Properties of Convergence and Oscillating Points of the Generalized Infinite Tetration Function

By
Ian Jung
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Abstract

The infinite tetration function, composed of an infinitely high tower of its base, is notable for its counterintuitive convergence interval. In this article, we explore the convergence properties of the generalized infinite tetration function, where the initial value in the iterative process defining the infinite tetration can be any arbitrary number. We start by examining the conditions for convergence within the convergence interval, then focus mainly on the properties of the oscillating points. This paper provides a novel proof of the uniqueness and existence of these oscillating points and examines the function’s convergence to them. Furthermore, we propose a direct relationship between the values of the oscillating points and x in the generalized infinite tetration function, using the two branches of the Lambert W function.
Keywords: infinite tetration function, infinite power tower function, fixed-point iteration, convergence, Lambert W function

1. Introduction

The infinite tetration function, also known as the infinite power tower function, takes the form
f(x) = x^{x^{x^{.^{.^{.}}}}},
which raises a base to its own power infinitely many times. Since the tetration is constructed downward, the function can also be recursively represented as
f(x) = \lim_{n \to \infty} y_n,
where y_0 = x and y_n = x^{y_{n-1}} for n = 1,2,3, \ldots1.

Due to its infinite nature, the function might seem to diverge at first glance. However, this function actually converges within a specific interval, which was first proven to be [ e^{-e}, e^{\frac{1}{e}}] by Euler2.

The convergence value is specifically given by a fixed point of the map y \mapsto x^y, which is the solution y^ to the equation x^{y^} = y^*.
Outside the lower bound of the convergence interval, the function exhibits an intriguing behavior in which it oscillates between different values during the iterative process; we denote these values as oscillating points.

The infinite power tower appears in many intriguing mathematical problems, such as finding values of x and y that satisfy the relation x^y = y^x3. Also, the concepts of fixed points and the convergence of iterations toward them—which play a significant role in analyzing the function—have numerous applications, including the analysis of equilibrium stability in game theory and the study of the behavior of dynamical systems in physics4

It is a well-known fact that the Lambert W function, denoted by W(z), is closely related to the infinite tetration function. The Lambert W function, defined on the complex plane, is the multivalued inverse function of f(z) = ze^z, where z \in \mathbb{C}. This function is useful when representing the solutions of equations involving exponentials or logarithms that cannot be expressed using elementary functions, and it has proven its significance in various fields such as enzyme kinetics and astrophysics5’2. For real arguments, the Lambert W function has two real branches, W_0(x) and W_{-1}(x), which arise from the fact that f(x) = xe^x is not injective. As we will see, these two branches can be used to analyze the convergence points of the infinite tetration function.

Figure 1: Plot of the Lambert W function

In this article, we consider a modified form of the infinite tetration function that allows for an arbitrary initial value a in the iterative process:
x^{.^{.^{.^{x^{a}}}}}.
We will refer to this modified function as the generalized infinite tetration. Toledo also examined this function6, investigating its convergence depending on the value of x and the initial value a. This investigation is briefly demonstrated in the initial sections of this article, addressing the convergence interval of the generalized infinite tetration function and the representation of the convergence values using the Lambert W function. However, we would like to remind you that this paper focuses on the properties of oscillating points of the generalized infinite tetration function. Although some articles explain the oscillating points7’8, the direct relationship between x in the function and the oscillating points has not been determined. Thus, in this paper, we first demonstrate that only 2-cycles of oscillating points can exist, and that such cycles must uniquely exist for each fixed x in the function; that is, the same cycle will occur for a given x regardless of the choice of the initial value a. Next, we prove the convergence to these oscillating points using the fixed-point theorem, and finally, we represent the values of the oscillating points using the two branches of the Lambert W function: W_0(x) and W_{-1}(x).

2. Convergence of the generalized infinite tetration function

The generalized infinite tetration function can alternatively be defined as the limit of an infinite sequence of exponentiations, as explained above. To determine whether this iterative sequence converges to a single value, we apply the fixed-point theorem, which provides the sufficient conditions for convergence of the fixed-point iteration.
[fixed-point theorem9]
If \alpha is a root of the equation x=f(x) where f(x) is a continuous and differentiable function, then the sequence of approximations {x_n}_{n \geq 0} by the fixed-point iteration will converge to the root \alpha provided the initial approximation x_0 is chosen in I, where I is an interval containing the point x=\alpha and |f'(x)|<1 for all x \in I.

Since \alpha is the root of x=f(x), \alpha=f(\alpha).
If x_{n-1} and x_n are two successive approximations to \alpha, then x_n=f(x_{n-1}).
It can also be written as
x_n-\alpha=f(x_{n-1})-f(\alpha).
By the mean value theorem, there exists c \in (x_{n-1},\alpha) such that
\frac{f(x_{n-1})-f(\alpha)}{x_{n-1}-\alpha}=f'(c).
Hence, x_n-\alpha=(x_{n-1}-\alpha)f'(c). Let k be the supremum of |f'(x)| in I. Then, |f'(x)| \leq k < 1.
Therefore,
|x_n-\alpha|=|x_{n-1}-\alpha||f'(c)| \leq k|x_{n-1}-\alpha|.
Similarly,
|x_{n-1}-\alpha| \leq k|x_{n-2}-\alpha| \implies |x_n-\alpha| \leq k^2|x_{n-2}-\alpha|.
Proceeding on, |x_n-\alpha| \leq k^n|x_0-\alpha|. Since k<1,
0 \leq \lim_{n \to \infty} |x_n-\alpha| \leq \lim_{n \to \infty} k^n|x_0-\alpha|=0.
Hence, \lim_{n \to \infty} |x_n-\alpha|=0, implying \lim_{n \to \infty} x_n=\alpha.
Therefore, {x_n} converges to \alpha.

[The recursive definition of the generalized infinite tetration function]
Let y_0 = a where a \in \mathbb{R} and y_n = x^{y_{n-1}} for n = 1,2,3, \cdots. The generalized infinite tetration function is defined as
P(x,a) = \lim_{n \to \infty} y_n.

Hence forth, we will denote the generalized infinite tetration function by P(x,a). By Lemma, the convergence of P(x,a) to a stable fixed point requires that \left| \frac{d(x^y)}{dy} \right| < 1 for a given x. Moroni7 showed that, for the stable fixed point y satisfying y = x^y, the condition holds for y \in (e^{-1}, e), and consequently, x \in (e^{-e},e^{\frac{1}{e}}). It is further demonstrated that P(x,a) can converge even when \left| \frac{d(x^y)}{dy} \right| = 1, specifically at the boundary points (x,y) = (e^{-e},e^{-1}) and (e^{\frac{1}{e}}, e). This is because f(y) = x^y has a “half stable” saddle fixed point when x = e^{\frac{1}{e}} and a unique stable fixed point when x = e^{-e}, which allows for convergence. Thus, it is established that P(x,a) can converge to some value in the interval [e^{-1},e] when e^{-e} \leq x \leq e^{\frac{1}{e}}. The condition on the initial value a for P(x,a) to converge to this stable fixed point, as proved by Toledo6, is

  • \forall a \in \mathbb{R} if x \in [e^{-e},1]
  • a < r if x \in (1, e^{\frac{1}{e}}]

where r = \frac{W_{-1}(-\ln x)}{-\ln x} represents the largest real solution y of the equation y = x^y for x \in (1, e^{\frac{1}{e}}].

3. Single-point convergence value of the infinite tetration function

Let k be the single-point convergence value of P(x,a) for x \in [e^{-e}, e^{\frac{1}{e}}], then k = x^k. Applying the natural logarithm to both sides, we have the following.
\ln k = k \ln x
\implies \ln x = \frac{\ln k}{k} = e^{-\ln k} \ln k
\implies -\ln x = e^{-\ln k} (-\ln k)
Apply the Lambert W function to both sides of the equation.
W(-\ln x) = -\ln k
By the identity W(x) = \ln(x/W(x)), we have
k = e^{-W(-\ln x)} = e^{-\ln(\frac{-\ln x}{W(-\ln x)})} = e^{\ln(\frac{W(-\ln x)}{-\ln x})} = \frac{W(-\ln x)}{-\ln x}.
However, in the domain [-e^{-1},0 ), the Lambert W function has two branches, namely W_0(x) and W_{-1}(x), as shown in Figure1. Note that -e^{-1} \leq -\ln x < 0 implies 1 < x \leq e^{\frac{1}{e}}. Thus, there are two possibilities, k = \frac{W_{0}(-\ln x)}{- \ln x} or k = \frac{W_{-1}(-\ln x)}{- \ln x}, that satisfy the equation k = x^k for x in (1,e^{\frac{1}{e}}]. \frac{W_{-1}(-\ln x)}{-\ln x} is decreasing on (1,e^{\frac{1}{e}}], since the denominator -\ln x is decreasing and negative and the numerator W_{-1}(-\ln x) is increasing and negative on that interval. Thus, the minimum value for \frac{W_{-1}(-\ln x)}{-\ln x} in (1,e^{\frac{1}{e}}] is
\frac{W_{-1}(-\ln x)}{-\ln x} \bigg\rvert_{x = e^{\frac{1}{e}}}= \frac{W_{-1}(-1/e)}{-1/e}= \frac{-1}{-1/e} = e.
In the previous section, it was shown that the stable fixed point y of P(x,a) should satisfy e^{-1} \leq y \leq e. Hence, \frac{W_{-1}(-\ln x)}{-\ln x} cannot be the stable fixed point except when x = e^{\frac{1}{e}}. However, the values of \frac{W_{-1}(-\ln x)}{-\ln x} and \frac{W_{0}(-\ln x)}{-\ln x} are equal when x = e^{\frac{1}{e}}. Therefore, the stable fixed point of P(x,a) generally has the form k = \frac{W_0(-\ln x)}{-\ln x} for x \in [e^{-e}, e^{\frac{1}{e}}]. It is also worth noting that a trivial case arises when the initial value a is equal to the unstable fixed point, i.e., a = \frac{W_{-1}(-\ln x)}{-\ln x}. In this case, P(x,a) trivially takes the value \frac{W_{-1}(-\ln x)}{-\ln x}6.
When 0 < x < e^{-e}, P(x,a) does not converge but oscillates, and the iterative process seems to converge to several points. We will discuss more on this property in the following sections.

4. Uniqueness and existence of oscillating points

When x is outside the interval [e^{-e},e^{\frac{1}{e}}], the infinite tetration function does not converge. In particular, when x \in (0,e^{-e}), the function appears to converge to oscillating points, points that appear alternately with each iteration in Definition. However, first looking into this function, it is hard to infer how many true oscillating points there are. In this section, we will prove the uniqueness of oscillating points for each x in the infinite tetration function.

4.1 Nonexistence of n-cycles of order greater than 2

Oscillating points form a cycle of distinct numbers for each fixed-point iteration. Let us assume that there is an n-cycle with n distinct oscillating points. Label its points in the order of iteration as a sequence
(y_1, y_2, \ldots, y_n)
where y_{i+1} = x ^ {y_i} for i = 1 , 2, \ldots, n with indices taken \bmod\,n so that y_{n+1} = y_{1}. Let us examine whether any sets of oscillating points exist within the range of x > e^{\frac{1}{e}} or x < e^{-e}.


x > e^{\frac{1}{e}} > 1

Since f(y) = x^y is strictly increasing, the map y \mapsto x ^ y is order-preserving. Applying this map thus has the effect of “shifting” every element in the ordered list one step. Now, suppose that the cycle contains at least two elements (i.e., it is nontrivial), and pick any two adjacent elements in the ordered list, say y_i and y_{i+1}. Since the elements are distinct, either y_i < y_{i+1} or y_i > y_{i+1} holds. Then, we have

    \[x^{y_i} < x^{y_{i+1}} \implies y_{i+1} < y_{i+2} $ for the first case and $ x^{y_i} > x^{y_{i+1}} \implies y_{i+1} > y_{i+2}\]

for the second case after applying the mapping once. We can repeat the same argument for y_{i+1} and y_{i+2}, and continue similarly for all adjacent pairs around the cycle in each case. Consequently, every element of the cycle is mapped strictly in the same direction, and we can never return to a previously visited value. Thus, no nontrivial cycle can be formed.

x< e^{-e} < 1

Since f(y) = x^y is strictly decreasing, (f \circ f)(y) = x^{x^y} is strictly increasing and the map y \mapsto x^{x^y} is order-preserving. Thus, applying the similar “shifting” logic, y \mapsto x^{x^y} cannot have a nontrivial cycle. If any cycle exists, it must be trivial, i.e., an initial point y_i satisfies y_i = x^{x^{y_i}}.

Now, consider a nontrivial cycle under y \mapsto x ^ y. Then, under y \mapsto x^{x^y}, the sequence will visit every second point in the original cycle. This sequence of (y_1, y_3, y_5, \ldots ) eventually returns to y_1, specifically when 2m \bmod n = 0 where m is the number of iterations. Since y \mapsto x^{x^y} only has trivial cycles, m = 1 must be a solution to that equation; that is, 2 \bmod n = 0. The only positive integers satisfying this are n = 1 and n = 2. Therefore, the oscillating points can have a period of at most 2.

4.2 Unique existence of 2-cycles

There is a possibility that depending on the initial value a of the iteration defining P(x,a), the process could converge to different 2-cycles. Thus, it is reasonable to consider the existence of several distinct 2-cycles. We will now prove that a 2-cycle must exist uniquely for each x in the interval (0,e^{-e}). Let (y_1, y_2) be a pair of oscillating points in the same 2-cycle, where we can assume y_1 < y_2 without loss of generality. We need to prove that there exists a unique pair (y_1,y_2) for each x in the interval (0,e^{-e}) which satisfies the two following conditions: \begin{cases} y_2 = x^{y_1}\ y_1 = x^{y_2} \end{cases} We observe that y_1, y_2 < 1 since x < 1. Combining the two equations gives us

(1)   \begin{equation*} y_1^{y_1} = y_2^{y_2} \end{equation*}

Let y_2 = p y_1 where p>1, since y_1 < y_2. Apply the natural logarithm to both sides of eq1.

    \[y_1 \ln y_1 = y_2 \ln y_2 = p y_1 \ln(p y_1)\]


    \[\implies \ln y_1 = p \left(\ln p + \ln y_1 \right)\]


    \[\implies (p-1)\ln y_1 = -p \ln p\]


(2)   \begin{equation*}\implies\begin{cases}y_1 = p^{\frac{p}{1-p}} y_2 = p^{\frac{1}{1-p}}\end{cases}\end{equation*}


This representation of y_1 and y_2 using p is also proved by Moroni. Now, apply y_2=x^{y_1} on eq1.

y_1^{y_1} = x^{y_1 y_2}

We can rearrange this equation in terms of \ln x

    \[\ln x = \frac{y_1 \ln y_1}{y_1 y_2} = \frac{\ln y_1}{p y_1} = \frac{p}{1-p} \ln p \cdot \frac{1}{(p \cdot p^{\frac{p}{1-p}})} = \frac{\ln p \cdot p^{\frac{p}{p-1}}}{1-p}\]

It is sufficient to show that a unique p exists for each x to prove the unique existence of (y_1,y_2) for each x, since the value of y_1 and y_2 can be uniquely determined by the unique value of p, as shown by eq2. For 0 < x < e^{-e}, \ln x is strictly monotone with the range of -\infty < \ln x < -e. If \dfrac{\ln p \cdot p^{p/(p-1)}}{1-p} is also strictly monotone for p > 1 and has the range containing { y \mid -\infty < y < -e }, then there must be one unique p for each fixed x in 0 < x < e^{-e} that satisfies \ln x = \dfrac{\ln p \cdot p^{\frac{p}{p-1}}}{1-p}.

Figure 2: Plot of y=\frac{lnx \cdot x^{\frac{x}{x-1}}}{1-x}


The function
f(x) = \dfrac{\ln x \cdot x^{\frac{x}{x-1}}}{1-x}
is strictly monotone, and has the range {y | -\infty < y < -e} for x > 1.

We will first prove the monotonicity of the function, and then determine its range. Let x = e^{2t}, where t > 0. Then we have

(3)   \begin{align*} \frac{\ln x \cdot x^{\frac{x}{x-1}}}{1-x} &= \frac{2t}{1-e^{2t}} e^{\frac{2te^{2t}}{e^{2t}-1}} = \frac{2t}{1-e^{2t}} e^{t\frac{2e^{t}}{e^{t}-e^{-t}}} = \frac{2t}{1-e^{2t}} e^{t \left( 1 + \frac{e^{t}+e^{-t}}{e^{t}-e^{-t}} \right) } \\ &= \frac{2te^t}{1-e^{2t}} e^{t \frac{(e^{t}+e^{-t})/2}{(e^{t}-e^{-t})/2} } = \frac{t}{(e^{-t}-e^{t})/2} e^{\frac{t \cosh(t)}{\sinh(t)} } \\ &= -\frac{t}{\sinh(t)} e^{\frac{t \cosh(t)}{\sinh(t)}} = -t \operatorname{csch}(t) e^{t \coth(t)}. \end{align*}


Since x=e^{2t} is strictly monotone for x>1 and t>0, it suffices to show that -t \operatorname{csch}(t) e^{t \coth(t)} is strictly monotone to prove that \frac{\ln x \cdot x^{x/(x-1)}}{1-x} is strictly monotone. Thus, We will now show that -t \operatorname{csch}(t) \cdot e^{t \coth(t)} is strictly monotone for t > 0. Differentiating the function with respect to t, we have

    \[\frac{d}{dt}(-t \operatorname{csch}(t) e^{t \coth(t)}) = e^{t \coth(t)} \operatorname{csch}(t)(t^2 \operatorname{csch}^2 (t)-1).\]

Since t^2 \operatorname{csch}^2 (t) = \frac{t^2}{\sinh^2 (t)} < 1 and \operatorname{csch}(t) > 0 and e^{t \coth(t)} > 0 holds for all t>0, the following also holds for all t>0.

    \[\frac{d}{dt}(-t \operatorname{csch}(t) e^{t \coth(t)}) < 0\]

Therefore, -t \operatorname{csch}(t) e^{t \coth(t)} is strictly decreasing for t > 0, which proves that \dfrac{\ln x \cdot x^{\frac{x}{x-1}}}{1-x} is also strictly decreasing for x > 1

Figure 3: Plot of y=-x \operatorname{csch}(x) e^{x \coth(x)}

Now we will determine the range of the function for x > 1 by finding the value of the limit of f(x) when x approaches 1 and \infty.

The supremum of f(x) is the value of the limit of f(x) when x approaches 1.

(4)   \begin{align*} \lim_{{x \to 1}} \left( \frac{{\ln x \cdot x^{x/(x-1)}}}{{1-x}} \right) &= \lim_{t \to 0} \left( -t \operatorname{csch}(t) e^{t \coth(t)} \right) \\ &= \lim_{t \to 0} \left( \frac{-t}{\sinh(t)} e^ {t \coth(t)} \right) \\ &= \lim_{t \to 0} \left( \frac{-t}{\sinh(t)} e^{\frac{t}{\sinh(t)} \cosh(t)} \right) \end{align*}


Since \lim_{t \to 0} \left( \frac{t}{\sinh(t)} \right) \overset{\text{ô: L'Hôpital}}{=} \lim_{t \to 0} \left( \frac{1}{\cosh(t)} \right) = 1,

    \[\lim_{t \to 0} \left( \frac{-t}{\sinh(t)} e^{\frac{t}{\sinh(t)} \cosh(t)} \right)= -e.\]


Thus, the supremum of f(x) is -e.

The infimum of f(x) is the value of the limit of f(x) when x approaches \infty.

    \begin{align*}\lim_{x\to\infty} \left( \frac{\ln x \cdot x^{x/(x-1)}}{1-x} \right)& = \lim_{t\to\infty} \left( -t \cdot \operatorname{csch}(t) \cdot e^{t \coth(t)} \right) \\ &= \lim_{t\to\infty} \left( -\frac{2t e^{t \coth(t)}}{e^t - e^{-t}} \right) \end{align*}

    \[= \lim_{t \to \infty} \left( -\frac{2t e^{t(\coth(t)-1)}}{1 - e^{-2t}} \right)= \lim_{t\to\infty} (-2t) = -\infty\]

Thus, the infimum of f(x) is - \infty.

We can now conclude the following for x<1: -\infty < \frac{\ln(x) \cdot x^{x/(x-1)}}{1-x} < -e.. The proof ensures that \ln x = \frac{\ln p \cdot p^{p/(p-1)}}{1-p} must have one unique solution of p (p > 1) for each fixed value of x in 0 < x < e^{-e}. Therefore, a unique 2-cycle of oscillating points (y_1, y_2) exist for each x in 0 < x < e^{-e}.

5. Convergence to oscillating points

In the previous section, we have shown that only a unique 2-cycle of oscillating points exist for each x in 0 < x < e^{-e}. However, we also need to show that the iteration defining P(x,a) converges to this 2-cycle of oscillation points.

5.1 Condition on oscillating points

The pair of oscillating points (y_1, y_2) satisfies x^{y_1} = y_2 and x^{y_2} = y_1, that is, x^{x^{y_1}} = y_1 and x^{x^{y_2}} = y_2. By Lemma, \left|\frac{d(x^{x^{y_1}})}{dy_1}\right| < 1 and \left|\frac{d(x^{x^{y_2}})}{dy_2}\right| < 1 are sufficient conditions for the generalized infinite tetration function (and also for the infinite tetration function) to converge to the 2-cycle of oscillating points (y_1,y_2). Computing the derivative, we have

    \[\frac{d(x^{x^{y_1}})}{dy_1}=x^{x^{y_1}} \cdot x^{y_1} (\ln x)^2 = y_1 y_2 (\ln x)^2.\]

Since x^{y_2}=y_1,

    \[y_2 \ln x = \ln y_1 \implies \ln x = \frac{\ln y_1}{y_2}.\]

Thus,

    \[\frac{d(x^{x^{y_1}})}{dy_1} = y_1 y_2 \left(\frac{\ln y_1}{y_2}\right)^2 = \frac{y_1 (\ln y_1)^2}{y_2}.\]

Let y_2 > y_1 and y_2 = py_1 where p>1. Using the relation in eq2, we have

    \[\frac{d(x^{x^{y_1}})}{dy_1} = \frac{y_1 (\ln y_1)^2}{y_2} = \frac{(\ln p)^2 p}{(1-p)^2}.\]


Let p=e^{2t} (t>0), then

    \[\frac{d(x^{x^{y_1}})}{dy_1} = \frac{(\ln p)^2 p}{(1-p)^2} = \frac{e^{2t} (4t^2)}{(1-e^{2t})^2} = \frac{t^2}{\sinh^2(t)}.\]


Since t < \sinh t for t>0,

    \[0 < \frac{d(x^{x^{y_1}})}{dy_1} = \frac{t^2}{\sinh^2(t)} < 1.\]

Thus, \left|\frac{d(x^{x^{y_1}})}{dy_1}\right|<1 holds when oscillating points exist. Similarly, it can be shown that \left|\frac{d(x^{x^{y_2}})}{dy_2}\right|<1. Therefore, y_1 and y_2 are stable fixed points of f(y) = x^{x^y}, and P(x,a) can converge to the 2-cycle of oscillating points (y_1, y_2).

5.2 Condition on initial value of iteration

Now, we will conduct a graphical analysis to identify the conditions on the initial value a that result in P(x,a) converging to the 2-cycle of oscillating points for any fixed x in (0,e^{-e}). For the sake of convenience, we define the following functions: f_1(y) = x^{x^y}, f_2(y) = x^y, and f_3(y) = y.

Consider the equation y = x^{x^{y}}. It is evident that the equation admits a trivial solution \beta satisfying \beta = x^\beta, since this implies x^{x^\beta} = x^\beta = \beta. Having previously established the unique existence of the 2-cycle of oscillating points, it follows that there are exactly two distinct solutions other than the trivial one. Denote these two nontrivial solutions by \alpha and \gamma (previously labeled y_1 and y_2), satisfying \alpha = x^\gamma and \gamma = x^\alpha. Without loss of generality, assume \alpha < \gamma. Since \beta is distinct from \gamma, either \beta > \gamma or \beta < \gamma must hold. The map y \mapsto x^y is order-reversing, so

    \[\beta > \gamma \implies x^\beta < x^\gamma \implies \beta < \alpha,\]

which contradicts the assumption \alpha < \gamma. On the other hand,

    \[\beta < \gamma \implies x^\beta > x^\gamma \implies \beta > \alpha,\]

and it follows that \alpha < \beta < \gamma; that is, the trivial solution \beta must lie strictly between the two nontrivial solutions. Accordingly, we have three points of intersection between f_1(y) and f_3(y), with the middle point also lying on f_2(y), as shown in Figure4.

Figure 4: Plot of f_1(y) = x^{x^y}, f_2(y) = x^y, and f_3(y) = y for x \in (0,e^{-e})}

Let us define the second value of the iteration in Definition as b, where b = x^a. It is important to note that P(x,a) converges to the 2-cycle of oscillating points when the fixed-point iteration on the map y \mapsto f_1(y) with the initial value of a converges to one point, while the same iteration with the initial value b converges a different point. To check this condition, we examine the limit of convergence for different ranges of the initial value a_0. Before doing so, we first analyze some key properties of f_1(y).

The first derivative of f_1(y) is given by

    \[f_1'(y) = \frac{d(x^{x^{y}})}{dy} = x^{x^y + y} \cdot (\ln x)^2,\]


so f_1'(y) > 0 for all y \in \mathbb{R} whenever x \in (0,e^{-e}). Hence f_1(y) is strictly increasing on \mathbb{R}. To determine the inflection point of f_1(y), we compute the second derivative:

    \[f_1''(y) = \frac{d}{dy} \left( x^{x^y + y} \cdot (\ln x)^2 \right) = x^{x^y + y} \cdot (x^y \ln x + 1) \cdot (\ln x)^3.\]

Setting f_1''(y) = 0, the only solution to this equation comes from the factor
x^y \ln x + 1 = 0,
which yields the unique inflection point at y = -\log_x(-\ln x). According to the mean value theorem, there exists c_1 \in (\alpha,\beta) such that

    \[f_1'(c_1) = \frac{f_1(\beta) - f_1(\alpha)}{\beta - \alpha} = \frac{\beta - \alpha}{\beta - \alpha} = 1\]

and there exists c_2 \in (\beta, \gamma) such that

    \[f_1'(c_2) = \frac{f_1(\gamma) - f_1(\beta)}{\gamma - \beta} = \frac{\gamma - \beta}{\gamma - \beta} = 1.\]

Since f_1'(c_1) = f_1'(c_2) = 1, Rolle’s theorem implies the existence of c \in (c_1, c_2) \subset (\alpha, \gamma) such that f_1''(c) = 0. It follows that c = -\log_x(-\ln x), so the unique inflection point lies in the interval (\alpha, \gamma). Observing f_1''(y), we see that f_1''(y) > 0 for y < c and f_1''(y) < 0 for y > c.

Case 1: a_0 \geq \gamma
For y > \gamma > c, we have f_1''(y) < 0, so f_1'(y) is decreasing. Since f_1'(y) > 0 for all y \in \mathbb{R} and f_1'(\gamma) < 1, it follows that

    \[0 < f_1'(y) \leq f_1'(\gamma) < 1\]

for all y \geq \gamma. Therefore, according to Lemma, the fixed-point iteration starting from a_0 \geq \gamma converges to \gamma.

Case 2: \beta < a_0 < \gamma Let {a_{n}}_{n \geq 0} be the sequence defined by a_{n+1}=f_{1}(a_{n}) with initial value \beta < a_{0}< \gamma. Since f_1(y) > y for all y \in (\beta, \gamma), we have a_{n+1} = f(a_n) > a_n, so the sequence is strictly increasing. Also, we can show that {a_{n}} is bounded above by \gamma using induction on n: a_0 < \gamma by the assumption and

    \[a_n < \gamma \implies f_1(a_n) < f_1(\gamma) \implies a_{n+1} < \gamma\]

since f_1(y) is strictly increasing. Thus, by the monotone convergence theorem, {a_{n}} converges to its supremum, L = \sup_n a_{n}. Since f_{1}(y) is continuous and increasing,

    \[f_{1}(L) = f_{1}\left( \sup_{n}a_{n} \right) = \sup_{n}f_{1}(a_{n}) = \sup_{n}a_{n+1} = L,\]

so L is a fixed point of f_{1}(y). Since \beta < a_0 < L, the only fixed point that satisfies this is \gamma. Therefore, L = \gamma, and the fixed-point iteration starting from \beta < a_0 < \gamma converges to \gamma.

Case 3: a_0 = \beta

It is evident that this fixed-point iteration converges to \beta, since \beta is a fixed point of f_1(y).

Case 4: \alpha < a_0 < \beta
Analogous to Case 2, the fixed-point iteration starting from \alpha < a_0 < \beta converges to \alpha.

Case 5: a_0 \leq \alpha
For y < \alpha < c, we have f_1''(y) > 0, so f_1'(y) is increasing. Similarly to Case 1, it follows that

    \[0 < f_1'(y) \leq f_1'(\alpha) < 1\]

for all y \leq \alpha. Therefore, the fixed-point iteration starting from a_0 < \alpha converges to \alpha.

Combining the results for all 5 cases, we can conclude that the fixed-point iteration on y \mapsto f_1(y) with the initial point a_0 converges to

    \[\begin{cases}\alpha & \text{if } a_0 < \beta,\ \beta & \text{if } a_0 = \beta,\ \gamma & \text{if } a_0 > \beta.\end{cases}\]

This repulsive property of the trivial fixed point y = \beta can be further explained by evaluating the derivative of f_1(y) at that point. Note that f_1(y) = x^{x^y} exhibits asymptotic behavior: for x < e^{-e} < 1, \lim_{y \to \infty} f_1(y) = 1 and \lim_{y \to -\infty} f_1(y) = 0. Given this behavior and the uniqueness of the inflection point of f_1(y), one can obtain that the f_1(y) cannot be tangent to f_3(y) = y and must intersect it transversely at each fixed point. Thus, f_1(y) - f_3(y) changes sign from negative to positive at y = \beta, and from positive to negative at y = \alpha and y = \gamma. Therefore,

    \[f_1'(\beta) - f_3'(\beta) > 0 \implies f_1'(\beta) > f_3'(\beta) = 1,\]

so y = \beta is an unstable fixed point.

Figure 5: Cobweb plot of the fixed-point iteration on f_1(y) with the initial value a_0

Now, returning to the original question, if a = \beta, then b = x^\beta = \beta, so the fixed-point iterations on y \mapsto f_1(y) starting from a and b both converge to the fixed point \beta. In fact, \beta = \frac{W_{0}(-\ln x)}{- \ln x}, which can be derived using the method presented in the previous sections. In this case, oscillating points do not exist, but P(x,\beta) rather converges to a single value, namely \beta = \frac{W_{0}(-\ln x)}{- \ln x}. In other cases, however, we have either

    \[a > \beta \implies x^a < x^\beta \implies b < \beta\]

or

    \[a < \beta \implies x^a > x^\beta \implies b > \beta,\]

so each iteration starting from a and b lies on opposite sides of \beta. Thus, the two iterations converge to separate fixed points, namely \alpha and \gamma.

In summary, for each fixed x in (0,e^{-e}), P(x,a) converges to the 2-cycle of oscillating points (y_1,y_2) = (\alpha, \gamma) for all a \in \mathbb{R} except when a = \beta. In that case, the function converges to a single point and P(x,\beta) = \beta holds for x in (0,e^{-e}).

6. Values of oscillating points

In this section, we will find the direct relation between the oscillating points (y_1,y_2) and x of the infinite tetration function. We can determine the limits of y_1 and y_2 as p approaches \infty and 1, by applying the natural logarithm to both sides of the equations in eq 2.

(5)   \begin{align*} \lim_{p \to \infty} \ln y_1 &= \lim_{p \to \infty} \frac{p}{1-p} \ln p = -\infty \implies \lim_{p \to \infty} y_1 = 0 \\ \lim_{p \to \infty} \ln y_2 &= \lim_{p \to \infty} \frac{1}{1-p} \ln p = 0 \implies \lim_{p \to \infty} y_2 = 1 \\ \lim_{p \to 1} \ln y_1 &= \lim_{p \to 1} \frac{p}{1-p} \ln p = -1 \implies \lim_{p \to 1} y_1 = \frac{1}{e} \\ \lim_{p \to 1} \ln y_2 &= \lim_{p \to 1} \frac{1}{1-p} \ln p = -1 \implies \lim_{p \to 1} y_2 = \frac{1}{e} \end{align*}


Since \frac{p}{1-p} \ln p and \frac{1}{1-p} \ln p are strictly decreasing and the natural logarithm is strictly increasing, the values of the limits are either the supremum or the infimum of the range of y_1 or y_2. Thus, the range of y_1 and y_2 is

(6)   \begin{equation*}\begin{cases}0 < y_1 < \frac{1}{e} \\\frac{1}{e} < y_2 < 1\end{cases}\end{equation*}

Note that y_2 \neq 1 holds, as mentioned in the section on the unique existence of 2-cycles, and y_1 \neq 0 since y_1 = x^{y_2}. We can also verify that the limits of x as p approaches \infty and 1 correspond to the infimum and supremum of the range of x where oscillating points emerge.

(7)   \begin{align*} \lim_{p \to 1} x &= \lim_{p \to 1} y_1^{\frac{1}{y_2}} = \left(\frac{1}{e}\right)^e = e^{-e}\ \lim_{p \to \infty} x &= \lim_{p \to \infty} y_1^{\frac{1}{y_2}} = 0 \end{align*}


We obtain the following relations by applying the natural logarithm to Equation 1 and using the identity x = e^{\ln x}.

    \[y_1 \ln y_1 = y_2 \ln y_2 = \ln y_1 \cdot e^{\ln y_1} = \ln y_2 \cdot e^{\ln y_2}\]

Hence, we have

(8)   \begin{align*}\ln y_1 \cdot e^{\ln y_1} &= y_2 \ln y_2, \\\ln y_2 \cdot e^{\ln y_2} &= y_1 \ln y_1. \label{eq5}\end{align*}


Since y_1 and y_2 are both in the interval (0,1), it follows that -1/e \leq y_1 \ln y_1 = y_2 \ln y_2 < 0. Thus, y_1 \ln y_1 and y_2 \ln y_2 are both in the domain of W_0(x) and W_{-1}(x) and we can apply the Lambert W function to Equation 4 and Equation 5. By the ranges of y_1 and y_2 in Eq. 3, we have -\infty < \ln y_1 < -1 and -1 < \ln y_2 < 0. Since W_{-1}(x) \leq -1 \leq W_0(x), we should apply W_{-1}(x) to Equation 4 and W_0(x) to Equation 5.

(9)   \begin{align*} \ln y_1 &= W_{-1}(y_2 \ln y_2) \implies y_1 = e^{W_{-1}(y_2 \ln y_2)} = \frac{y_2 \ln y_2}{W_{-1}(y_2 \ln y_2)}\\ \ln y_2 &= W_0(y_1 \ln y_1) \implies y_2 = e^{W_0(y_1 \ln y_1)} = \frac{y_1 \ln y_1}{W_0(y_1 \ln y_1)} \end{align*}


The identity W(x) = \ln \left( \dfrac{x}{W(x)} \right) was used in the final step. Since x^{y_2} = y_1, we obtain the following result:

(10)   \begin{align*} \ln x &= \frac{\ln y_1}{y_2} = \frac{\ln y_1}{(y_1 \ln y_1)/(W_0(y_1 \ln y_1))} = \frac{W_0(y_1 \ln y_1)}{y_1} \quad \ & \implies x = e^{\frac{W_0(y_1 \ln y_1)}{y_1}} \end{align*}

Similarly, using x^{y_1} = y_2, we have

(11)   \begin{align*} \ln x &= \frac{\ln y_2}{y_1} = \frac{\ln y_2}{(y_2 \ln y_2)/(W_{-1}(y_2 \ln y_2))} = \frac{W_{-1}(y_2 \ln y_2)}{y_2} \ & \implies x = e^{\frac{W_{-1}(y_2 \ln y_2)}{y_2}} \end{align*}

Therefore, although we cannot express y_1 and y_2 as functions of x using previously defined elementary or special functions, we can conclude that the following relation holds.

(12)   \begin{equation*}x = e^{\frac{W_0(y_1 \ln y_1)}{y_1}} = e^{\frac{W_{-1}(y_2 \ln y_2)}{y_2}}\end{equation*}

Figure 6: Convergence behavior of P(x,a): y_1 and y_2 indicate oscillating points, while y represents the single-point convergence value \frac{W_0(-\ln x)}{-\ln x}

7. Conclusion

In this article, we presented a novel proof for the properties of oscillating points in the generalized infinite tetration function, which are observed within the range 0 < x < e^{-e}. Using the fixed-point theorem and other calculus techniques, we proved the uniqueness and existence of these oscillating points for each fixed x, and examined the function’s convergence to them. Additionally, we conducted a graphical analysis to identify the conditions on the initial value required for convergence of the iteration. Finally, we derived a direct relationship between the values of the oscillating points and x, using the two branches of the Lambert W function. All in all, the convergence values and oscillating points of P(x,a) are given as follows:

    \[P(x,a) =\begin{cases}\dfrac{W_0(-\ln x)}{-\ln x} \quad \text{if }\begin{cases}x \in (0,e^{-e}) \text{ and } a = \frac{W_0(-\ln x)}{-\ln x}, \\\text{or } x \in [e^{-e},1] \text{ and } a \in \mathbb{R}, \\\text{or } x \in (1,e^\frac{1}{e}] \text{ and } a < \frac{W_{-1}(-\ln x)}{-\ln x}\end{cases}\dfrac{W_{-1}(-\ln x)}{-\ln x} \quad \text{if } x \in (1,e^\frac{1}{e}] \text{ and } a = \frac{W_{-1}(-\ln x)}{-\ln x} \\(y_1, y_2) \quad \text{if } x \in (0, e^{-e}) \text{ and } a \neq \frac{W_0(-\ln x)}{-\ln x}\end{cases}\]


where y_1 and y_2 are given by final Equation.

While our results establish the properties of the oscillating points and convergence conditions of the generalized infinite tetration function P(x,a), they are limited to real inputs. Future work could further generalize the function by either extending the domain of the function to complex values of x and a, or by allowing more values in the power tower to be arbitrarily chosen. Investigating the possibility of cycles of order greater than 2 in these generalizations may provide a more comprehensive view of the oscillating points of the infinite tetration.

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