# Fixed Points, Symmetries, and Bounds for Basins of Attraction of Complex Trigonometric Functions

Authors: Kasey Bray; Jerry Dwyer; Roger W. Barnard; G. Brock Williams
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

The dynamical systems of trigonometric functions are explored, with a focus ontz=tanz and the fractal image created by iterating the Newton map, Ftz, of tz. The basins of attraction created from iterating Ftz are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x- and y-axis symmetry of the Newton map and explore the nature of the fractal images.


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## Body

## 1. Introduction

Newton’s method is a very well-known iterative procedure for finding roots. Rather than applying Newton’s method to one point at a time, we can instead consider the Newton map:(1)Ffz=z–fzf′z,of the function fz. Iterating Ffz from a given starting value yields the familiar sequence produced by Newton’s method. Iterating the function Ffz yields a sequence of maps with fascinating dynamics.The dynamics of the Newton map have been studied for the cases when the original functionf is a polynomial or rational function (see Dwyer et al. [1]; Dwyer et al. [2]). In this paper, we consider the case fz=tanz.We first present an outline of the primary definitions and theorems, with an emphasis on the concepts used in this paper. Next, we describe the Newton maps ofsinz and cosz before moving to the much more complex case of tanz. We illustrate the symmetries of the Newton map for tanz and provide bounds for the basins of attraction.


### 1.1. Preliminaries in Complex Analysis

An expression of the formz=x+iy, where x and y are real numbers, is a complex number and ℂ is the set of all complex numbers. We call x the real part, denoted as Rez, and y the imaginary part, denoted as Imz. The modulus (or absolute value) of z, z2=x2+y2, is a real number which measures the distance from the origin, and z¯=x−iy is the complex conjugate of z.Each complex numberz in ℂ can be identified with the unique point Rez,Imz in the plane ℝ2. We can establish polar coordinates, r and θ, for z=x+iyby notingx=rcosθ, y=rsinθ, r=x2+y2, and θ is the angle between the positive real axis and the line segment from 0 to z in the counterclockwise direction. Hence, the complex number z=x+iy can be written in the polar form z=reiθ, and using Euler’s equation we obtain z=rcosθ+isinθ.A complex-valued functionfz=fx+iy assigns to each z in the domain exactly one complex number w=fz. Just as z decomposes into real and imaginary parts, each complex-valued function can be written as fz=ux,y+ivx,y, where u and v are each real-valued functions. In essence, fz is a pair of real functions of two real variables that maps regions from its domain in the complex plane onto its range in another copy of the complex plane.The derivative of a complex function is defined by an extension of the definition of the real case. IfG is an open set in the complex plane and f:G⟶ℂ, then f is differentiable at a point z0∈ℂ if limz⟶z0fz−fz0/z−z0 exists. The value of this limit is denoted as f′z and is called the derivative of fatz0.It is possible for a complex function to be differentiable solely at isolated points, thus analyticity, a property defined over open sets, is a stronger condition. A complex-valued functionfz is said to be analytic on an open set G if it has a derivative at every point of G. If fz is analytic on ℂ, then it is said to be entire.The complex numberz is a zero (or root) of the function fz if it is a solution to the equation fz=0.Singularities of functions can result in particularly interesting fractal images. LetBra denote the ball of radius r about a. Then, we have the following: a function f has an isolated singularity at z=a if there exists r>0 such that f is defined and analytic in Bra\a but not inBra. That is, f is an analytic in some neighborhood of a but not at a itself. In addition, if limz⟶afz=∞, then a is called a pole of f. The point a is called a removable singularity if there is an analytic function g:Bra⟶ℂ such that gz=fz for 0<z−a<r. If an isolated singularity is neither a pole nor a removable singularity, it is called an essential singularity.A functionf is said to be meromorphic in a domain G if at every point of G it is either analytic or has a pole. In particular, we regard analytic functions on G as being special cases of meromorphic functions, and in this paper, we consider only analytic and meromorphic functions.


### 1.2. Preliminaries in Complex Dynamical Systems

For an analytic functionf and given z0 in ℂ, the orbit of z0 is the sequence of iterates z0,fz0,f2z0,…,fnz0,… where f2z0 means ffz0 and fnz0 is the nth application of the function f to the value z0. The initial value z0 is called the seed value, and dynamics is interested in the fate of orbits, that is, the behavior of fnz0 as n⟶∞. Do they converge, diverge, cycle, or behave chaotically? A fixed point occurs when fz=z. If fnz=z for some n∈ℤ, and z,fz,…,fn−1z are distinct points, then z is a periodic point with periodn, defined as the smallest n for which fnz=z. The set z,fz,…,fn−1z is called a n-cycle for f. If the orbit of z contains preliminary values before settling at either a fixed point (fn+1z=fz for some n>1) or a periodic orbit (fn+pz=fnz for some n>1, where p is the period of the periodic orbit), then z is called an eventually fixed point or eventually periodic, respectively.For example, consider the complex functionfz=z2. Fixed points of f occur when z2=z, and thus the fixed points are 0 and 1. Seed values −1,i, and −i have orbits −1,1, i,−1,1, and −i,−1,1, respectively. Since each orbit lands on the fixed point 1 after several iterations, these are all eventually fixed points. Similarlyz
0
=
e
2
π
i
/
3 and z0=e4πi/3 are periodic fixed points of period two. If z0>1, then fnz0⟶∞ as n⟶∞, and if z0<1, then fnz0⟶0 asn⟶∞.Theorem 1.
Letf be a (real or complex) continuous function from its domain set to itself. Suppose a and z0 are both in the domain set and fnz0⟶a, then fa=a.Proof.
Since the sequencefnz0⟶a and f is continuous, we must have(2)fa=flimn⟶∞fnz0=limn⟶∞fn+1z0=a.


### 1.3. Newton’s Method

For a real-valued functionfx, x∈ℝ, with root a, we define the Newton map of f as(3)Ffx=x–fxf′x.The sequence of iteratesFfnx0 converges to the sought-after root, a, for an initial guess x0 “close enough” to a. Note that when fx=0, Fx=x; therefore, the convergence of Ffn to a root of fx can be thought of as a convergence to a fixed point of Fx. Henceforth, we will be concerned with the orbits Ffnz0 for various analytic or meromorphic functions f and seed values z0 and explore the fractal nature of the images created by these iterations.Example 1.
Consider the functionfz=z−2z−3, which has zeros at z=2 and z=3, and Newton map Ffz=z−z2−5z+6/2z−5=z2−6/2z−5. For seed valuez0=1/2, the iterates are z0=0.5,z1=1.4375,z2=1.8511,z3=1.9829,z4=1.9997,z5=1.9999,… which are clearly converging to root z=2. Experiments with different values of z0 will reveal that seed values which are closer to the root 2 will produce orbits that converge to 2, and likewise, points closer to 3 will iterate to 3 under Ff. Note that if we choose z0=5/2, Newton’s method will fail. Analytically, we can see this because 5/2 leads to a zero in the denominator of F and informally we see that the point 5/2 lies directly halfway between roots 2 and 3 and is thus pulled equally in both directions by each root. That is, 5/2 separates those points on the real line which will iterate to 2 and those which will iterate to 3, so it seems our method should fail at this point. Figure 1 shows the dynamics for Ff. Iterates of seed values from the green region will converge to 2, whereas seed values from the blue region will iterate to 3.Figure 1
The dynamics of the Newton map offz=z−2z−3, as discussed in Example 1.For any real or complex functionf and any real or complex number w, we define Afw, the attracting basin of w, under the function f, to be the set of all starting points whose iterates limit to the point w. That is,(4)Afw=z∈G:fnz⟶w.For any real or complex functionf and any real or complex number w, we define Af∗w, the primary (or immediate) basin of attraction of w under f, to be the largest set containing w that lies in the basin of attraction of w. Equivalently, the primary basin of attraction is the connected component of the basin of attraction containing w.Theorem 2 (attracting property of Newton’s method).
Given any real or complex analytic functionf with root a, there exists r>0 such that all points within a distance r of a are necessarily in the set AFa, where F is the Newton map of f. That is, for all initial values z0 that are “close enough” to a, the orbit Fnz0 converges toa. An outline of the proof is discussed by Brilleslyper et al. [3].Note thatAF∗w⊆AFw and there is no universal value for r, as it depends heavily on the function f. The value r gives a lower bound on how close a seed value must be in order for guaranteed convergence under Newton’s method. Furthermore, as f becomes more complicated, so do the dynamics of F, and it may not simply be the case that all seed values close to a certain root converge to that root. Consider Figure 2, where the picture of the dynamics for fz=z3−1 is shown.Figure 2
The global dynamics ofFz=−z3+3z2+1/3z, the Newton map for fz=z3−1. The blue region represents AF1, green represents AFe2πi/3, and yellow represents AFe−2πi/3.


### 1.4. Classification of Fixed Points

The behavior of a function near its fixed points can vary, and we use this to classify fixed points. In general, if the action off is to move points closer to the fixed point, we call it attracting. Conversely, a fixed point may be repelling, in which case, no matter how close the seed is to the point, iterates of the seed will diverge from the fixed point. Formally, we have the following definition.Definition 1.
Letf be a map from its domain set G (a subset of either ℝ or ℂ) into itself.(a)
A finite fixed pointa in ℂ is an attracting fixed point (of f) if there exists a neighborhood U of a such that for any point z∈G∩U\a, we have fz−a<z−a.(b)
We call ∞ anattracting fixed point of f if there exists a neighborhood U of ∞ such that for any point z∈G∩U\∞, we have fz>z. That is, the action of f is to move each point in G∩U\∞ closer to ∞ (as measured by the spherical metric).(c)
A finite fixed pointa in ℂ is called a repelling fixed point (of f) if there exists a neighborhood U of a such that for any point z∈G∩U\a, we havefz−a>z−a.(d)
We call ∞ arepelling fixed point of f if there exists a neighborhood U of ∞ such that for any point z∈G∩U\∞, we have fz<z.
Observe that in the case of (a) and (b) in Definition1, UAF∗a⊆AFa. Thus, if a fixed point is attracting, the iterates of any seed value in the neighborhood U converge monotonically to the fixed point, while points in AFa\U will eventually converge to a. The following theorem is helpful in determining the classification of a given fixed point.Theorem 3.
Letfz be an analytic map on a domain G ⊂ ℂ such that fa=a for some a in G. Then, we have the following:(a
)a is an attracting fixed point if and only if f′a<1,(b)
a is a repelling fixed point if and only if f′a>1.We note that iff′a=0, a is called a superattracting fixed point, and if f′a=1, a is a neutral fixed point. The behavior of neutral fixed points is complicated: sometimes they can exhibit both a partial attracting nature and a partial repelling nature. For example, for the complex function fz=sinz, we have a neutral fixed point at a=0. For real-valued seeds, a has attracting properties, but for purely imaginary-valued seeds, a=0 has a repelling nature. Indeed, for z=bi, b∈ℝ, sinbi=isinhb. Now, since sinhb is an increasing function, for allb∈ℝ+, sinnbi⟶∞ as n⟶∞, and for all b∈ℝ−, sinnbi⟶−∞ as n⟶∞. Similarly, neutral fixed points can act in neither an attracting nor repelling fashion.We can see that periodic points correspond exactly to fixed points of higher iteratesfn of f, so the subsequent classifications follow from Theorem 3.Definition 2.
Supposez0,fz0,…,fn−1z0 forms a p-cycle for the map f. That is, fnz0=z0. Let λ=d/dzfnz|z=z0, then the p-cycle of f is called(a)
Superattracting ifλ=0,(b)
Attracting if0<λ<1,(c)
Repelling ifλ>1,(d)
Neutral ifλ=1.


## 1.1. Preliminaries in Complex Analysis

An expression of the formz=x+iy, where x and y are real numbers, is a complex number and ℂ is the set of all complex numbers. We call x the real part, denoted as Rez, and y the imaginary part, denoted as Imz. The modulus (or absolute value) of z, z2=x2+y2, is a real number which measures the distance from the origin, and z¯=x−iy is the complex conjugate of z.Each complex numberz in ℂ can be identified with the unique point Rez,Imz in the plane ℝ2. We can establish polar coordinates, r and θ, for z=x+iyby notingx=rcosθ, y=rsinθ, r=x2+y2, and θ is the angle between the positive real axis and the line segment from 0 to z in the counterclockwise direction. Hence, the complex number z=x+iy can be written in the polar form z=reiθ, and using Euler’s equation we obtain z=rcosθ+isinθ.A complex-valued functionfz=fx+iy assigns to each z in the domain exactly one complex number w=fz. Just as z decomposes into real and imaginary parts, each complex-valued function can be written as fz=ux,y+ivx,y, where u and v are each real-valued functions. In essence, fz is a pair of real functions of two real variables that maps regions from its domain in the complex plane onto its range in another copy of the complex plane.The derivative of a complex function is defined by an extension of the definition of the real case. IfG is an open set in the complex plane and f:G⟶ℂ, then f is differentiable at a point z0∈ℂ if limz⟶z0fz−fz0/z−z0 exists. The value of this limit is denoted as f′z and is called the derivative of fatz0.It is possible for a complex function to be differentiable solely at isolated points, thus analyticity, a property defined over open sets, is a stronger condition. A complex-valued functionfz is said to be analytic on an open set G if it has a derivative at every point of G. If fz is analytic on ℂ, then it is said to be entire.The complex numberz is a zero (or root) of the function fz if it is a solution to the equation fz=0.Singularities of functions can result in particularly interesting fractal images. LetBra denote the ball of radius r about a. Then, we have the following: a function f has an isolated singularity at z=a if there exists r>0 such that f is defined and analytic in Bra\a but not inBra. That is, f is an analytic in some neighborhood of a but not at a itself. In addition, if limz⟶afz=∞, then a is called a pole of f. The point a is called a removable singularity if there is an analytic function g:Bra⟶ℂ such that gz=fz for 0<z−a<r. If an isolated singularity is neither a pole nor a removable singularity, it is called an essential singularity.A functionf is said to be meromorphic in a domain G if at every point of G it is either analytic or has a pole. In particular, we regard analytic functions on G as being special cases of meromorphic functions, and in this paper, we consider only analytic and meromorphic functions.


## 1.2. Preliminaries in Complex Dynamical Systems

For an analytic functionf and given z0 in ℂ, the orbit of z0 is the sequence of iterates z0,fz0,f2z0,…,fnz0,… where f2z0 means ffz0 and fnz0 is the nth application of the function f to the value z0. The initial value z0 is called the seed value, and dynamics is interested in the fate of orbits, that is, the behavior of fnz0 as n⟶∞. Do they converge, diverge, cycle, or behave chaotically? A fixed point occurs when fz=z. If fnz=z for some n∈ℤ, and z,fz,…,fn−1z are distinct points, then z is a periodic point with periodn, defined as the smallest n for which fnz=z. The set z,fz,…,fn−1z is called a n-cycle for f. If the orbit of z contains preliminary values before settling at either a fixed point (fn+1z=fz for some n>1) or a periodic orbit (fn+pz=fnz for some n>1, where p is the period of the periodic orbit), then z is called an eventually fixed point or eventually periodic, respectively.For example, consider the complex functionfz=z2. Fixed points of f occur when z2=z, and thus the fixed points are 0 and 1. Seed values −1,i, and −i have orbits −1,1, i,−1,1, and −i,−1,1, respectively. Since each orbit lands on the fixed point 1 after several iterations, these are all eventually fixed points. Similarlyz
0
=
e
2
π
i
/
3 and z0=e4πi/3 are periodic fixed points of period two. If z0>1, then fnz0⟶∞ as n⟶∞, and if z0<1, then fnz0⟶0 asn⟶∞.Theorem 1.
Letf be a (real or complex) continuous function from its domain set to itself. Suppose a and z0 are both in the domain set and fnz0⟶a, then fa=a.Proof.
Since the sequencefnz0⟶a and f is continuous, we must have(2)fa=flimn⟶∞fnz0=limn⟶∞fn+1z0=a.


## 1.3. Newton’s Method

For a real-valued functionfx, x∈ℝ, with root a, we define the Newton map of f as(3)Ffx=x–fxf′x.The sequence of iteratesFfnx0 converges to the sought-after root, a, for an initial guess x0 “close enough” to a. Note that when fx=0, Fx=x; therefore, the convergence of Ffn to a root of fx can be thought of as a convergence to a fixed point of Fx. Henceforth, we will be concerned with the orbits Ffnz0 for various analytic or meromorphic functions f and seed values z0 and explore the fractal nature of the images created by these iterations.Example 1.
Consider the functionfz=z−2z−3, which has zeros at z=2 and z=3, and Newton map Ffz=z−z2−5z+6/2z−5=z2−6/2z−5. For seed valuez0=1/2, the iterates are z0=0.5,z1=1.4375,z2=1.8511,z3=1.9829,z4=1.9997,z5=1.9999,… which are clearly converging to root z=2. Experiments with different values of z0 will reveal that seed values which are closer to the root 2 will produce orbits that converge to 2, and likewise, points closer to 3 will iterate to 3 under Ff. Note that if we choose z0=5/2, Newton’s method will fail. Analytically, we can see this because 5/2 leads to a zero in the denominator of F and informally we see that the point 5/2 lies directly halfway between roots 2 and 3 and is thus pulled equally in both directions by each root. That is, 5/2 separates those points on the real line which will iterate to 2 and those which will iterate to 3, so it seems our method should fail at this point. Figure 1 shows the dynamics for Ff. Iterates of seed values from the green region will converge to 2, whereas seed values from the blue region will iterate to 3.Figure 1
The dynamics of the Newton map offz=z−2z−3, as discussed in Example 1.For any real or complex functionf and any real or complex number w, we define Afw, the attracting basin of w, under the function f, to be the set of all starting points whose iterates limit to the point w. That is,(4)Afw=z∈G:fnz⟶w.For any real or complex functionf and any real or complex number w, we define Af∗w, the primary (or immediate) basin of attraction of w under f, to be the largest set containing w that lies in the basin of attraction of w. Equivalently, the primary basin of attraction is the connected component of the basin of attraction containing w.Theorem 2 (attracting property of Newton’s method).
Given any real or complex analytic functionf with root a, there exists r>0 such that all points within a distance r of a are necessarily in the set AFa, where F is the Newton map of f. That is, for all initial values z0 that are “close enough” to a, the orbit Fnz0 converges toa. An outline of the proof is discussed by Brilleslyper et al. [3].Note thatAF∗w⊆AFw and there is no universal value for r, as it depends heavily on the function f. The value r gives a lower bound on how close a seed value must be in order for guaranteed convergence under Newton’s method. Furthermore, as f becomes more complicated, so do the dynamics of F, and it may not simply be the case that all seed values close to a certain root converge to that root. Consider Figure 2, where the picture of the dynamics for fz=z3−1 is shown.Figure 2
The global dynamics ofFz=−z3+3z2+1/3z, the Newton map for fz=z3−1. The blue region represents AF1, green represents AFe2πi/3, and yellow represents AFe−2πi/3.


## 1.4. Classification of Fixed Points

The behavior of a function near its fixed points can vary, and we use this to classify fixed points. In general, if the action off is to move points closer to the fixed point, we call it attracting. Conversely, a fixed point may be repelling, in which case, no matter how close the seed is to the point, iterates of the seed will diverge from the fixed point. Formally, we have the following definition.Definition 1.
Letf be a map from its domain set G (a subset of either ℝ or ℂ) into itself.(a)
A finite fixed pointa in ℂ is an attracting fixed point (of f) if there exists a neighborhood U of a such that for any point z∈G∩U\a, we have fz−a<z−a.(b)
We call ∞ anattracting fixed point of f if there exists a neighborhood U of ∞ such that for any point z∈G∩U\∞, we have fz>z. That is, the action of f is to move each point in G∩U\∞ closer to ∞ (as measured by the spherical metric).(c)
A finite fixed pointa in ℂ is called a repelling fixed point (of f) if there exists a neighborhood U of a such that for any point z∈G∩U\a, we havefz−a>z−a.(d)
We call ∞ arepelling fixed point of f if there exists a neighborhood U of ∞ such that for any point z∈G∩U\∞, we have fz<z.
Observe that in the case of (a) and (b) in Definition1, UAF∗a⊆AFa. Thus, if a fixed point is attracting, the iterates of any seed value in the neighborhood U converge monotonically to the fixed point, while points in AFa\U will eventually converge to a. The following theorem is helpful in determining the classification of a given fixed point.Theorem 3.
Letfz be an analytic map on a domain G ⊂ ℂ such that fa=a for some a in G. Then, we have the following:(a
)a is an attracting fixed point if and only if f′a<1,(b)
a is a repelling fixed point if and only if f′a>1.We note that iff′a=0, a is called a superattracting fixed point, and if f′a=1, a is a neutral fixed point. The behavior of neutral fixed points is complicated: sometimes they can exhibit both a partial attracting nature and a partial repelling nature. For example, for the complex function fz=sinz, we have a neutral fixed point at a=0. For real-valued seeds, a has attracting properties, but for purely imaginary-valued seeds, a=0 has a repelling nature. Indeed, for z=bi, b∈ℝ, sinbi=isinhb. Now, since sinhb is an increasing function, for allb∈ℝ+, sinnbi⟶∞ as n⟶∞, and for all b∈ℝ−, sinnbi⟶−∞ as n⟶∞. Similarly, neutral fixed points can act in neither an attracting nor repelling fashion.We can see that periodic points correspond exactly to fixed points of higher iteratesfn of f, so the subsequent classifications follow from Theorem 3.Definition 2.
Supposez0,fz0,…,fn−1z0 forms a p-cycle for the map f. That is, fnz0=z0. Let λ=d/dzfnz|z=z0, then the p-cycle of f is called(a)
Superattracting ifλ=0,(b)
Attracting if0<λ<1,(c)
Repelling ifλ>1,(d)
Neutral ifλ=1.


## 2. Newton’s Method and Trigonometric Functions

As discussed by Alexander et al. [4], the beginning of the study of complex dynamics dates back to 1870 where Ernst Schröder studied iterative equation solving algorithms in the complex plane and hence the nature of the infinite sequence z,fz,f2z,…,fnz,… from a theoretical viewpoint. Schröder’s interest in iterations led to the cornerstones of complex dynamics: Schröder’s fixed-point theorem and fixed point classifications. Schröder used a heuristic approach in his proof of the fixed-point theorem that relied on the Taylor series expansion of f about an attracting fixed point and led to an explanation as to why Newton’s method works. He also studied convergence rates. Namely, for simple roots, a, of a polynomial, Newton’s method converges quadratically in a neighborhood U of a. For roots of multiplicity greater than1, Schröder modified Newton’s method to maintain this desirable convergence, and later, developed a family of similar root-solving algorithms which would either increase the rate of convergence, or have convergence of an arbitrary order.Schröder was successful in developing fundamental tools of complex dynamics and applying them to the Newton map ofqz=z2−1: “Schröder observed, first, that there were periodic points of Fqz of every order on the imaginary axis.His second observation was that if z was on the imaginary axis, but not eventually periodic, then Fqkz takes on infinitely many values” [4]. In other words, the forward orbit of z consists of infinitely many distinct points. Yet, his work raised an unanswered fundamental question: how far away can an arbitrary point z be from an attracting fixed point a of a function f, such that fnz⟶a? It was not until the emergence of independent work by Pierre Fatou and Gaston Julia near the end of WWI that solutions to this query were explored. Furthermore, Schröder was unable to extend his results to higher-degree polynomials, in particular, he failed in an attempt to understand Newton’s method for the cubic cz=z3−1 and it took more than 45 years before the dynamics of Fcz were well understood [4].The 1918Grand Prix des Sciences Mathematiques competition was devoted to the study of the iteration of complex functions and hence sparked a flurry of papers published by Fatou and Julia on the global behavior of iterates of complex rational functions. During this time, they were able to bring about a substantial global theory of rational dynamics, but little was known about the global iterative behavior of other kinds of complex functions [4]. Unfortunately, it was not until the onset of the personal computer that complex dynamics received considerable public attention again, and in the early 1980s the field exploded as computer-generated images of the Mandelbrot set and Newton’s method on cubic polynomials circulated widely.As shown later, the main focus of this paper will be the iteration of the functionz−sinzcosz, so a look at literature concerning the dynamics of trigonometric functions will be of use. Unfortunately, this particular set of research is more limited than that of polynomials and rational functions. Schubert [5] proves that the area of the Fatou (or stable) set of the sine function in a vertical strip of width 2π is finite. He also references the dynamical work, both historical and recent, carried out with functions such as sinz,cosz,hsinz+a for 0<h<1 real, sinaz+b for a≠0, and λsinz.Devaney [6] discusses the special class of meromorphic functions fz whose Schwarzian derivative:(5)Sfz=f‴zf′z−32f″zf′z2,is a polynomial including the family of functions λtanz. As we know, the fate of asymptotic and critical values under iteration plays a crucial role in determining dynamics, and the main property of maps with polynomial Schwarzian derivatives is that they have a finite number of asymptotic values (all of which are isolated) and no critical values. In particular, for Tλz=λtanz=λ/ieiz−e−iz/eiz+e−iz, where λ>0, we have STλz=2, Tλ has asymptotic values at λi and –λi, and Tλ preserves the real axis.The Julia setJTλ is the closure of the set of repelling periodic points, or equivalently, JTλ is the closure of the set which consists of the union of all of the preimages of the poles of Tλ. Furthermore, all the poles and their preimages are dense in the Julia set. Devaney shows that JTλ is not a fractal set, indeed, it is a smooth manifold of ℂ. If λ=1, then JTλ=ℝ, and all points z=x+iy such that y≠0 tend asymptotically to the neutral fixed point 0. When λ<−1, Tλ has an attracting periodic cycle of period two, and points in the upper and lower half planes hop back and forth as they are attracted to the cycle. And, since Tλ′x>1 for all x∈ℝ, JTλ=ℝ for λ<−1 [6].Devaney’s results show that for0<λ<1, zero is an attracting fixed point for Tλ and JTλ breaks up into a Cantor set. In fact, the basin of zero is infinitely connected, contrasting the situation for polynomial or entire maps (such as z−sinzcosz) in which finite attracting fixed points always have a simply connected immediate basin of attraction (Devaney [6]). A full picture of the parameter plane for the tangent family is presented by Keen and Kotus [7]; however, an extensive literature search has failed to find work on the dynamics of Newton’s method applied to trigonometric functions.


### 2.1. Introduction to the Dynamics of Trig Functions

This paper focuses on the dynamics of the Newton map oftz=tanz, so as a means of comparison, we first look at some of the basic properties and dynamics of the entire functions sz=sinz and cz=cosz.First,sz has a Newton map:(6)Fsz=z−sinzcosz=zcosz−sinzcosz.For alln∈ℤ, the fixed points of Fs are zn:=nπ and Fs′zn=tan2nπ=0 thus, zn is a superattracting fixed point of Fs.Similarly,cz has a Newton map:(7)Fcz=z+coszsinz=zsinz+coszsinz,which has fixed points at z˜n:=π/2+nπ and Fc′z˜n=cot2π/2+nπ=0. Thus, z˜n is a superattracting fixed point of Fc.The images produced from the iteration ofFs and Fc look as one might expect: with strips of width π about each fixed point, where points within this strip will converge (under iterations of the Newton map) to each respective root (see Figure 3(a)). However, we can see fractal properties about the boundaries and the dynamics are far from trivial. Notice that the Newton maps of sinz and cosz have singularities at π/2+nπ and nπ, respectively, which leads to the boundary behavior shown in Figure 3(b).Figure 3
(a) Global dynamics ofFsz and (b) the fractal boundary of Fsz at z=π/2+iy.
(a)
(b)Now, consider the meromorphic functiontz=tanz which yields the corresponding Newton map:(8)Ftz=z−tanztan′z.Note thattan′z=1/cos2z is never zero, so Ftz=z−sinzcosz for all z∈ℂ and Ft is an entire function. Furthermore, the roots of tz are the roots of sinz,zn=nπ for all n∈ℕ; however, the fixed points of Ft occur at both zn=nπ andz˜n=π/2+nπ. Taking the derivative of the Newton map:(9)Ft′z=1−cos2z−sin2z=2sin2z,we see that Ft′zn=0 and Ft′z˜n=2>1; thus, by Theorem 3, for all n∈ℤ, zn is a superattracting fixed point of Ft and z˜n is a repelling fixed point of Ft. Figure 4 shows the computer-generated global dynamics of Ftz. Each root zn lies at the center of one of the main colored bulbs. Seed values found outside of this strip of bulbs fail to iterate to any root under Ft.Figure 4
Global dynamics ofFtz.Recall that forFtz=z−sinzcosz, we have(10)AFzn=z∈ℂFtkz⟶zn,fork=1,2,…,and AF∗zn is the largest connected component of AFzn containing the root zn.A closer inspection of any one primary basin of attraction gives greater insight into what is happening. Continuously zooming in on the boundary of eachAF∗zn reveals the seemingly infinite nature of the fractal image. Each bulb consists of a boundary of bulbs, and it appears that there are an infinite number of them, each consisting of seed values which converge to a different root under iterations of Ft. Figure 5(a) shows a closer view of the dynamics about z0=0. Every seed value from AF∗0 will iterate to 0. Furthermore, notice that from this distance, we see no more subsets of AFz0 about the boundary of AF∗z0. Indeed, it seems that none of the bulbs that touch AF∗zn contain seed values that converge to zn.Figure 5
(a) Primary basin of attraction ofFt about z0=0. (b) The largest bulb attached to AF∗z0 in quadrant I.
(a)
(b)Figure5(b) displays a close-up of the largest bulbs stemming from AF∗z0 in the first quadrant. Notice that this close-up imitates the broader picture of Figure 5(a) in shape and bulb placement. Namely, the hexagonal-like shape of this bulb, while slightly distorted, resembles that of AF∗z0 with the positioning of boundary bulbs in the same general area. Moreover, these loose properties can be observed in any bulb one chooses to zoom in on.


### 2.2. Symmetry ofFtz

Our first exploration into the dynamics seen in Figure5(a) is in the symmetry of the Newton map. We employ the following standard properties for all complex numbers z:(i)
sin−z=−sinz and cos−z=cosz,(ii)
sinz¯=sinz¯ and cosz¯=cosz¯.We first show thatFtz is symmetric about the x−axis for all z∈ℂ. Let Ftz=z−sinzcosz=z1. Then,(11)Ftz¯=z¯−sinz¯cosz¯=z¯−sinzcosz¯=z−sinzcosz¯=z1¯=Ftz¯,that is, for any z=x+iy, if Ft takes z to z1=x1+iy1, then it takes z¯=x−iy to z1¯=x1−iy1.In a similar manner, it can be shown that ifFt takes z=x+iy to z1=x1+iy1, then it takes −z¯=−x+iy to −z1¯=−x1+iy1. Hence, Ft is symmetric about the y−axis.The symmetry ofFt means that an exploration of the dynamics which occur in the first quadrant is sufficient to understand the global dynamics of tanz under Newton’s method (see Figure 6).Figure 6
x- and y-axis symmetry of Ftz.


### 2.3. Bounding the Primary Basins

What can we say about the basins of attraction these images present us with? We would like to, in some way, bound the different sets of seed values converging to distinct roots, and in this section, we will focus on the primary basins about rootszn. Recall that the function being iterated is Ftz=z−sinzcosz, sinz=eiz−e−iz/2i, and cosz=eiz+e−iz/2. So, for purely imaginary seed valuesz=bi, b∈ℝ, we have(12)Ftbi=bi–sinbicosbi=bi−14ie−b−ebe−b+e−b=bi+i4e−2b−e2b=ib+14e2b−e2b,that is, points on the imaginary axis remain on the imaginary axis under iteration. Similarly, points on the real axis remain on the real axis since for z=x∈ℝ, Ftx=x−sinxcosx is in ℝ.Now, according to Definition1, for each attracting fixed point zn, there exists a neighborhood U about zn such that all points in U converge monotonically to zn. We will first examine sets of real points and points of the form z=zn+iy.Proposition 1.
Along the real axis,U=−π/2+zn,π/2+zn. That is, for allz∈ℝ∩−π/2+zn,π/2+zn\zn, n∈ℤ, we have(13)Ftz−zn<z−zn.Proof.
Before proving this result, note the following:(i)
Fora,z∈ℝ, if 0<a<z, then z−a<z hence z−a<z,(ii)
Fora,z∈ℝ, if z<a<0, then z−a>z hence z−a<z,(iii)
Forz∈ℝ, if 0<z<π, then z>sinzcosz, and if −π<z<0, then z<sinzcosz,(iv)
Forz∈ℝ,sinzcosz has period π.
Now, we first prove the result forn=0. That is, we will show that for all(14)z∈ℝ∩−π2,π2\0,Ftz<z.
Suppose thatz∈0,π/2. Then, z>0, sinzcosz>0, and z>sinzcosz, so by (28a), z−sinzcosz<z. Supposez∈−π/2,0. Then,z<0, sinzcosz<0, and z<sinzcosz. Hence, by (28b), z−sinzcosz<z. Thus, for allz∈U=−π/2,π/2, we have z−sinzcosz<z, that is, Ftz<|z.
Let us now consider the general case. Forz∈zn,π/2+zn, let w=z−zn=z–nπ. Then, 0<w<π/2 implies w>sinwcosw and sinwcosw>0. Thus, by the above remarks, we have(15)w−sinwcosw<ww−−sinw−cosw<w⟺w−sinw+nπcosw+nπ<wz−zn−sinzcosz<z−zn⟺Ftz−zn<z−zn.
Similarly, forz∈−π/2+zn,zn, let w=z−zn. Then, −π/2<w<0 and by the same reasoning, we have(16)w−sinwcosw<w⟺Ftz−zn<z−zn.
Therefore, for allz∈ℝ∩−π/2+zn,π/2+zn\zn, n∈ℤ, we have(17)Ftz–zn<z−zn.
To place similar bounds along the imaginary axis, we define the set:(18)βn:zn+bi:b∈ℝ,b≠0,andsinh2b<4b,and we have the following result.Proposition 2.
For allz∈ℂ∩βn\zn, we have Ftz−zn<z−zn.Proof.
We will first prove the result for the attracting fixed pointz0=0, and then extend it to the general case zn=nπ. For all z∈ℂ∩β0\z0, we have(19)sinzcosz=sinbicosbi=14ie−b−ebe−b+e−b=−i4e−2b−e2b.
Thus,(20)z−sinzcosz−z0<z−z0⟺bi−sinbicosbi<bi⟺bi+i4e−2b−e2b<bib+14e−2b−e2b<b⟺b+14e−2b−e2b<bandb+14e−2b−e2b>−b.
We first consider the case whereb>0:(21)b+14e−2b−e2b<b⟺14e−2b−e2b<0e−2b<e2b,b∈0,∞,b+14e−2b−e2b>−b−14e−2b−e2b<2b14e2b−e−2b<2b,⟺sinh2b<4b.
Hence, the setb:sinh2b<4b satisfies the desired inequality.
Now, suppose thatb<0 and let b=−a, where a>0. Then, we have(22)b+14e−2b−e2b<b−a+14e2a−e−2a<−a=a⟺sinh2a<4a−sinh2b<−4bsinh2b>4b,b+14e−2b−e2b>−b−a+14e2a−e−2a>−−a=−a⟺14e2a−e−2a>0e2a>e−2ae−2b>e2b,b∈−∞,0.
Therefore, for the attracting fixed pointz0, for all z∈ℂ∩β0\z0, we have(23)Ftz−z0<z−z0.
We now extend this result to the general case. Letz∈ℂ∩βnzn and let w=z−zn. Sincez=zn+bi, we havew=bi for all b such thatsinh2b<4b, hence by the first part we have(24)w−sinwcosw<ww−sinw+nπcosw+nπ<w⟺z−sinzcosz−zn<z−zn.
Thus, for the attracting fixed pointzn, for all z∈ℂ∩βn\zn, we have(25)Ftz−zn<z−zn.
The proof of the last result of this section makes use of both Schwarz’s lemma and the maximum modulus principle.Lemma 1 (Schwarz).
LetD=z:z<1, and suppose f is an analytic on D with(a)
fz≤1forz∈D,(b)
f0=0.Then,f’0≤1 and fz≤z for all z in the open disk D. Moreover, if f’0=1 or if fz=z for some z≠0, then there is a constant c,c=1, such that fw=cw ∀w∈D.Theorem 4 (Maximum modulus).
LetG be a bounded open set in ℂ and suppose f is a continuous function on the closure of G, G¯, and analytic in G. Then,(26)maxfz:z∈G¯=maxfz:z∈∂G,where ∂G is the boundary of G.The maximum modulus principle says that in a bounded domain, an analytic function that has a continuous extension to the boundary attains its maximum modulus on the boundary. Equivalently, if the modulus of an analytic function achieves its maximum value at some pointinside the domain G, then the function is constant in G.We can now show that the unit disk is contained in the primary basin of attraction.Theorem 5.
For allz∈ℂ∩Dn\zn, where Dn≔z:z−zn<1, we have(27)Ftz−zn<z−zn.Proof.
This proof will make use of the following equivalencies for allz∈ℂ:(28a)sinzcosz=12sin2z,(28b)sin2z¯=sin2xcos2iy−cos2xsin2iy=sin2xcosh2y−icos2xsinh2y,(28c)Rezsinz¯=xsin2xcosh2y+ycos2xsinh2y,(28d)sin2z2=14sin22xe2y+e−2y2+cos22xe2y−e−2y2=14e4y+e−4y+2−4cos22x=12cosh4y+1−2cos22x=12cosh4y−cos4x.
We will first prove this result for then=0 case, i.e., we consider z0=0 and the open unit disk D0.
F
t
z
=
z
−
sin
z
cos
z is analytic and continuous on all of ℂ, in particular, Ft is continuous on D0¯ and analytic on D0. Thus, by the maximum modulus principle, ∀z∈D0¯, Ftz=z−sinzcosz attains its maximum value when z=1⟺y=±1−x2. Thus, we have(29)z−sinzcosz2=z2−2Rezsinz¯cosz¯+sinzcosz2=z2−Rezsin2z¯+14sin2z2=z2−xsin2xcosh2y−ycos2xsinh2y+18cosh4y−cos4x≤1−xsin2xcosh21−x2−1−x2cos2xsinh21−x2+18cosh41−x2−cos4x,for−1≤x≤1.
Now, let us definegx:=1−xsin2xcosh21−x2−1−x2cos2xsinh21−x2+1/8cosh41−x2−cos4x. A direct computation shows that on the interval −1,1, g′x=0 if and only if x=0. This can also be seen informally by graphing either gx or g′x. Therefore, g takes on its maximum value at x=0 and we have(30)z−sinzcosz2≤gx≤g0=1−sinh2+18cosh4−1,for all<1−1≤x≤1.
Thus, the maximum ofFtz is less than 1 for all z∈D0¯, so certainly Ftz<1 for all z∈D0. Furthermore, F0=0, thus by Schwarz’s lemma, for all z∈D0, Ftz≤z. Note that the only point for which Ftz=z−sinzcosz=z is when z=z0, so excluding z0 from the domain will yield a strict inequality. That is, for all z∈ℂ∩D0\z0, we have Ftz<z, i.e., Ftz−z0<z−z0.
We now extend this to the general case wherezn=nπ∀n∈ℤ. Consider Dn=z:z−zn<1 and let w=z−zn for z∈Dn¯. Then, we have w≤1, Ft0=0 and Ftw=w−sinwcosw<1 by the same reasoning as above, and thus by Schwarz’s lemma,(31)Ftw≤wz−zn−sinz−zn≤z−zn⟺z−sinzcosz−zn≤z−znfor allz∈Dn.
Hence,(32)Ftz−zn<z−zn∀z∈ℂ∩Dn\zn.
Based on Figure4, further exploration could entail an attempt to bound monotonically converging seed values in AF∗0 within the ellipse x/π/22+y/β2<1, where β:=x:sinh2x=4x, or equivalently, z+c+z−c<π where(33)c=π22–β2.


## 2.1. Introduction to the Dynamics of Trig Functions

This paper focuses on the dynamics of the Newton map oftz=tanz, so as a means of comparison, we first look at some of the basic properties and dynamics of the entire functions sz=sinz and cz=cosz.First,sz has a Newton map:(6)Fsz=z−sinzcosz=zcosz−sinzcosz.For alln∈ℤ, the fixed points of Fs are zn:=nπ and Fs′zn=tan2nπ=0 thus, zn is a superattracting fixed point of Fs.Similarly,cz has a Newton map:(7)Fcz=z+coszsinz=zsinz+coszsinz,which has fixed points at z˜n:=π/2+nπ and Fc′z˜n=cot2π/2+nπ=0. Thus, z˜n is a superattracting fixed point of Fc.The images produced from the iteration ofFs and Fc look as one might expect: with strips of width π about each fixed point, where points within this strip will converge (under iterations of the Newton map) to each respective root (see Figure 3(a)). However, we can see fractal properties about the boundaries and the dynamics are far from trivial. Notice that the Newton maps of sinz and cosz have singularities at π/2+nπ and nπ, respectively, which leads to the boundary behavior shown in Figure 3(b).Figure 3
(a) Global dynamics ofFsz and (b) the fractal boundary of Fsz at z=π/2+iy.
(a)
(b)Now, consider the meromorphic functiontz=tanz which yields the corresponding Newton map:(8)Ftz=z−tanztan′z.Note thattan′z=1/cos2z is never zero, so Ftz=z−sinzcosz for all z∈ℂ and Ft is an entire function. Furthermore, the roots of tz are the roots of sinz,zn=nπ for all n∈ℕ; however, the fixed points of Ft occur at both zn=nπ andz˜n=π/2+nπ. Taking the derivative of the Newton map:(9)Ft′z=1−cos2z−sin2z=2sin2z,we see that Ft′zn=0 and Ft′z˜n=2>1; thus, by Theorem 3, for all n∈ℤ, zn is a superattracting fixed point of Ft and z˜n is a repelling fixed point of Ft. Figure 4 shows the computer-generated global dynamics of Ftz. Each root zn lies at the center of one of the main colored bulbs. Seed values found outside of this strip of bulbs fail to iterate to any root under Ft.Figure 4
Global dynamics ofFtz.Recall that forFtz=z−sinzcosz, we have(10)AFzn=z∈ℂFtkz⟶zn,fork=1,2,…,and AF∗zn is the largest connected component of AFzn containing the root zn.A closer inspection of any one primary basin of attraction gives greater insight into what is happening. Continuously zooming in on the boundary of eachAF∗zn reveals the seemingly infinite nature of the fractal image. Each bulb consists of a boundary of bulbs, and it appears that there are an infinite number of them, each consisting of seed values which converge to a different root under iterations of Ft. Figure 5(a) shows a closer view of the dynamics about z0=0. Every seed value from AF∗0 will iterate to 0. Furthermore, notice that from this distance, we see no more subsets of AFz0 about the boundary of AF∗z0. Indeed, it seems that none of the bulbs that touch AF∗zn contain seed values that converge to zn.Figure 5
(a) Primary basin of attraction ofFt about z0=0. (b) The largest bulb attached to AF∗z0 in quadrant I.
(a)
(b)Figure5(b) displays a close-up of the largest bulbs stemming from AF∗z0 in the first quadrant. Notice that this close-up imitates the broader picture of Figure 5(a) in shape and bulb placement. Namely, the hexagonal-like shape of this bulb, while slightly distorted, resembles that of AF∗z0 with the positioning of boundary bulbs in the same general area. Moreover, these loose properties can be observed in any bulb one chooses to zoom in on.


## 2.2. Symmetry ofFtz

Our first exploration into the dynamics seen in Figure5(a) is in the symmetry of the Newton map. We employ the following standard properties for all complex numbers z:(i)
sin−z=−sinz and cos−z=cosz,(ii)
sinz¯=sinz¯ and cosz¯=cosz¯.We first show thatFtz is symmetric about the x−axis for all z∈ℂ. Let Ftz=z−sinzcosz=z1. Then,(11)Ftz¯=z¯−sinz¯cosz¯=z¯−sinzcosz¯=z−sinzcosz¯=z1¯=Ftz¯,that is, for any z=x+iy, if Ft takes z to z1=x1+iy1, then it takes z¯=x−iy to z1¯=x1−iy1.In a similar manner, it can be shown that ifFt takes z=x+iy to z1=x1+iy1, then it takes −z¯=−x+iy to −z1¯=−x1+iy1. Hence, Ft is symmetric about the y−axis.The symmetry ofFt means that an exploration of the dynamics which occur in the first quadrant is sufficient to understand the global dynamics of tanz under Newton’s method (see Figure 6).Figure 6
x- and y-axis symmetry of Ftz.


## 2.3. Bounding the Primary Basins

What can we say about the basins of attraction these images present us with? We would like to, in some way, bound the different sets of seed values converging to distinct roots, and in this section, we will focus on the primary basins about rootszn. Recall that the function being iterated is Ftz=z−sinzcosz, sinz=eiz−e−iz/2i, and cosz=eiz+e−iz/2. So, for purely imaginary seed valuesz=bi, b∈ℝ, we have(12)Ftbi=bi–sinbicosbi=bi−14ie−b−ebe−b+e−b=bi+i4e−2b−e2b=ib+14e2b−e2b,that is, points on the imaginary axis remain on the imaginary axis under iteration. Similarly, points on the real axis remain on the real axis since for z=x∈ℝ, Ftx=x−sinxcosx is in ℝ.Now, according to Definition1, for each attracting fixed point zn, there exists a neighborhood U about zn such that all points in U converge monotonically to zn. We will first examine sets of real points and points of the form z=zn+iy.Proposition 1.
Along the real axis,U=−π/2+zn,π/2+zn. That is, for allz∈ℝ∩−π/2+zn,π/2+zn\zn, n∈ℤ, we have(13)Ftz−zn<z−zn.Proof.
Before proving this result, note the following:(i)
Fora,z∈ℝ, if 0<a<z, then z−a<z hence z−a<z,(ii)
Fora,z∈ℝ, if z<a<0, then z−a>z hence z−a<z,(iii)
Forz∈ℝ, if 0<z<π, then z>sinzcosz, and if −π<z<0, then z<sinzcosz,(iv)
Forz∈ℝ,sinzcosz has period π.
Now, we first prove the result forn=0. That is, we will show that for all(14)z∈ℝ∩−π2,π2\0,Ftz<z.
Suppose thatz∈0,π/2. Then, z>0, sinzcosz>0, and z>sinzcosz, so by (28a), z−sinzcosz<z. Supposez∈−π/2,0. Then,z<0, sinzcosz<0, and z<sinzcosz. Hence, by (28b), z−sinzcosz<z. Thus, for allz∈U=−π/2,π/2, we have z−sinzcosz<z, that is, Ftz<|z.
Let us now consider the general case. Forz∈zn,π/2+zn, let w=z−zn=z–nπ. Then, 0<w<π/2 implies w>sinwcosw and sinwcosw>0. Thus, by the above remarks, we have(15)w−sinwcosw<ww−−sinw−cosw<w⟺w−sinw+nπcosw+nπ<wz−zn−sinzcosz<z−zn⟺Ftz−zn<z−zn.
Similarly, forz∈−π/2+zn,zn, let w=z−zn. Then, −π/2<w<0 and by the same reasoning, we have(16)w−sinwcosw<w⟺Ftz−zn<z−zn.
Therefore, for allz∈ℝ∩−π/2+zn,π/2+zn\zn, n∈ℤ, we have(17)Ftz–zn<z−zn.
To place similar bounds along the imaginary axis, we define the set:(18)βn:zn+bi:b∈ℝ,b≠0,andsinh2b<4b,and we have the following result.Proposition 2.
For allz∈ℂ∩βn\zn, we have Ftz−zn<z−zn.Proof.
We will first prove the result for the attracting fixed pointz0=0, and then extend it to the general case zn=nπ. For all z∈ℂ∩β0\z0, we have(19)sinzcosz=sinbicosbi=14ie−b−ebe−b+e−b=−i4e−2b−e2b.
Thus,(20)z−sinzcosz−z0<z−z0⟺bi−sinbicosbi<bi⟺bi+i4e−2b−e2b<bib+14e−2b−e2b<b⟺b+14e−2b−e2b<bandb+14e−2b−e2b>−b.
We first consider the case whereb>0:(21)b+14e−2b−e2b<b⟺14e−2b−e2b<0e−2b<e2b,b∈0,∞,b+14e−2b−e2b>−b−14e−2b−e2b<2b14e2b−e−2b<2b,⟺sinh2b<4b.
Hence, the setb:sinh2b<4b satisfies the desired inequality.
Now, suppose thatb<0 and let b=−a, where a>0. Then, we have(22)b+14e−2b−e2b<b−a+14e2a−e−2a<−a=a⟺sinh2a<4a−sinh2b<−4bsinh2b>4b,b+14e−2b−e2b>−b−a+14e2a−e−2a>−−a=−a⟺14e2a−e−2a>0e2a>e−2ae−2b>e2b,b∈−∞,0.
Therefore, for the attracting fixed pointz0, for all z∈ℂ∩β0\z0, we have(23)Ftz−z0<z−z0.
We now extend this result to the general case. Letz∈ℂ∩βnzn and let w=z−zn. Sincez=zn+bi, we havew=bi for all b such thatsinh2b<4b, hence by the first part we have(24)w−sinwcosw<ww−sinw+nπcosw+nπ<w⟺z−sinzcosz−zn<z−zn.
Thus, for the attracting fixed pointzn, for all z∈ℂ∩βn\zn, we have(25)Ftz−zn<z−zn.
The proof of the last result of this section makes use of both Schwarz’s lemma and the maximum modulus principle.Lemma 1 (Schwarz).
LetD=z:z<1, and suppose f is an analytic on D with(a)
fz≤1forz∈D,(b)
f0=0.Then,f’0≤1 and fz≤z for all z in the open disk D. Moreover, if f’0=1 or if fz=z for some z≠0, then there is a constant c,c=1, such that fw=cw ∀w∈D.Theorem 4 (Maximum modulus).
LetG be a bounded open set in ℂ and suppose f is a continuous function on the closure of G, G¯, and analytic in G. Then,(26)maxfz:z∈G¯=maxfz:z∈∂G,where ∂G is the boundary of G.The maximum modulus principle says that in a bounded domain, an analytic function that has a continuous extension to the boundary attains its maximum modulus on the boundary. Equivalently, if the modulus of an analytic function achieves its maximum value at some pointinside the domain G, then the function is constant in G.We can now show that the unit disk is contained in the primary basin of attraction.Theorem 5.
For allz∈ℂ∩Dn\zn, where Dn≔z:z−zn<1, we have(27)Ftz−zn<z−zn.Proof.
This proof will make use of the following equivalencies for allz∈ℂ:(28a)sinzcosz=12sin2z,(28b)sin2z¯=sin2xcos2iy−cos2xsin2iy=sin2xcosh2y−icos2xsinh2y,(28c)Rezsinz¯=xsin2xcosh2y+ycos2xsinh2y,(28d)sin2z2=14sin22xe2y+e−2y2+cos22xe2y−e−2y2=14e4y+e−4y+2−4cos22x=12cosh4y+1−2cos22x=12cosh4y−cos4x.
We will first prove this result for then=0 case, i.e., we consider z0=0 and the open unit disk D0.
F
t
z
=
z
−
sin
z
cos
z is analytic and continuous on all of ℂ, in particular, Ft is continuous on D0¯ and analytic on D0. Thus, by the maximum modulus principle, ∀z∈D0¯, Ftz=z−sinzcosz attains its maximum value when z=1⟺y=±1−x2. Thus, we have(29)z−sinzcosz2=z2−2Rezsinz¯cosz¯+sinzcosz2=z2−Rezsin2z¯+14sin2z2=z2−xsin2xcosh2y−ycos2xsinh2y+18cosh4y−cos4x≤1−xsin2xcosh21−x2−1−x2cos2xsinh21−x2+18cosh41−x2−cos4x,for−1≤x≤1.
Now, let us definegx:=1−xsin2xcosh21−x2−1−x2cos2xsinh21−x2+1/8cosh41−x2−cos4x. A direct computation shows that on the interval −1,1, g′x=0 if and only if x=0. This can also be seen informally by graphing either gx or g′x. Therefore, g takes on its maximum value at x=0 and we have(30)z−sinzcosz2≤gx≤g0=1−sinh2+18cosh4−1,for all<1−1≤x≤1.
Thus, the maximum ofFtz is less than 1 for all z∈D0¯, so certainly Ftz<1 for all z∈D0. Furthermore, F0=0, thus by Schwarz’s lemma, for all z∈D0, Ftz≤z. Note that the only point for which Ftz=z−sinzcosz=z is when z=z0, so excluding z0 from the domain will yield a strict inequality. That is, for all z∈ℂ∩D0\z0, we have Ftz<z, i.e., Ftz−z0<z−z0.
We now extend this to the general case wherezn=nπ∀n∈ℤ. Consider Dn=z:z−zn<1 and let w=z−zn for z∈Dn¯. Then, we have w≤1, Ft0=0 and Ftw=w−sinwcosw<1 by the same reasoning as above, and thus by Schwarz’s lemma,(31)Ftw≤wz−zn−sinz−zn≤z−zn⟺z−sinzcosz−zn≤z−znfor allz∈Dn.
Hence,(32)Ftz−zn<z−zn∀z∈ℂ∩Dn\zn.
Based on Figure4, further exploration could entail an attempt to bound monotonically converging seed values in AF∗0 within the ellipse x/π/22+y/β2<1, where β:=x:sinh2x=4x, or equivalently, z+c+z−c<π where(33)c=π22–β2.


## 3. Conclusion

The fractal image created from iterating the Newton map oftz=tanz is symmetric about both the x- and y−axis as well as with respect to each attracting fixed point. In general, that which can be said about the dynamics surrounding z0=0 can be said about the dynamics about zn=nπ. Indeed, as was shown, monotonically convergent seed values are bounded inside each primary basin of attraction by a circle of radius at least one centered at zn. These bounds can be extended along the real axes to −π/2+zn and π/2+zn, and along the line z=zn+iy to sinh2b<4b for b∈ℝ. This leads one to conjecture that the unit circle is a minimal bound which could be extended to the ellipse: z−c+z−c<π where c=π/22–β2. Furthermore, one could possibly show that each connected component of AFnπ is bounded by a scaled version of this ellipse.Indeed, as we have highlighted by the exploration of the Newton maps of two (seemingly) simple functions (sinz and tanz), complex dynamical systems have numerous interesting properties available for study.



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*Source: 1853467-2020-04-23.xml*