pole z = 0 and a new parameter μ ∈ C ⧹ { 0 } to the holomorphic family g λ ( z ) = λ sin z , λ ∈ C ⧹ { 0 } . The family g λ has interesting dynamical properties related to the Fatou and Julia sets. In [7], Domínguez and Sienra described the Fatou set for values of the parameter λ inside the unit disc, obtaining the result that the Fatou set consists of a simply connected attracting and completely invariant domain. For values of λ on the unit circle of parabolic type, λ = e i 2 π θ , θ = p / q , ( p , q ) = 1 , it was proved that: (i) if q is even, then there is one q -cycle of Fatou components and (ii) if q is odd, then there are two q -cycles of Fatou components. Moreover, the Fatou components of such cycles are bounded. Zhang in [8] proved that for typical rotation numbers 0 < θ < 1 , the boundary of the Siegel disk of f θ ( z ) = e i 2 π θ sin z is a Jordan curve that passes through exactly two critical points π / 2 and − π / 2 . Herman rings are ruled out by the maximum principle.

In this paper, we are interested in the dynamic behavior of f λ , μ ( z ) = λ sin ( z ) + μ z , λ , μ ∈ C ⧹ { 0 } , that is, we are interested in a perturbation of g λ = λ sin z . For the family g λ , the point 0 plays an important role, as an attracting fixed point of g λ , to get a simply connected attracting and completely invariant domain in the Fatou set, for 0 < ∣ λ ∣ < 1 . For the family f λ , μ , zero cannot be a fixed point. Because 0 is a non-omitted pole of f λ , μ and belongs to the Julia set, the dynamical behavior of f λ , μ is quite different from g λ . Another difference between the two families is that the set of singular values of f λ , μ is countable infinite while for g λ , it is finite. In this paper, we present some symmetric properties of f λ , μ and prove the invariance of the imaginary axis, which belongs to the Julia set, when ∣ λ ∣ ≥ 1 . Fixing the parameter λ = 1 and moving the parameter μ , a slice of the parameter plane is defined. For a slice, we prove that the Fatou set contains two attracting domains for μ ∈ ( 0 , 2 ) . Also, we give conditions on the parameters λ , μ to have super-attracting or attracting domains, parabolic domains and Siegel discs and present examples of these domains. Also, we present an example where an attracting and a parabolic domains coexist in the Fatou set, for λ , μ given. We were not able to give conditions on parameters λ , μ to have a Herman ring. We remark that the family f λ , μ depends holomorphically on two parameters λ , μ . The main results are stated as follows.

Theorem A

Let f λ , μ ( z ) = λ sin ( z ) + μ z , λ ∈ C ⧹ { 0 } and μ ∈ R ⧹ { 0 } sufficiently small. For ∣ λ ∣ ≥ 1 , the imaginary axis belongs to the Julia set of f λ , μ .

Theorem B

Let f λ , μ ( z ) = λ sin ( z ) + μ z . There exists a set of complex parameters ( λ , μ ) , such that the Fatou set of f λ , μ contains two super-attracting domains.

Theorem C

If f μ ( z ) = sin ( z ) + μ z , z ∈ R ⧹ { 0 } and μ ∈ ( 0 , 2 ) , then the Fatou set of f μ contains two attracting domains.

Theorem D

Let f λ , μ ( z ) = λ sin ( z ) + μ z . There exists parameters λ , μ , such that f λ , μ has fixed points ± π / 2 and the Fatou set contains two attracting domains, two parabolic domains and two Siegel discs.

The parameters given in Theorems C and D are different for the attracting case. Section 2 is concerned with the iteration of transcendental meromorphic functions and properties of their Fatou and Julia sets. Section 3 contains the study of some fixed point and critical values of f λ , μ , which are used in the proof of the theorems mentioned earlier. In Section 4, we prove Theorems A, B, C, and D and discuss examples of the domains mentioned in the theorems. Section 4 closes with an example of f λ , μ where the Fatou set has two types of domains, for λ , μ given.

2 Preliminaries

We denote by ℳ the class of transcendental meromorphic functions f : C → C ^ with at least one non-omitted pole. The n-iterate of f ∈ ℳ is defined as the composition of f with itself n times, that is, f ∘ f ∘ … ∘ f ︸ n times = f n with f 0 = I d and n ∈ N .

The class ℳ is not closed under composition, that is, f ∈ ℳ does not imply that f n ∈ ℳ .

Definition 2.1

[9] Let D be a domain in C ^ . The rank n of a point z ∈ D is the greatest integer n , such that f n ( z ) can be defined as f ( f n − 1 ( z ) ) and f j ( z ) ∈ D , for j = 1 , … , n − 1 , while f n ( z ) ∉ D . The case when n = ∞ is allowed.

Remark 2.2

The points of rank at least n form an open set D n and we have

D = D 1 ⊃ D 2 ⊃ D 3 ⊃ … ⊃ D ∞ = ⋂ n = 1 ∞ D n .

For n finite D n is the domain of definition of f n .

The following result is due to Radström, see [9] for a proof.

Lemma 2.3

The set C ^ ⧹ D ∞ consists of C ^ ⧹ D together with all predecessors of points of C ^ ⧹ D .

Let f : D ⊂ C → C ^ be a function in class ℳ . For z 0 ∈ D , we define, respectively, the forward orbit of z 0 under f , the backward orbit of z 0 under f , and the grand orbit of z 0 under f as O + ( z 0 ) = { f n ( z 0 ) : n ∈ N } , O − ( z 0 ) = ∪ n = 1 ∞ f − n ( z 0 ) = ∪ n = 1 ∞ { z : f n ( z ) = z 0 } and O ( z 0 ) = O + ( z 0 ) ∪ { z 0 } ∪ O − ( z 0 ) .

We define E ( f ) as the set of Fatou exceptional values of f , that is, points whose O − ( z 0 ) is finite. Observe that O − ( ∞ ) is an infinite set. Indeed, f − 3 ( ∞ ) is infinite as a consequence of the Picard’s little theorem. Thus, the largest open set where all iterates of a function f ∈ ℳ are well defined is C ^ \ O − ( ∞ ) ¯ . We remark that by f − 1 ( z ) , we mean all the branches of the inverse function. We say that a set U is invariant under f if f ( U ) ⊂ U and that U is completely invariant if f − 1 ( U ) = U = f ( U ) .

The set of singular values of f ∈ ℳ is defined as SV ( f ) = CV ∪ AV ¯ , where CV is the set of critical values of f and AV is the set of asymptotic values of f . We recall that a critical value is the image of a critical point. A point a ∈ C ^ is called an asymptotic value of f if there is a path γ ( t ) → ∞ as t → ∞ , such that f ( γ ( t ) ) → a .

Let f ∈ ℳ and S p ( f ) = ⋃ k = 0 p − 1 f k ( ( SV ) ⧹ A k ( f ) ) , where A k ( f ) = { z ∈ C : f k is not analytic at z } . The set P ( f ) = ⋃ p = 1 ∞ S p ( f ) is defined as the post-singular set of f , that is, the post-singular set of a function f ∈ ℳ is formed by the singular values of f and their images under iteration, except at points where f is not analytic. Let sing ( f − p ) be the set of singularities of the inverse of the p -th iterate of f . We have sing ( f − p ) ⊂ S p ( f ) ⊂ S p + 1 ( f ) .

Let f ∈ ℳ and D ⊂ C . We say that z 0 ∈ D is a periodic point of period n of f if f n ( z 0 ) = z 0 , n ∈ N , and f i ( z 0 ) ≠ z 0 for i = 1 , … , n − 1 . When n = 1 , the point z 0 is called a fixed point of f . If z 0 is a periodic point of period n of f , then { z 0 , f ( z 0 ) , … , f n − 1 ( z 0 ) } is called a cycle of periodic points. The periodic points have associated a complex number which is called the multiplier and its definition is as follows.

Definition 2.4

[10] Let f ∈ ℳ and z 0 ≠ ∞ be a periodic point of f of period n . The multiplier of z 0 is λ = ( f n ) ′ ( z 0 ) = ∏ j = 0 n − 1 f ′ ( f j ( z 0 ) ) . For the case when z 0 = ∞ , the multiplier is defined by λ = d d z 1 f n 1 z z = 0 .

The classification of a periodic point z 0 ∈ C of period n of f ∈ ℳ is given in terms of its multiplier:

if λ = 0 , the point z 0 is called super-attracting;
if 0 < ∣ λ ∣ < 1 , the point z 0 is called attracting;
if ∣ λ ∣ > 1 , the point z 0 is called repelling;
if ∣ λ ∣ = 1 and λ is a root of unity, the point z 0 is called rationally indifferent; and
if ∣ λ ∣ = 1 and λ is not a root of unity, the point z 0 is called irrationally indifferent.

It follows from the chain rule that all the points in a cycle of periodic points have the same multiplier. Observe that if f and g are conjugate by a homeomorphism ϕ and z 0 ∈ C is a periodic point of f ∈ ℳ , then ϕ ( z 0 ) is a periodic point of g with the same period and multiplier of f .

The Fatou set of f ∈ ℳ , denoted by F = F ( f ) , is defined by

F ( f ) = { z ∈ C : { f n } n ≥ 1 is well defined and normal in some neighborhood of z } .

The Julia set of f ∈ ℳ , denoted by J = J ( f ) , is the complement of the Fatou set, that is, J = C ⧹ F ( f ) .

Some properties of the Fatou and Julia sets for f ∈ ℳ are the following: (i) the Fatou set is open, so the Julia set is closed; (ii) the Julia set is perfect and non-empty; (iii) F ( f ) and J ( f ) are completely invariant, that is, z ∈ F ( f ) if and only if f ( z ) ∈ F ( f ) and z ∈ J ( f ) if and only if f ( z ) ∈ J ( f ) ; and (iv) F ( f n ) = F ( f ) and J ( f n ) = J ( f ) , for all n ∈ N ; and (v) the repelling periodic points are dense in J ( f ) ; see [2] for the proofs.

The following result gives a characterization of the Julia set, see [10] for a proof.

Lemma 2.5

Let f ∈ ℳ . For any w ∈ J ( f ) and w ∉ E ( f ) , w is an accumulation point of J = O − ( w ) ¯ .

For f ∈ ℳ , the classification of a Fatou component U can be periodic, pre-periodic, or wandering. The classification of periodic Fatou component is as follows: attracting, parabolic, or rotation domains; the rotation domains are either Siegel discs or Herman rings; see [3] for details.

The following theorems were proved by Baker et al. in [2,4].

Theorem 2.6

If f ∈ ℳ and U is an invariant component in the Fatou set of f , then the component U has connectivity 1 , 2 , or ∞ . Moreover, if U is doubly connected, then U is a Herman ring.

Theorem 2.7

Let f ∈ ℳ and U be a component in the Fatou set of f .

If U is an attracting component or a parabolic component, then U contains a singular value of f .
If U is a Siegel disc or a Herman ring, then ∂ U ⊂ O + ( SV ( f ) ) ¯ .

Eremenko and Lyubich in [11,12] defined the classes of functions S and ℬ for transcendental entire functions, also called of finite singular type and bounded singular type, respectively. For functions in class ℳ , similar definitions apply.

Definition 2.8

The class S is the set of functions f ∈ ℳ of finite singular type, that is, the class S consists of functions f ∈ ℳ such that the set of singular values SV ( f ) is finite.

Definition 2.9

The class ℬ is the set of functions f ∈ ℳ of bounded singular type, that is, the class ℬ consists of functions f ∈ ℳ such that the set of singular values SV ( f ) is contained in a bounded set in C .

Examples of families of functions in ℳ ∩ S and ℳ ∩ ℬ are the following.

The family T λ ( z ) = λ tan ( z ) , λ ∈ C ⧹ { 0 } , has a finite set of singular values which are the asymptotic values { i λ , − i λ } . Thus, T λ belongs to the class S .
The family h λ ( z ) = λ sin ( z ) z , λ ∈ C ⧹ { 0 } , has an asymptotic value in z = 0 and an infinite set of bounded critical values w . Thus, h λ ( z ) = λ sin ( z ) z belongs to the class ℬ .

3 Properties, fixed points and singular values of f λ , μ

In this section, we present some geometrical properties, some fixed points which are classified in terms of their multiplier and critical values of f λ , μ ( z ) = λ sin ( z ) + μ z , for some λ , μ given. The fixed points and the singular values of f λ , μ will depend holomorphically on the parameters.

3.1 Properties of f λ , μ

Functions that belong to the family f λ , μ have some symmetries due to the properties of trigonometric complex functions and the complex inversion with respect to the origin. These symmetries are stated in the following lemma.

Lemma 3.1

Let f λ , μ ( x + i y ) = λ sin ( x + i y ) + μ x + i y , λ ∈ C ⧹ { 0 } and μ > 0 be a fixed real number sufficiently small. The following conditions are satisfied:

f λ , μ ( x − i y ) + f λ , μ ( x + i y ) = 2 ℜ ( f λ , μ ( x + i y ) ) , where ℜ is the real part of f λ , μ ;
f λ , μ ( − x + i y ) + f λ , μ ( x + i y ) = 2 ℑ ( f λ , μ ( x + i y ) ) , where ℑ is the imaginary part of f λ , μ ;
f λ , μ ( − x − i y ) = − f λ , μ ( x + i y ) ;
∣ f λ , μ ′ ( − x + i y ) ∣ = ∣ f λ , μ ′ ( x + i y ) ∣ .

Proof

We only prove ( c ) , since the proofs of the other properties are similar. Without loss of generality, we consider λ ∈ R ⧹ { 0 } . Evaluating f λ , μ at z = x + i y ∈ C and separating its real and imaginary parts, we obtain the following:

f λ , μ ( x + i y ) = λ sin ( x + i y ) + μ x + i y = λ ( sin ( x ) cos ( i y ) + sin ( i y ) cos ( x ) ) + μ x + i y = λ sin ( x ) cosh ( y ) + i λ sinh ( y ) cos ( x ) + μ x − i μ y x 2 + y 2 = λ sin ( x ) cosh ( y ) + μ x x 2 + y 2 + i λ sinh ( y ) cos ( x ) − μ y x 2 + y 2 .

On the other hand, evaluating f λ , μ at − z = − x − i y , we have

f λ , μ ( − x − i y ) = λ sin ( − x − i y ) + μ − x − i y = λ ( sin ( − x ) cos ( − i y ) + sin ( − i y ) cos ( − x ) ) + μ − x − i y = − λ sin ( x ) cosh ( y ) − μ x x 2 + y 2 + i μ y x 2 + y 2 − λ sinh ( y ) cos ( x ) − = − λ sin ( x ) cosh ( y ) + μ x x 2 + y 2 − i λ sinh ( y ) cos ( x ) − μ y x 2 + y 2 = − f λ , μ ( x + i y ) . □

Observe from (d) in Lemma 3.1, that it is possible to reduce the study of some dynamical results of f λ , μ to the subset of C where ℜ ( z ) ≥ 0 . We state the following result.

Lemma 3.2

If λ , μ ∈ R + ⧹ { 0 } and μ is sufficiently small, then (i) f λ , μ ( z ) = λ sin ( z ) + μ z maps the region R ≔ { z ∈ C : − π < x < 0 } into the semi-plane { f λ , μ ( z ) : ℜ ( f λ , μ ) < 0 } and (ii) f λ , μ ( z ) = λ sin ( z ) + μ z maps the region R ′ ≔ { z ∈ C : 0 < x < π } into the semi-plane { f λ , μ ( z ) : ℜ ( f λ , μ ) > 0 } .

Proof

Let z ∈ R and consider λ , μ ∈ R + ⧹ { 0 } . Then, f λ , μ can be written as follows:

f λ , μ ( z ) = λ sin ( z ) + μ z = λ sin ( x + i y ) + μ x + i y = λ ( sin ( x ) cos ( i y ) + cos ( x ) sin ( i y ) ) + μ x + i y = λ sin ( x ) cosh ( y ) + i λ cos ( x ) sinh ( y ) + μ x x 2 + y 2 − i μ y x 2 + y 2 .

Thus, the real part of f λ , μ ( z ) is u ( x , y ) = λ sin ( x ) cosh ( y ) + μ x x 2 + y 2 . Now, since z ∈ R , if we assume − π < ℜ ( z ) < 0 , then both λ sin ( x ) cosh ( y ) and the quotient μ x x 2 + y 2 are negative for μ > 0 sufficiently small. Therefore, ℜ ( f λ , μ ( z ) ) < 0 . By similar arguments, we can prove (ii).□

The invariance of the imaginary axis is given by the following lemma.

Lemma 3.3

Let f λ , μ with λ ∈ R ⧹ { 0 } and μ ∈ R ⧹ { 0 } . The set ℑ = { i y : y ∈ R ⧹ { 0 } } is invariant under f λ , μ n , for n ∈ N .

Proof

We proceed by induction. Take z ∈ ℑ and n = 1 , we shall prove that z = i y is invariant under f λ , μ . Observe that

f λ , μ ( i y ) = λ sin ( i y ) + μ i y = λ i sinh ( y ) − i μ y 2 = i λ sinh ( y ) − μ y 2 = i m 1 ,

where m 1 = λ sinh ( y ) − μ y 2 is a real function of y . Now suppose we have already proved that f λ , μ k ( i y ) = i m k , where m k is a real function of y . Next take

f λ , μ k + 1 ( i y ) = f λ , μ k ( f λ , μ ( i y ) ) = f λ , μ k ( i m 1 ) = i m k + 1 ,

where m k + 1 is a real function of y . This completes the inductive proof.□

3.2 Fixed points of f λ , μ

In this section, by placing some restrictions on λ , μ , we can find different types of fixed points of the functions f λ , μ . Also, we describe the singular values of f λ , μ which will be used to prove some results in Section 4.

Lemma 3.4

Let f λ , μ ( z ) = λ sin ( z ) + μ z and 0 < μ < 5 . If z = x ∈ R + ⧹ { 0 } and 0 < λ < 2 π , then f λ , μ ( x ) has a real fixed point w . Moreover, if z = x ∈ R − ⧹ { 0 } and − 2 π < λ < 0 , then there is a real fixed point − w , which is symmetric to w with respect to the imaginary axis.

$Figure 1 H ( x ) H\left(x) for λ = 0.3 , 0.7 , 1.5 , 2 , 4 , 5 , 6 , 7 \lambda =0.3,0.7,1.5,2,4,5,6,7 and μ = 2.4 \mu =2.4 .$

Figure 1

H ( x ) for λ = 0.3 , 0.7 , 1.5 , 2 , 4 , 5 , 6 , 7 and μ = 2.4 .

$Figure 2 Close up of the graph H ( x ) H\left(x) .$

Figure 2

Close up of the graph H ( x ) .

Proof

Take H ( x ) = f λ , μ ( x ) − x . Figures 1 and 2 show some graphs of H ( x ) , in blue, for λ = 0.3 , 0.7 , 1.5 , 2 , 4 , 5 , 6 and μ = 2.4 and a close up near to the origin. We consider two cases: (i) t ∈ ( 0 , ∞ ) and (ii) t ∈ ( − ∞ , 0 ) .

(i) Let t ∈ ( 0 , ∞ ) such that t 2 < μ and 0 < λ < 2 π . Evaluating H ( x ) , for instance, at t = 0.02 , we obtain

H ( 0.02 ) = λ sin ( 0.02 ) + μ 0.02 − 0.02 > μ 0.02 − 0.02 > 0 .

Now, take t ∈ ( 0 , ∞ ) such that t 2 > μ and 0 < λ < 2 π . Evaluating H ( x ) at t = 2 π , we have

H ( 2 π ) = λ sin 2 π + μ 2 π − 2 π = μ 2 π − 2 π < 0 .

Since H ( t ) is a continuous function for t ∈ ( 0 , ∞ ) and satisfies that H ( 0.02 ) > 0 and H ( 2 π ) < 0 , then by the intermediate value theorem, there exists w such that H ( w ) = 0 . This implies that w is a fixed point of f λ , μ ( x ) . By analogous arguments, the case (ii) works (or by properties of symmetry); thus, − w is a real fixed point of f λ , μ .□

Proposition 3.5

Let f λ , μ ( z ) = λ sin ( z ) + μ z . If λ ∈ C ⧹ { 0 } and μ = π 2 2 − λ π 2 , then f λ , μ has fixed points at z = ± π 2 .

Proof

Indeed, evaluating f λ , π 2 4 − λ π 2 ( z ) at z = π 2 (similar calculations for − π / 2 ), we obtain

f λ , π 2 4 − λ π 2 π 2 = λ sin π 2 + π 2 2 − λ π 2 π 2 = π 2 . □

In particular, for λ , μ non-zero real positive parameters, such that 0 < λ < π / 2 and μ = π 2 2 − λ π 2 , the point π / 2 is a fixed point. More generally, if 0 < λ < ( 2 k − 1 ) π 2 and μ = ( 2 k − 1 ) π 2 2 − λ ( 2 k − 1 ) π 2 , k ∈ N , then ( 2 k − 1 ) π 2 is a fixed point of f λ , μ .

For the case when ( 2 k − 1 ) π 2 < λ < 0 , k ∈ Z − , it follows from Lemma 3.1 (c) that − ( 2 k − 1 ) π 2 is a fixed point of f λ , μ .

Observation 1

Throughout the document, we deal with fixed points of period one.
The fixed points in Proposition 3.5 depend holomorphically of the parameters λ and μ .

(a) Super-attracting and attracting fixed points

We will analyze some cases of super-attracting and attracting fixed points of f λ , μ for different parameters λ , μ .

(1) Super-attracting fixed points of f λ , μ

To identify the set of real parameters ( λ , μ ) different from ( 0 , 0 ) , such that f λ , μ has fixed points we have to solve the following equation:

(3.1) λ sin ( w ) + μ w = w .

If w 0 is a fixed point that satisfies ∣ f λ , μ ′ ( w 0 ) ∣ = 0 , then

(3.2) λ cos ( w 0 ) − μ w 0 2 = 0 ⇔ λ cos ( w 0 ) − μ w 0 2 = 0 .

From equations (3.1) and (3.2), we can express λ and μ in terms of the fixed point w 0 as follows:

(3.3) λ = w 0 sin ( w 0 ) + w 0 cos ( w 0 ) ; μ = w 0 3 cos ( w 0 ) sin ( w 0 ) + w 0 cos ( w 0 ) .

If λ and μ are as in the equations of (3.3), then f λ , μ has a super-attracting fixed point at w 0 . Moreover, from the symmetries of f λ , μ , we obtain that − w 0 is also a fixed point of f λ , μ .

We define the parametric curve

ℒ ( w ) = w sin ( w ) + w cos ( w ) , w 3 cos ( w ) sin ( w ) + w cos ( w ) ,

with w ∈ R and sin ( w ) + w cos ( w ) ≠ 0 . Figure 3 shows the graph of ℒ , where the horizontal axis corresponds to λ -axis and the vertical axis corresponds to μ -axis. The points in ℒ are the set of real parameters ( λ , μ ) such that f λ , μ has a super-attracting fixed point. Figure 4 shows a close up of the graph ℒ when 0 < λ < π / 2 and μ > 0 sufficiently small.

$Figure 3 Graph of ℒ {\mathcal{ {\mathcal L} }} in the ( λ , μ ) \left(\lambda ,\mu ) -plane.$

Figure 3

Graph of ℒ in the ( λ , μ ) -plane.

$Figure 4 Close up of the graph ℒ {\mathcal{ {\mathcal L} }} for μ > 0 \mu \gt 0 sufficiently small.$

Figure 4

Close up of the graph ℒ for μ > 0 sufficiently small.

Lemma 3.4 and the curve ℒ assure the existence of fixed points of f λ , μ , but we do not know the exact value of each fixed point. If we give appropriate values to the parameters λ , μ , then we will be able to know the exact value of some fixed points of f λ , μ .

Observe that if we assume that w 0 = π / 2 is a super-attracting fixed point of f λ , μ that satisfies the equations in (3.3), then we can calculate that λ = π 2 and μ = 0 . Because μ ≠ 0 , we conclude that λ = π 2 cannot be a super-attracting fixed point of f λ , μ . A similar argument works for points of the form ( 2 k − 1 ) π / 2 , k ∈ Z .

In (2) below, we shall give parameters λ , μ ∈ C ⧹ { 0 } , such that the points ± π / 2 are attracting fixed points of f λ , μ .

(2) Attracting fixed points of f λ , μ

From Proposition 3.5, we know that f λ , μ has real fixed points at ± π / 2 , for λ ∈ C ⧹ { 0 } and μ = π 2 2 − λ π 2 . To make them attracting, we take

f λ , μ ′ ± π 2 = λ cos ( ± π / 2 ) − μ ( ± π / 2 ) 2 = − μ ( π / 2 ) 2 = ∣ μ ∣ ( π / 2 ) 2 < 1 .

This gives a condition on λ which depends on μ .

In the following case, we will give different values to the parameters λ , μ , such that the points ± π / 2 are always fixed points, but they can be characterized as rationally indifferent or irrationally indifferent.

(b) Indifferent fixed points of f λ , μ

Let λ ∈ C and μ = π 2 2 − λ π 2 . By Proposition 3.5, we have that f λ , π 2 2 − λ π 2 has fixed points at ± π 2 . Denote by ψ , the multiplier of the fixed point π 2 ; that is, ψ = f λ , π 2 2 − λ π 2 ′ π 2 (similar for − π 2 ). Assume that ψ = e 2 π i θ , thus

− μ ( π / 2 ) 2 = e 2 π i θ ,

and the parameter λ satisfies

λ = ( e 2 π i θ + 1 ) π 2 .

Taking

∣ ψ ∣ = f λ , π 2 2 − λ π 2 ′ π 2 = − μ ( π / 2 ) 2 = ∣ e 2 π i θ ∣ = 1 ,

we have the following classification.

The fixed points ± π 2 are rationally indifferent, if θ ∈ Q .
The fixed points ± π 2 are irrationally indifferent, if θ ∈ R ⧹ Q .

3.3 Singular values of f λ , μ

We recall that the set of singular values of f λ , μ contains the asymptotic values and the critical values of f λ , μ .

3.3.1 Asymptotic values

Claim. f λ , μ ( z ) = λ sin ( z ) + μ z has no finite asymptotic values, for λ ∈ C ⧹ { 0 } and μ ∈ R + ⧹ { 0 } .

Consider any path Γ : [ 0 , ∞ ) → C ^ which tends to ∞ . To obtain that lim Γ ( t ) → ∞ f λ , μ ( Γ ( t ) ) = a we have to use that λ sin ( Γ ( t ) ) → Γ → ∞ L , but this is only satisfied if L = ∞ , since the family λ sin ( z ) has no finite asymptotic values. Therefore, f λ , μ has no finite asymptotic values.

3.3.2 Critical values

To obtain the critical values of f λ , μ , we need to find the critical points of f λ , μ = λ sin ( z ) + μ z . Thus, we take the derivative of f λ , μ and calculate the points z ∈ C , such that

(3.4) f λ , μ ′ ( z ) = λ cos ( z ) − μ z 2 = 0 .

The problem is to find the exact solutions of the non-linear equation (3.4), but we can use numerical approximations to obtain approximations of critical points of f λ , μ .

Observe from equation (3.4) that there are solutions of λ z 2 cos ( z ) − μ = 0 , for z ≠ 0 . We know that λ z 2 cos ( z ) = 0 , when z = ( 2 k − 1 ) π / 2 , k ∈ Z , and λ ∈ R ⧹ { 0 } . We say that the points ( 2 k − 1 ) π / 2 are the critical points of f λ , μ with an error of ∣ μ ∣ = ε μ → 0 , for ∣ μ ∣ sufficiently small. We denote the critical points of f λ , μ as

c k = ( 2 k − 1 ) π / 2 + ε μ , k ∈ Z .

For instance, taking the case when 0 < λ < π / 2 and μ = π 2 2 − λ π 2 , Figure 5 shows the intersection of the graphs λ cos ( x ) , in red, and μ x 2 , in blue, for λ = 0.3 , 0.5 , 0.7 , 1 . We can observe in Figure 5 that the graphs intersect each other approximately at ± 3 π / 2 , ± 5 π / 2 , ± 7 π / 2 , … , that is, for k = 0 , − 1 and 0 < λ < π / 2 , the curve λ cos x never touches the curve μ x 2 .

$Figure 5 The graphs λ cos ( x ) \lambda \cos \left(x) in red and μ x 2 \frac{\mu }{{x}^{2}} in blue for λ = 0.3 , 0.5 , 0.7 , 1 \lambda =0.3,0.5,0.7,1 and μ = π 2 2 − λ π 2 \mu ={\left(\frac{\pi }{2}\right)}^{2}-\lambda \frac{\pi }{2} .$

Figure 5

The graphs λ cos ( x ) in red and μ x 2 in blue for λ = 0.3 , 0.5 , 0.7 , 1 and μ = π 2 2 − λ π 2 .

Evaluating f λ , μ at the critical points c k = ( 2 k + 1 ) π 2 + ε μ , k ∈ Z , we obtain the critical values of f λ , μ , that is,

f λ , μ ( c k ) = λ sin ( 2 k + 1 ) π 2 + ε μ + μ ( 2 k + 1 ) π 2 + ε μ = ± λ cos ( ε μ ) + μ ( 2 k + 1 ) π 2 + ε μ , k ∈ Z .

We denote by cv k the set of critical values of f λ , μ . Observe that f λ , μ ( c k ) → ± λ , when ∣ μ ∣ is sufficiently small and k → ∞ , k ∈ Z ⧹ { 0 } . Thus, λ and − λ are accumulation points of the critical values of f λ , μ . The set of critical values of f λ , μ is countable infinite, so it does not belong to the class S defined in Section 2.

We can obtain a characterization of the critical values of f λ , μ for z = x and λ , μ non-zero real parameters as follows: if p is a critical point of f λ , μ , then

f λ , μ ( p ) = λ sin ( p ) + μ p = λ 2 − μ 2 p 4 + μ p .

Proposition 3.6

f λ , μ belongs to the class ℬ , for λ ∈ R ⧹ { 0 } and ∣ μ ∣ sufficiently small.

Proof

The set of singular values of f λ , μ is the set of critical values c v k = ± λ cos ( ε μ ) + μ ( 2 k + 1 ) π 2 + ε μ , k ∈ Z , which is an infinite set. Now, take a bounded set B ⊂ C , such that it contains the accumulation points ± λ of the critical values c v k , k ∈ Z . For ∣ μ ∣ sufficiently small and large values of k , we conclude that f λ , μ belongs to the class ℬ .□

4 Dynamics of f λ , μ

In this section, we give conditions on the parameters λ , μ to prove Theorems A, B, C, and D.

Theorem A

Let f λ , μ ( z ) = λ sin ( z ) + μ z , λ ∈ C ⧹ { 0 } and μ ∈ R ⧹ { 0 } sufficiently small. For ∣ λ ∣ ≥ 1 , the imaginary axis belongs to the Julia set of f λ , μ .

Proof

Let U be a neighborhood with center ζ = i y 1 , y 1 ∈ R ⧹ { 0 } , and radius 0 < r < ∣ y 1 ∣ . We shall prove that the family f λ , μ is not point-wise bounded in U .

Consider w = i y 2 , with y 2 ∈ R ⧹ { 0 } , such that w ≠ ζ and either (i) μ > 0 , y 1 y 2 > 0 and y 1 > y 2 or (ii) μ < 0 , y 1 y 2 > 0 and y 2 > y 1 . By Lemma 3.3, the images of the points ζ , w and its iterates under f λ , μ are in the imaginary axis. Taking the distance between f λ , μ ( ζ ) and f λ , μ ( w ) , we have

∣ f λ , μ ( ζ ) − f λ , μ ( w ) ∣ = λ sin ( ζ ) + μ ζ − λ sin ( w ) + μ w = λ sin ( i y 1 ) + μ i y 1 − λ sin ( i y 2 ) + μ i y 2 = i λ sinh ( y 1 ) − i μ y 1 − i λ sinh ( y 2 ) + i μ y 2 = ∣ i ∣ λ ( sinh ( y 1 ) − sinh ( y 2 ) ) + μ 1 y 2 − 1 y 1 > ∣ λ ∣ ∣ ( sinh ( y 2 ) − sinh ( y 1 ) ) ∣ .

Now, observe that the real function g ( x ) = sinh ( x ) is continuous in [ y 1 , y 2 ] and differentiable on ( y 1 , y 2 ) , so by the Mean Value Theorem, there exists y 3 ∈ ( y 1 , y 2 ) , such that

∣ sinh ( y 2 ) − sinh ( y 1 ) ∣ = cosh ( y 3 ) ∣ y 2 − y 1 ∣ .

Therefore, when ∣ λ ∣ > 1 , we obtain

∣ λ ∣ ∣ sinh ( y 2 ) − sinh ( y 1 ) ∣ = ∣ λ ∣ cosh ( y 3 ) ∣ y 2 − y 1 ∣ > ∣ λ ∣ ∣ y 2 − y 1 ∣ > ∣ y 2 − y 1 ∣ .

For the case when ∣ λ ∣ = 1 , we have ∣ sinh ( y 2 ) − sinh ( y 1 ) ∣ = cosh ( y 3 ) ∣ y 2 − y 1 ∣ > ∣ y 2 − y 1 ∣ . Thus, it follows that the family f λ , μ ( z ) = λ sin ( z ) + μ z has an expanding behavior in the imaginary axis; that is, f λ , μ takes arbitrarily large values under iteration; hence, the family f λ , μ is not point-wise bounded. We conclude that the family f λ , μ is not normal in the imaginary axis, and therefore, the Julia set of f λ , μ ( z ) = λ sin ( z ) + μ z contains the imaginary axis when ∣ λ ∣ ≥ 1 .□

Theorem B

Let f λ , μ ( z ) = λ sin ( z ) + μ z . There exists a set of complex parameters ( λ , μ ) such that the Fatou set of f λ , μ contains two super-attracting domains.

Proof

First, we identify the set of complex parameters ( λ , μ ) , such that f λ , μ has two super-attracting fixed points. By (a)-(1) in Section 3.2, the curve ℒ contains complex values of ( λ , μ ) such that f λ , μ has two super attracting fixed points, thus they must have attracting basins which are components of the Fatou set.□

Observe that the curve ℒ exists as a complex curve in the parameter space, so moving the parameters ( λ , μ ) within the curve ℒ gives a “slice” where there are always two super-attracting domains.

An example that satisfies Theorem B is the following.

Example 1

Let f λ , μ and λ = ( 2 π ) / ( 3 3 + π ) , μ = π 3 / ( 9 ( 3 3 + π ) ) in the curve ℒ . The function

2 π 3 3 + π sin ( z ) + π 3 / ( 9 ( 3 3 + π ) ) z

has two fixed points at ζ = π / 3 and − ζ = − π / 3 , that is,

f λ , μ π 3 = 2 π 3 3 + π sin π 3 + π 3 / ( 9 ( 3 3 + π ) ) π 3 = 3 π 3 3 + π + π 2 3 ( 3 3 + π ) = π 3 .

By similar calculations − ζ = − π 3 is also a fixed point of f λ 3 , μ 3 . The two fixed points are super-attracting since evaluating f λ , μ ′ at ζ = π 3 , we obtain

f λ , μ ′ π 3 = 2 π 3 3 + π cos π 3 − π 3 / ( 9 ( 3 3 + π ) ) π 3 2 = π 3 3 + π − π 3 3 + π = 0 .

By the symmetries of f λ , μ , the point − ζ = − π 3 is a super-attracting fixed point of f λ , μ .

If z = x ∈ R ⧹ { 0 } , then f 2 π 3 3 π , π 3 9 ( 3 3 + π ) ( x ) has two orbits that converge to the super-attracting fixed points ± π 3 which are shown in Figure 6 in red color.

$Figure 6 An orbit converging to ζ 1 = π 3 {\zeta }_{1}=\frac{\pi }{3} , the orbit of − ζ 1 = − π 3 -{\zeta }_{1}=-\frac{\pi }{3} is similar.$

Figure 6

An orbit converging to ζ 1 = π 3 , the orbit of − ζ 1 = − π 3 is similar.

The dynamical plane of f 2 π 3 3 π , π 3 9 ( 3 3 + π ) ( z ) = 2 π 3 3 + π sin ( z ) + π 3 / ( 9 ( 3 3 + π ) ) z is shown in Figure 7. Observe that there are two disjoint super-attracting domains, say D + and D − , in the Fatou set (in black), where each one them contains a super-attracting fixed point and the domains are separated by the imaginary axis, which is in the Julia set (in yellow). By the periodicity, the components of the basins of the fixed points alternate along the real line.

$Figure 7 The Fatou set in black, the Julia set in yellow and the orbit of a point in the super-attracting domain D − {D}^{-} in white.$

Figure 7

The Fatou set in black, the Julia set in yellow and the orbit of a point in the super-attracting domain D − in white.

4.1 Proof of Theorem C

If we set λ = 1 in f λ , μ and vary the parameter μ ∈ C ⧹ { 0 } , we can define a slice of the space of parameters as follows:

PL = { μ : f μ n ( c k ) is bounded, where c k is a critical point } .

Figures 8 shows the slice PL, and Figure 9 shows a close up of a “cardioid.”

$Figure 8 The slice PL of f λ , μ {f}_{\lambda ,\mu } .$

Figure 8

The slice PL of f λ , μ .

Figure 9

Close up of the “cardioid.”

It is possible to give an approximation of the zeros of f μ ( z ) = sin ( z ) + μ z ; that is, for z = k π , we have sin ( k π ) + μ k π ≈ 0 with an error ε k π = ∣ μ ∣ k π → 0 , for k ∈ Z ⧹ { 0 } big enough and ∣ μ ∣ sufficiently small.

Looking at the Figure 9, we state the following conjecture.

Conjecture

In the slice PL, there exist a cardioid curve C with cusp at 0 such that for all μ ∈ C ˚ , the family f 1 , μ has two attracting fixed points and C ˚ ∩ R = ( 0 , 2 ) , where C ˚ is the interior of C .

For z real and μ ∈ ( 0 , 2 ) , we have the following result.

Lemma 4.1

f μ ( z ) = sin ( z ) + μ z has only two real attracting fixed points when μ ∈ ( 0 , 2 ) and z ∈ R ⧹ { 0 } .

Proof

By Lemma 3.4 we know that f μ has two real fixed points, say ± t , when z = x ∈ R ⧹ { 0 } . We claim that if t > 0 , then t ∈ ( 0 , 2 ] , and it is the unique attracting fixed point of f μ , for μ ∈ ( 0 , 2 ) . The case when t ∈ [ − 2 , 0 ) is similar.

Suppose that t > 2 and take H ( x ) = sin x + μ / x − x . Since 0 < μ < 2 and t > 2 , so μ / t < 1 and we have that H ( t ) = sin t + μ / t − t ≤ 1 + μ / t − t < 1 + 1 − 2 = 0 . Then, t is not a fixed point, because H ( t ) must be zero, hence t ∈ ( 0 , 2 ] .

Now, if t is attracting must satisfy cos t − μ t 2 < 1 . Observe that

− 1 − μ t 2 < ∣ cos t ∣ − μ t 2 ≤ cos t − μ t 2 < 1 ,

then

− 1 − μ t 2 < 1 .

Hence, μ > − 2 t 2 , we recall that μ ∈ ( 0 , 2 ) and t ∈ ( 0 , 2 ] . To finish the proof of the claim, it remains to show that t is the unique real positive fixed point of f μ , for μ ∈ ( 0 , 2 ) .

Assume that there are two different real positive attracting fixed points of f μ , say t and t 1 , that is,

sin t + μ t = t and sin t 1 + μ t 1 = t 1 for t , t 1 > 0 .

Clearing μ in the equations above, we obtain

t 2 − t sin t = t 1 2 − t 1 sin t 1 .

Then,

( t 2 − t 1 2 ) − ( t sin t − t 1 sin t 1 ) = 0 ,

thus t = t 1 , which contradicts the assumption. Therefore, t is the unique real positive fixed point in ( 0 , 2 ] . The case for the real negative attracting fixed point t ∈ [ − 2 , 0 ) is similar. Thus, the claim is proved.□

$Figure 10 f μ n ( 1 ) → t {f}_{\mu }^{n}\left(1)\to t , for μ = 1.5 \mu =1.5 .$

Figure 10

f μ n ( 1 ) → t , for μ = 1.5 .

$Figure 11 f μ n ( − 1 ) → − t {f}_{\mu }^{n}\left(-1)\to -t , for μ = 1.5 \mu =1.5 .$

Figure 11

f μ n ( − 1 ) → − t , for μ = 1.5 .

Theorem C

If f μ ( z ) = sin ( z ) + μ z , z ∈ R ⧹ { 0 } and μ ∈ ( 0 , 2 ) , then the Fatou set of f μ contains two attracting domains.

Proof

From Lemma 4.1, the family f μ has only two real attracting fixed points ± t ≠ 0 , which are symmetric with respect to the imaginary axis. Now, take neighborhoods U ± of ± t , respectively, such that they do not intersect the imaginary axis which is in the Julia set by Theorem A. The critical values of f μ accumulate to ± 1 , where f μ n ( 1 ) → t and f μ n ( − 1 ) → − t . Figures 10 and 11 show an example of the orbits of − 1 , and 1, when z = x and μ = 1.5 . Thus, there exist D ± , such that U ± ⊂ D ± , so the Fatou set of f μ contains two attracting domains D ± , for μ ∈ ( 0 , 2 ) . Observe that D ± contains all the singular values of f μ , so there are no other Fatou components.

To prove that f μ n ( 1 ) → t set f μ n = a n , we want to prove that lim n → ∞ a n = t . Let I 1 = [ 1 , π / 2 ] = [ x 1 , x 1 ′ ] and take a 1 ∈ I 1 , so we have

a 2 = sin a 1 + μ a 1 , for μ ∈ ( 0 , 2 ) .

Since sin x + μ x ∣ [ 1 , π / 2 ] = I 1 is a decreasing function, we have

a 2 ∈ sin [ f ( π / 2 ) , f ( 1 ) ] = [ f ( x 1 ′ ) , f ( x 1 ) ] = sin x 1 ′ + μ x 1 ′ , sin x 1 + μ x 1 = [ x 2 , x 2 ′ ] = I 2 ,

Thus, a 2 ∈ I 2 ≔ [ x 2 , x 2 ′ ] . We can do the same with a 3 , a 4 , … and obtain a n ∈ I n = [ x n , x n ′ ] , where

x n = sin x n − 1 + μ x n − 1 and x n ′ = sin x n − 1 ′ + μ x n − 1 ′ .

The function sin x < x , for x ∈ [ 1 , π / 2 ] , so sin x + μ x < x + μ x . The sequence x n is decreasing and has a lower bound, thus x n converges to a limit, say L ; that is, lim n → ∞ x n = L (similar argument is used for x n ′ ). Moreover, since sin x + μ x is continuous we have sin x n + μ x n → sin L + μ L as n → ∞ . But x n + 1 = sin x n + μ x n , then lim n → ∞ x n + 1 = L , which implies that sin L + μ L = L . Therefore, L is a fixed point of f μ . Since t is the unique fixed point, L = t . The previous argument shows that lim n → ∞ a n = t . Similar argument can be used to prove that f μ n ( − 1 ) → − t . Thus, Theorem C is proved.□

Examples of the Fatou and Julia sets of Theorem C should be similar to those shown in Figure 7, for μ ∈ ( 0 , 2 ) .

Conjecture

If there exists the cardioid curve C , then for all μ ∈ C ˚ , the Fatou set of f 1 , μ contains two attracting domains.

4.2 Proof of Theorem D

To prove Theorem D, we will need three lemmas.

Lemma 4.2

Let f λ , μ ( z ) = λ sin ( x ) + μ z . There exist non-zero complex parameters λ , μ such that the fixed points of f λ , μ are ± π / 2 and the Fatou set of f λ , μ contains two attracting domains.

Take the parameters μ = ( π / 2 ) 2 − λ ( π / 2 ) and λ ∈ C ⧹ { 0 } . By (a)-(2) in Section 3.2, f λ , μ has two fixed points at ± π 2 , which are attracting, that is, they satisfy

f λ , μ ′ ± π 2 = ∣ μ ∣ ( π / 2 ) 2 < 1 .

The pole z = 0 is in the Julia set of f λ , μ ( x ) , so the real line is divided into two sets B + (right) and B − (left) which contains the real fixed points π 2 and − π 2 , respectively. The set of critical values c v k , k ∈ Z , of f λ , μ is on the real line and accumulates to ± λ .

Let U + ⊂ B + be a neighborhood of the fixed point π 2 . There is at least one critical value of f λ , μ in U + , because the set of critical values c v k → λ , when k → ∞ , k ∈ Z . Thus, U + belongs to an attracting domain, say D + of the Fatou set of f λ , μ on the right half-plane B + . By similar arguments, there is an attracting domain, say D − , of the Fatou set of f λ , μ on the left half-plane B − .

Example 2

Let f λ , μ , μ = π 2 2 − λ π 2 and λ = π / 4 . Figure 12 shows the two attracting domains in the Fatou set.

Figure 12

The Fatou set in black, the Julia set in yellow and the orbit of a point in the attracting domain in white.

In the following result, we give conditions on the parameters λ , μ , such that f λ , μ has two parabolic components in the Fatou set and give an example.

Lemma 4.3

Let f λ , μ ( z ) = λ sin ( z ) + μ z . If θ ∈ Q , λ , μ are non-zero complex parameters such that λ = ( e i θ + 1 ) ( π / 2 ) and μ = π 2 2 − λ π 2 , then ± π / 2 are fixed points and the Fatou set of f λ , μ contains two parabolic domains.

From Section 3.2(b)-(i), the points z 1 = π 2 and z 2 = − π 2 are rationally indifferent fixed points of f λ , π 2 2 − λ π 2 . Then, the Fatou set contains two parabolic domains.

As an example of Lemma 4.3, if we take the parameters λ = ( e i θ + 1 ) ( π / 2 ) = π , where θ = 2 π , and μ = π 2 2 − π 2 ( π ) = − π 2 4 , we can define the real-valued polynomial P ( x ) = x 2 − π x + π 2 4 . Observe that P ( x ) has a root of multiplicity 2 at x = π 2 . Thus, P ( x ) > 0 for x ∈ R ⧹ π 2 .

For x > 0 , λ = π and μ = − π 2 2 , we have

λ sin ( x ) + μ x ≤ x ⇔ λ x sin ( x ) + μ < x 2 ⇔ π x + π 2 2 ≤ x 2 ⇔ 0 ≤ x 2 − π x − π 2 2 = P ( x ) .

Thus, for x > 0 with x ≠ π 2 , λ = π and μ = − π 2 2 , we have that λ sin ( x ) + μ x < x . Then, f π , − π 2 2 ( x ) has only one fixed point ζ ∈ R + at ζ = π 2 . Similar arguments for ζ ∈ R − at ζ = − π 2 . The points are shown in Figure 13.

$Figure 13 Graph of f π , − π 2 4 ( x ) {f}_{\pi ,-\tfrac{{\pi }^{2}}{4}}\left(x) with two rationally indifferent fixed points.$

Figure 13

Graph of f π , − π 2 4 ( x ) with two rationally indifferent fixed points.

The Fatou set of f π , − π 2 2 ( z ) = π sin ( z ) − π 2 2 contains two parabolic invariant domains. Figure 14 shows the two parabolic domains in the Fatou set, the orbit of a point in one of the parabolic domains and the imaginary axis which is contained in the Julia set.

$Figure 14 The parabolic domains in the Fatou set of f π , − π 2 4 ( z ) {f}_{\pi ,\tfrac{-{\pi }^{2}}{4}}\left(z) in black and the orbit of a point in white.$

Figure 14

The parabolic domains in the Fatou set of f π , − π 2 4 ( z ) in black and the orbit of a point in white.

In the following lemma, we show that the Fatou set of f λ , μ ( z ) = λ sin ( z ) + μ z contains two Siegel discs for the parameters given in Section 3.2(b)-(ii).

Lemma 4.4

Let f λ , μ ( z ) = λ sin ( z ) + μ z . If θ ∈ R ⧹ Q is a Brujno number, λ , μ non-zero are complex parameters such that λ = ( e i θ + 1 ) ( π / 2 ) and μ = π 2 2 − λ π 2 , then ± π / 2 are fixed points and the Fatou set of f λ , μ contains two Siegel discs.

Proof

From Section 3.2(b)-(ii), the points z 1 = π 2 and z 2 = − π 2 are irrationally indifferent fixed points of f λ , π 2 2 − λ π 2 . Therefore, the Fatou set of f λ , π 2 2 − λ π 2 contains a Siegel disc with center z 1 = π 2 . Similar for the fixed point z 2 = − π 2 .□

Example 3

Let f λ , μ ( z ) , θ = 1 + 5 2 , λ = ( e i ( 1 + 5 ) 2 + 1 ) π 2 and μ = π 2 2 − λ π 2 = π 2 2 − ( e i ( 1 + 5 ) 2 + 1 ) π 2 π 2 . The hypothesis of Lemma 4.4 is satisfied, thus the function

f λ , π 2 2 − λ π 2 ( z ) = λ sin ( z ) + π 2 − 2 λ π 4 z

has two Siegel discs in the Fatou set which are symmetric with respect to the imaginary axis. Figure 15 shows the Siegel discs in black, the Julia set in scale of yellow and the orbit of a point in the Siegel disc in white.

$Figure 15 The Siegel discs of f λ , π 2 2 − λ π 2 {f}_{\lambda ,{\left(\tfrac{\pi }{2}\right)}^{2}-\tfrac{\lambda \pi }{2}} , for λ = e i ( 1 + 5 ) 2 + 1 π 2 \lambda =\left({e}^{\tfrac{i\left(1+\sqrt{5})}{2}}+1\right)\left(\frac{\pi }{2}\right) .$

Figure 15

The Siegel discs of f λ , π 2 2 − λ π 2 , for λ = e i ( 1 + 5 ) 2 + 1 π 2 .

Theorem D follows from Lemmas 4.2, 4.3, and 4.4.

4.3 Example where the Fatou set has two types of domains

The Fatou set can have different types of domains in the same dynamical plane. We present an example where an attracting domain and a parabolic domain coexist in the Fatou set of f λ , μ .

Let f λ , μ , λ = − 2 and μ = π 2 . Making some computations, we can see that the function f − 2 , π 2 ( z ) = − 2 sin ( z ) + π 2 z has two rationally indifferent fixed points at ζ 1 = π and ζ 2 = − π . For the parameters given the function has two extra fixed points, which can be obtained by numerical methods, that is, by using Newton’s method these points are approximately ζ 3 ≈ 3.919317 and ζ 4 ≈ − 3.919317 . Evaluating ∣ f − 2 , π 2 ′ ( z ) ∣ at ζ 3 and at ζ 4 , we obtain that they are attracting fixed points.

Figure 16 shows the parabolic and the attracting domains in black, the orbit of a point in the parabolic domain in white and the Julia set in scale of yellow.

$Figure 16 The attracting and parabolic domains in the Fatou set for λ = − 2 \lambda =-2 and μ = π 2 \mu ={\pi }^{2} .$

Figure 16

The attracting and parabolic domains in the Fatou set for λ = − 2 and μ = π 2 .

Acknowledgements

The authors would like to thank the referees for many helpful suggestions to improve the article and to Renato Leriche for his computational support.

Funding information: The research was supported by Benemérita Universidad Autónoma de Puebla.
Conflict of interest: The authors state no conflict of interest.

References

[1] I. N. Baker, J. Kotus, and L. Yinian, Iterates of meromorphic functions II: Examples of wandering domains, J. Lond. Math. Soc. 42 (1990), no. 2, 267–278. 10.1112/jlms/s2-42.2.267Search in Google Scholar

[2] I. N. Baker, J. Kotus, and L. Yinian, Iterates of meromorphic functions I, Ergodic Theory Dynam. Syst. 11 (1991), no. 2, 241–248. 10.1017/S014338570000612XSearch in Google Scholar

[3] I. N. Baker, J. Kotus, and L. Yinian, Iterates of meromorphic functions III: Preperiodic domains, Ergodic Theory Dynam. Syst. 11 (1991), no. 4, 603–618. 10.1017/S0143385700006386Search in Google Scholar

[4] I. N. Baker, J. Kotus, and L. Yinian, Iterates of meromorphic functions IV: Critical finite functions, Results Math. 22 (1992), 651–656. 10.1007/BF03323112Search in Google Scholar

[5] R. Devaney and L. Keen, Dynamics of tangent, Dynamical Systems (College Park, MD, 1986–87), Lecture Notes in Mathematics, vol. 1342, Springer, Berlin, 1988, pp. 105–111. 10.1007/BFb0082826Search in Google Scholar

[6] L. Keen and J. Kotus, Dynamics of the family lambda tan z, Conform. Geom. Dyn. 1 (1997), 28–57. 10.1090/S1088-4173-97-00017-9Search in Google Scholar

[7] P. Domínguez and G. Sienra, A study of the dynamics of λsinz, Int. J. Bifur. Chaos 12 (2002), no. 12, 2869–2883. 10.1142/S0218127402006199Search in Google Scholar

[8] S. Zhang, On PZ type Siegel disk of the sine family, Ergodic Theory Dynam. Syst. 36 (2016), 973–1006. 10.1017/etds.2014.89Search in Google Scholar

[9] P. Bhattacharya, Iteration of Analytic Functions, Ph.D. thesis, Imperial College London, 1969. Search in Google Scholar

[10] W. Bergweiler, An introduction to complex dynamics, Textos de Matemática, Universidad de Coimbra, Série B, 1995. Search in Google Scholar

[11] A. E. Eremenko and M. Y. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier 42 (1992), no. 4, 989–1020. 10.5802/aif.1318Search in Google Scholar

[12] A. E. Eremenko and M. Y. Lyubich, The dynamics of analytic transformations, Leningrad Math. J. 1 (1990), no. 3, 563–634. Search in Google Scholar

Received: 2021-05-20

Revised: 2021-10-13

Accepted: 2021-12-28

Published Online: 2022-03-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0002

Keywords for this article

iteration; fixed points; Fatou set; Julia set

Creative Commons

BY 4.0