FIXED POINTS OF A FAMILY OF EXPONENTIAL MAPS

We consider the family of functions f λ ( z ) = exp( iλz ) , λ real. With the help of MATLAB computations, we show f λ has a unique attracting ﬁ xed point for several values of λ . We prove there is no attracting periodic orbit of period n ≥ 2 .


Introduction
In this note we show the existence and uniqueness of an attracting fixed point for the map f λ (z) = exp(iλz), z ∈ C, for certain (real) values of the parameter λ.The proofs depend on MATLAB calculations, and as such can be viewed as computer-assisted proofs.By contrast, the exponential map z 7 → exp(z) admits no attracting fixed point.( [2], see Remark 12) We began with a MATLAB program in which one inputs a value λ, a point z 0 a positive integer N , and a tolerance .The program outputs f n (z 0 ), where n is the least integer k ≤ N for which if there is such an n.Here f n (z) is the n−fold iterate of f at z: thus, f 1 (z) = f (z), and f n (z) = f (f n−1 (z)) for n > 1.
We experimented with various values of λ and the computations indicated the function f λ had a fixed point inside the circle |z| ≤ 1/|λ| for |λ| ≤ 1.96 (approximately).However in order to prove the existence of a fixed point, one must use a theorem, which usually involves an invariant domain.Since it was not easy to find such a domain, we took another approach, using the Maximum Modulus Theorem.While the analysis is elementary, the computer-assisted proofs have yielded new results.
Suppose that z 0 is an attracting fixed point of the map f (z).Then for z sufficiently close to z 0 , the iterates z, f (z), f 2 (z), . . .converge to z 0 .
Let's begin by stating some facts.Let f be the function Conversely, any fixed point in the disk |z| < 1/|λ| is attracting. Proof.
The conclusion is an easy exercise.2 Fact 3. The region R λ defined by the inequalities Proof. 1  |λ| log(|λ|) < sgn( λ)y ≤ |z| < 1 |λ| , so log(|λ|) < 1, or |λ| < e. 2 Our first goal was to prove the existence of a fixed point inside the circle |z| = 1/|λ|.We did this using the Maximum Modulus Theorem, or rather a Corollary, called the Minimum Modulus Theorem.Our MATLAB calculations indicated the attracting fixed point should lie in the intersection of |z| ≤ 1/λ with the first quadrant (for λ > 0), denoted Q λ (or simply Q if λ is fixed), so it was in that region we applied the the Minimum Modulus Principle.
A more conventional approach to the existence of a fixed point using, say, the Brouwer Theorem, requires a having region which is mapped into itself by f .But neither the last equality resulting from the fact the power series has real coefficients.Also, it is an easy observation that z 0 is an attracting fixed point for f −λ iff z 0 is an attracting fixed point for f λ .Notation 1.We use both ' exp(•) ' and 'e • ' to denote the exponential function.Let λ ∈ R be one of the values 0.1, 0.2, 0.3, . . ., 1.8, 1.9, 1.95, or 1.96.Let Q be the intersection of the first quadrant <z ≥ 0, =z ≥ 0, with the closed disk |z| ≤ 1/λ.Then there exists a fixed point z in the interior Q o of Q for the map f (z) = exp(iλz).

Existence and Uniqueness of the Fixed Point
Proof.Our calculations using MATLAB show that The idea of the proof is to show |z − f (z)| is bounded below along the boundary of Q by some constant which is greater than 10 −6 .Theorem would then assert the existence of a fixed point for f in Q. Writing z = x + iy, for x, y ∈ R we have Denote the right hand side of the above by g(x, y).We now have to check the values of g(x, y) along the boundaries of Q.We will start with y = 0. Then for 0 Since 0 < λ < e we have g(x, 0) ≥ (1 − e −1 ) 2 on 0 ≤ x ≤ e −1 and by 2e Finally we need to check the boundary on the quarter circle; it is convenient to convert to polar coordinates ánd using the identity cos(α ± β) = cos α cos β ∓ sin α sin β we see that Since both (λ −1 − e − sin θ ) 2 and 2 λ e − sin θ (1 − cos (θ − cos θ)) are nonnegative, we must see where both terms are zero: the second expression is zero only when cos θ = θ.Call this θ 0 ; so θ 0 is approximately .739085.Putting θ = θ 0 into the first term and setting it to zero yields λ = λ 0 := e √ 1−θ 2 0 , or approximately 1.96131.

And for
One might infer from our results that f λ has a attracting fixed point for all values of λ, 0 < λ < λ 0 .But our method of proof can only be applied to finitely many λ.Remark 6.For θ 0 , λ 0 as in the proof of the theorem, the proof shows that the function f 0 (z) = exp(iλ 0 z) has a fixed point at z = z 0 := λ −1 0 e iθ 0 .Since z 0 lies on the circle |z| = 1/λ 0 , it follows |f 0 0 (z 0 )| = 1.Such a point is called a nonhyperbolic, or neutral fixed point.Remark 7. It is possible to express the fixed point z of f λ as an analytic function of λ.Solving z = f λ (z) for λ yields λ = −i log(z)/z.The inverse function is given by z = g(λ) := iW (−iλ)/λ where W is the Lambert Wfunction, or the principal branch of the inverse of w → we w .Since the values of W are not easy to calculate, this does not simplify the question of deciding when the fixed point z = g(λ) is attracting, i.e., when it satisfies |g(λ)| < 1 |λ| .(Cf [1].)Our MATLAB computation indicates the fixed point is unique.That is indeed the case, as we now prove.
Theorem 8. Let λ be one of the values in the Table .Then the map f (z) = exp(iλz) has a unique attracting fixed point.

Proof.
Let R be the region in C determined by the two inequalities x 2 + y 2 < 1/λ 2 and y > 1 λ log(λ).The region R is convex, and by Facts 1, 2, and 3 it is nonempty, and contains all attracting fixed points of the map f .Suppose now that z 1 , z 2 are two distinct fixed points of f .Then z 1 , z 2 lie in R, and if C is a contour joining z 1 and z 2 , If C is the straight line contour joining z 1 and z 2 , then C ⊂ R so that |f 0(z)| < 1 for z ∈ C, and since C is compact, max Remark 9.An alternative, more conventional approach to the existence of a fixed point may be possible using a standard fixed point theorem, such as the Brouwer Theorem.Assume that z 0 satisfies |f (z 0 ) − z 0 | < 10 −6 , and (where C is the line segment joining z and z 0 ) is valid as long as 10 −6 < 2 δ.
A tolerance finer than 10 −6 may be required.We have not carried out these calculations.However, we do not see how the critical value λ 0 could be obtained through this approach.

Theorem 3 .
Minimum Modulus Principle ([3], p. 313) Let U be a bounded open set in C, and let f be analytic on U and continuous on the closure Ū. Assume that f never vanishes on Ū .Then the minimum value of |f | on Ū occurs on the boundary, ∂U.Theorem 4.
Table, it follows from the Minimun Modulus Principle that the function f (z) = exp(iλz) has a fixed point. 2 Corollary 5. From Fact 1, any fixed point for f inside the circle |z| < 1 λ is attracting, so that Theorem 4 establishes the existence of an attracting fixed point in Q • .By Fact 2, the fixed point lies in the intersection Q • ∩ R.