Organization

The thesis consists of six chapters. Basic definitions and a brief literature survey of the dynamics of transcendental entire and meromorphic functions are provided in Chapter 1.

The organization of the rest of the chapters is as follows.

In Chapter 2, dynamics of functions in the one parameter family {λtanh(e^z) : λ ∈ R\{0}}is investigated. Certain dynamically relevant properties ofλtanh(e^z) are discussed in Section 2.1. Section 2.2 deals with the existence and nature of all the real periodic points of λtanh(e^z). In Section 2.3, bifurcation in the dynamics of λtanh(e^z) is proved to occur at a critical parameter λ^∗ ≈ −3.2946. Section 2.4 describes the change in topology of the Fatou components as a result of above mentioned bifurcation. Finally, it is established in Section 2.5 that the Lebesgue measure of the Julia set of λtanh(e^z) is zero for all nonzero λ.

Chapter 3 deals with the dynamics of functions in a class of entire transcendental functions which are of bounded type. We define a class E of transcendental entire functions f(z) =

X∞

n=0

anzⁿ for which (i) an ≥ 0 for all n, (ii) f(x) > 0 for x < 0 and (iii) the set of all singular values Sf is a bounded subset of R. For eachf ∈E, the dynamics of functions in the one parameter family {λf : λ > 0} is investigated. Let E0 ≡ {f ∈E : f(0) = 0} and E₁ ≡ {f ∈ E : f(0) 6= 0} . We have shown in Section 3.1 that both the classes E0 and E1 are closed under composition and that the compositions of certain kind of polynomials with the functions in E yield functions belonging to E. The real dynamics of λf(z) is determined separately for f ∈ E0 and f ∈ E1 in Section 3.2. It is shown in Section 3.3 that, forf ∈E1, there exists a positive real number λ^∗ (depending on f) such that a similar phenomena as chaotic burst occurs in the dynamics of functions in the one parameter family {λf : λ > 0} at λ =λ^∗. For f ∈ E0, it is shown that the Fatou set of λf is always the union of the attracting basin of the superattracting fixed 0 and possibly

wandering domains forλ >0. It is proved that the Fatou set of λf is connected when it is an attracting basin and all the singular values of λf lie in the immediate attracting basin.

Lastly, some examples of functions are discussed in Section 3.4.

In Chapter 4, the dynamics of functions in certain class of meromorphic functions of bounded type is investigated. Consider a class E of entire functions h such that (i) h(z) =

X∞

n=0

anzⁿ for z ∈ C where an ≥ 0 for all n > 0, (ii) a0 = h(0) ≥ 1, (iii) h(x) >

0 for all x < 0 and (iv) the closure of all the singular values of h is a bounded subset of {x ∈ R : x 6= 0}S

{z ∈ C : |z| = 1 and z 6= ±i}. Then, we consider the class M ≡ {f(z) = Jⁿ(h(z)) for z ∈ C : n ∈ N and h ∈ E} where Jⁿ denotes the n-times composition of the Joukowski functionJ(z) = z+¹_z. The change in dynamics of functions in the one parameter family S ≡ {fλ = λf : λ > 0} for f ∈ M is mainly explored in this chapter. General properties of fλ are first proved in Section 4.1. For instance, it is proved that Sp(fλ) is bounded for each p and for each natural number m, there exists a function inMhaving exactlym singular values. The dynamics offλ(x) for x∈Ris found in Section 4.2 and a similar phenomena as chaotic burst occurs in the dynamics of func- tion in S is shown at some critical value λ^∗ (depending on f) in Section 4.3. An example λJ(e^z+ 1)∈Sis discussed in detail in Section 4.4. Besides chaotic burst, we establish that the Julia set ofλJ(e^z+ 1) contains infinitely many bounded but not singleton components along with unbounded components whenever it is not equal to C. Further, the Julia setb can be expressed as a disjoint union of two completely invariant subsets one of which is totally disconnected.

Chapter 5 is devoted to the study of dynamics of certain meromorphic functions which are not of bounded type. Define N = n

f(z) = ^zm

sinh^mz for z ∈C: m∈No . The one parameter family S ={fλ(z) = λf(z) : λ ∈ R\ {0}} for f ∈ N is considered. The functions fλ ∈ S are proved to be not of bounded type in Section 5.1 along with other dynamically relevant properties of fλ. Since fλ and f₋λ are conformally conjugate and

hence have same dynamics, we only investigate the dynamics of fλ ∈Sfor λ >0. The real dynamics of fλ, λ > 0 is investigated in Section 5.2. The bifurcation in the dynamics of functions fλ ∈ S, λ > 0 at a critical parameter λ^∗ is proved in Section 5.3. Finally, the effect of this bifurcation on the topology of Fatou components is investigated in Section 5.4.

The dynamics of certain real meromorphic functions is studied in Chapter 6. Let R be a class of real meromorphic functions f satisfying (i) f(z) =

X∞

k=−∞

Ak

1

ak−z − 1 ak

, (ii) Ak > 0, ak 6= 0 for k ∈ Z and (iii)

X∞

k=−∞

Ak

a²_k converges. Then, a subclass R^∗ of R is considered that contains those functions f for which (i) f(z) =

X∞

k=1

Akz

a²_k−z², (ii) Ak >

0, ak 6= 0 fork ∈Nand (iii) X∞

k=1

Ak

a²_k converges. The change in the nature of the Fatou set of functions in the familyS≡ {ha,b,c(z)≡a+bz−_z^c+f(z) : a, b, c∈R, b, c≥0 andf ∈R} is investigated. Letha(z)≡ha,0,0(z) = a+f(z) wheref ∈R. The dynamics ofhais mainly investigated in Section 6.1. A sufficient condition for f ∈ R^∗ for being bounded on the imaginary axis is provided and the Fatou sets of hb,c(z) ≡ h0,b,c(z) = bz− ^c_z +f(z) and h_b(z) ≡ h_0,b,0(z) =bz+f(z) are explored in Section 6.2 when f ∈ R^∗ is bounded on the imaginary axis. Several examples are discussed in Section 6.3. In Section 6.4, the dynamics of a+ tanz is explored for a∈C\R though these functions are not real meromorphic.

Dynamics of f _λ (z ) = λtanh (e ^z )

The dynamics of transcendental meromorphic functions in the one parameter family M={fλ(z) =λ f(z) : f(z) = tanh(e^z) forz ∈Candλ∈R\ {0} }

is studied in the present chapter. The dynamics of the transcendental entire functions λe^z,(λ ∈ C\ {0}) have been extensively studied and a number of interesting properties of the Julia set of λe^z are proved [15–17^,27^,28^,39–41^,45^,47^,48^,89^,97^,102^,107^,115^,116^, 135^,136]. Devaney and Keen [42–44] studied the dynamics of meromorphic functions with constant/polynomial Schwarzian derivatives and in particular, the dynamics of functions in the one parameter family {λtanz : λ ∈ R\ {0} }. Jiang [73], Keen and Kotus [80] furthered the study of dynamics of the tangent family. It is worth to note that the dynamics of λtanhz is essentially same as the dynamics of λtanz, since tanhz and tanz are conformally conjugate by the conjugating map ψ(z) =iz.

The real meromorphic function tanz maps the upper half-plane into itself and this simplifies the determination of the dynamics of λtanhz, λ ∈ R\ {0}. But, the mapping properties of the real meromorphic function tanh(e^z) are comparatively more complicated.

Further, the function tanh(e^z) is a meromorphic function having non-rational Schwarzian derivative and the order is infinite. However, the functionsλe^z and λtanhz have constant Schwarzian derivatives and finite orders. Even though the order of fλ(z) = λtanh(e^z) is

infinite, it has only 3 finite asymptotic values, namely ±λ and 0. The asymptotic values of e^z and tanhz are logarithmic that are the simplest kind of direct singularities of the respective inverse functions. However, the asymptotic values±λoffλ are logarithmic, and over the asymptotic value 0 there lie a direct singularity as well as ordinary points of the inverse function of f_λ. Thus, the properties of the function f_λ differ in many ways from those of λe^z and λtanhz and this suggests that we explore the dynamics offλ.

The change in the dynamical behaviour of the functions fλ ∈Mis also investigated in this chapter. In the dynamics of functions fλ ∈ M, we show that the bifurcation occurs only at one critical parameter λ^∗ ≈ −3.2946. Then, certain topological properties of the Fatou sets of fλ are proved. Finally, it is shown that the measure of the Julia set of fλ is zero.

2.1 Properties of f

_λ

In this section, we prove some basic results about the functions fλ ∈M that are relevant in the study of the dynamics of fλ. The function fλ(z) = λtanh(e^z) is periodic of minimal period 2πiand maps the real lineRonto (0, λ) forλ >0. The poles offλ(z) are the zeros of cosh(e^z), and hence they satisfye^2ez =−1 = e^iπ(2k+1) fork ∈Z. Therefore, the set of poles off_λ(z) is{z =x+iy∈C : x= ln^π

2(2k+ 1) and y= ^π₂(2l+1) wherek ∈Zandl ∈Z}. Further, all the poles are simple and lie in the right half-plane {z ∈ C : <(z) ≥ ln(^π₂)}. Observe thatfλ(z) = λtanh(e^z) = λ(1 +_e2ez⁻²+1) and order of _e2ez⁻²+1 is infinity ( [101], page- 216) and therefore, the order of the function fλ(z) = λtanh(e^z) is infinite.

Lemma 2.1.1. Let g be a meromorphic function, h be a non-constant entire function and g, h∈S. Let F(z) =g(h(z)) be the composition function. If a is a finite asymptotic value of F(z), then either a is an asymptotic value of g or there exists b ∈C such that g(b) =a and b is an asymptotic value of h. Consequently, the number of finite asymptotic values of the composite function F is at most the sum of the numbers of finite asymptotic values of

the individual functions g and h.

Proof. Letγ : [0, ∞)→C be an asymptotic path corresponding to an asymptotic valuea for the function F(z). Let M denote the collection of all limit points of the set

{h(γ(tk)) : {tk}is any sequence of positive real numbers which tends to∞ask → ∞}. Observe that g(z) =a for every z ∈ M. Since g is a non-constant meromorphic function, the set M cannot have any limit point in C. Therefore, MT

C is a discrete subset of C.

Now we claim thatM contains only one element inC. If possible, the setb M contains more than one element inC. Suppose thatb m₁ andm₂ are inM withm₁ 6=m₂. Then, there exist open discs B₁(m₁) and B₂(m₂) such that B₁(m₁)∩M ={m₁} and B₂(m₂)∩M ={m₂}. The curve h(γ(t)) intersects the discs B1(m1) and B2(m2) infinitely many times and also the boundaries C1 = ∂B1(m1) and C2 = ∂B2(m2) of these discs infinitely many times.

Note that, if {h(γ(t)) : t ≥ 0} ∩Ci is a finite set S (say), then ST

M 6= ∅ which is a contradiction to Bi(mi)∩M ={mi} for i= 1, 2. Suppose that {h(γ(t)) : t≥0} ∩Ci is an infinite set. Then, the intersecting points{h(γ(t)) : t≥0} ∩Ci will have a limit point, li (say), since C1 and C2 are compact. This implies that li ∈ M which is a contradiction toB_i(m_i)∩M ={m_i} fori= 1, 2. Therefore,M is a singleton set in C.b

If M ={b} where b ∈ C, then a =g(b) and b is an asymptotic value of h(z). If M = {∞}, then a is an asymptotic value of g(z). Therefore, in both the cases, a corresponds either to an asymptotic value of h or to that of g. This completes the proof.

The following proposition determines all the singular values of fλ ∈M.

Proposition 2.1.1. Let fλ ∈M. Then fλ(z) has only three (finite) asymptotic values and no critical values.

Proof. Since f_λ⁰(z) = λ e^z sech²(e^z) 6= 0 for anyz ∈C, it follows that fλ(z) has no critical points and hence it has no critical values.

Turning to asymptotic values, by Lemma 2.1.1, it follows that fλ(z) will have at most 3 finite asymptotic values, since e^z has only one finite asymptotic value, namely, 0 and λtanh(z) has two finite asymptotic values, namely, λ and −λ.

If γ₁(t) = −t for t ∈ [0, ∞), then lim

t→∞f_λ(γ₁(t)) = 0. If γ₂(t) = t for t ∈ [0, ∞), then

tlim→∞f_λ(γ₂(t)) =λ. When γ₃(t) = t+iπ for t ∈ [0, ∞), lim

t→∞f_λ(γ₃(t)) =−λ. Therefore, 0 and ±λ are the three finite asymptotic values of fλ(z).

Two transcendental meromorphic functions f, g : C → Cb are called conformally con- jugate if there is an analytic homeomorphism ψ on Cb such that ψ(f(z)) = g(ψ(z)) for all z ∈ C. Sinceb f is transcendental, the function ψ(f(z)) is undefined only at ∞ which means that g(ψ(z)) is undefined only at ∞. Therefore, any conformal conjugacy ψ exist- ing between two meromorphic functions satisfiesψ(∞) = ∞and hence will be of the form ψ(z) =az+b where a and b are complex constants with a6= 0. In the following, we show that no two functionsfλ1 andfλ2 inMare conformally conjugate. Suppose that there exists an analytic homeomorphismψ(z) = az+b for allz ∈Cb between two functionsfλ1 andfλ2

inMwith λ₁ 6=λ₂. In the proof of Proposition 2.1.1, it is shown that the function f_λi has three finite asymptotic values, namely, 0, λi and −λi for i= 1, 2. Note that ±λi are the exceptional values offλi. Now, the conjugacy mapψ is required to take the set{λ1, −λ1} to {λ2, −λ2}. That is, either ψ(λ1) = λ2, ψ(−λ1) = −λ2 or ψ(λ1) = −λ2, ψ(−λ1) = λ2. This implies that b = 0 and consequently, ψ(z) =az. Therefore, afλ1(z) =fλ2(az) for all z ∈C. It follows thataf_λ⁰₁(0) = af_λ⁰₂(0) andλ1 =λ2 which is not true.

Define φ : R → R by φ(x) = _f(x)^x + _f0¹(x). Now, we discuss some properties of φ and thereby define the critical parameterλ^∗ that will be used later in this chapter. Note that, φ(x) = ^xf_f(x)f^0(x)+f(x)_0(x) = _tanh(e¹ x)

x+ ^tanh(ex)

e^xsech^2(ex)

= _tanh(e¹ x)

x+^e2ex_4e^−ex^−2ex

= _4ex tanh(e^{1 x})

(4xe^x + e^2ex − e^−2ex) . Letting φ1(x) = 4xe^x +e^2ex −e^−2ex, we observe that φ⁰₁(x) = 2e^x(2x+ 2 +e^2ex +e^−2ex) = 2e^xφ2(x), where φ2(x) = 2x+ 2 +e^2ex +e^−2ex. The function φ⁰₂(x) = 2+ 2e^x(e^2ex−e^−2ex)>0 forx <0. This implies thatφ2(x) is strictly increasing for

x <0. Sinceφ2(x)→ −∞asx→ −∞andφ2(x)→2 +e²+e⁻² >0 asx→0, there exists a pointx2 <0 such thatφ2(x)<0 forx < x2,φ2(x2) = 0 andφ2(x)>0 forx2 < x <0 and consequently, φ⁰₁(x) <0 for x < x2, φ⁰₁(x2) = 0 and φ⁰₁(x) >0 for x2 < x <0. Therefore, φ₁(x) is decreasing for x < x₂ and, is increasing for x₂ < x < 0. This shows that the function φ₁(x) attains the minimum value at the point x₂ and the minimum value φ₁(x₂) is negative, because φ1(x) →0 as x → −∞. Since φ1(x) →e² −e⁻² > 0 asx → 0, there exists a unique point x^∗ with x2 < x^∗ < 0 such that φ1(x) < 0 for x < x^∗, φ1(x) = 0 for x=x^∗ and φ1(x)>0 for x^∗ < x <0; and consequently,

φ(x)





<0 for x < x^∗ <0

= 0 for x=x^∗

>0 for x^∗ < x <0

(2.1)

Observe thatφ(x)>0 for x≥0. Define λ^∗ = x^∗

f(x^∗) = −1

f⁰(x^∗) (2.2)

where x^∗ is the unique real root of the equation φ(x) = _f(x)^x + _f0¹(x) = 0. Numerically, it is found that x^∗ ≈ −1.0789 and λ^∗ ≈ −3.2946. In this chapter, x^∗ and λ^∗ denote the numbers as defined by Equation (2.1) and Equation (2.2) respectively.