Iteration of Exponentials with Sign Changes

TL;DR

This paper investigates the behavior of infinite iterations of signed exponential functions with a fixed base, demonstrating convergence within a specific interval and the ability to reach any real number as a limit.

Contribution

It provides a detailed analysis of the convergence properties of signed exponential iterations and characterizes the possible limits for different base values.

Findings

Convergence occurs for bases within a specific interval.

Any real number can be achieved as a limit within this interval.

Additional results are provided for bases outside the main interval.

Abstract

In this paper we consider the iteration of infinitely many signed exponentials with the same base but the signs may vary. We show that for every base in an explicit interval this iteration converges for any sequence of signs and all the real numbers are possible limits. We give some more results for a base outside this interval.