The strange properties of the infinite power tower

Abstract: In this article we investigate some "unexpected" properties of the "Infinite Power Tower". [y = f(x) = {x^{{x^{{x^{{x^ {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} }}}}}}}] The material collected here is also intended as a potential guide for teachers of high-school/undergraduate students interested in planning an activity of investigative mathematics in the classroom, where the knowledge is gained through the active, creative and cooperative use of diversified mathematical tools (and some ingenuity). The activity should possibly be carried on with a laboratorial style, with no preclusions on the paths chosen and undertaken by the students and with little or no information imparted from the teacher's desk. The teacher should then act just as a guide and a facilitator. The mathematical requisites to follow this path are: functions, properties of exponentials and logarithms, sequences, limits and derivatives. The topics presented should then be accessible to undergraduate or "advanced high school" students.

• The paper establishes that the infinite power tower converges for base values in [e^(-e), e^(1/e)] while diverging or oscillating outside this range.
• It uses algebraic methods and cobweb diagrams to analyze fixed-point stability and a pitchfork bifurcation at x = e^(1/e).
• The findings extend tetration theory by linking recursive exponentiation to historical mathematical insights and suggesting new research directions.

An Investigation of the Infinite Power Tower: Mathematical Properties and Convergence

In the paper titled "The strange properties of the infinite power tower," the author, Luca Moroni, explores the mathematical intricacies of the Infinite Power Tower, also known as tetration when discussing its iteration to infinite heights. The primary focus of the study is to explore the unexpected properties, stability, and convergence issues of this complex function defined recursively by the process of exponentiation where the height of the exponentiation tower tends to infinity.

Core Discussion and Theoretical Insights

The Infinite Power Tower, expressed functionally as $y = f(x) = x^{x^{x^\cdots}}$, raises fundamental questions given its recursive nature and the unconventional transition from a familiar operation, exponentiation, to its repetitive extension. The convergence of such a function is not straightforward and depends significantly on the base $x$.

The sequence that defines the power tower can be generalized as $y_{n+1} = x^{y_n}$, where $y_1 = x$. The fixed-point equation for convergence then becomes $y = x^y$. One key finding is that the convergence of the Infinite Power Tower is guaranteed in the interval $e^{-e} \leq x \leq e^{1/e}$, where it stabilizes to values within the bounded set $1/e \leq y \leq e$.

Convergence and Fixed Points

Examining the stability of fixed points is central to the paper’s investigation. The paper utilizes both algebraic and graphical methods, including cobweb diagrams, to explore the behavior of sequences under various base values $x$. For $x > e^{1/e}$, the sequence diverges. Conversely, for $x < e^{-e}$, a peculiar phenomenon occurs where the sequence fails to converge but rather oscillates between two values forming a 2-cycle.

A further intriguing aspect is the transition at $x = e^{1/e}$, where the fixed point's stability markedly changes—a case described as a "pitchfork bifurcation" in dynamical systems terms.

Implications and Historical Context

Conceptually, this exploration of tetration extends the understanding of hyperoperations—a hierarchy of operations starting from successor, addition, multiplication, and exponentiation, progressively defined by iteration. The recursion underlying tetration aligns with recursive mathematical thinking akin to Peano's axioms.

From a historical perspective, the work touches on the Lambert W function, notable for solving specific transcendental equations without closed-form expressions. Notably, the paper also refers to contributions by Johann Lambert and Leonhard Euler, establishing a mathematical context for iterative exponentiations and their broader implications.

Speculations and Future Directions

The investigation offers a rich ground for further theoretical exploration, particularly in terms of numerical methods to solve complex dynamical sequences and series. Possible future research could explore multidimensional or complex extensions of the tetration and its varying stability landscapes under perturbations.

In conclusion, Moroni’s paper demystifies aspects of the Infinite Power Tower, contributing valuable insights into recursive functions, their convergence properties, and fixed-point theories. The interplay between empirical and theoretical mathematical tools enriches the understanding of this enigmatic function and highlights the depth yet to be fully uncovered in this field of mathematical investigation.