A Class of Stationary Sequences

TL;DR

This paper investigates conditions under which sequences defined by polynomial or real function iterations become constant after finite steps, providing a classification of such sequences based on initial values and functions.

Contribution

It introduces a new class of stationary sequences generated by polynomial and real function iterations, characterizing when they stabilize.

Findings

Sequences become constant for specific initial values and polynomials.

Generalization from polynomial to real functions expands the class of stationary sequences.

Provides criteria for sequence stabilization based on initial conditions and functions.

Abstract

We define a class of sequences ${a_n}$ by $a_1=a$ and $a_{n+1}=P(a_n)$, where $P(x)$ is a polynomial with real coefficients. We then find out for which values $a$ and for which polynomials $P(x)$ these sequences will be constant after a certain rank. Then we generalize it from polynomials $P(x)$ to real functions $f(x)$.