Combinatorial Relationship Between Finite Fields and Fixed Points of Functions Going Up and Down

Abstract

We explore a combinatorial bijection between two seemingly unrelated topics: the roots of irreducible polynomials of degree $m$ over a finite field $F_p$ for a prime number $p$ and the number of points that are periodic of order $m$ for a continuous piece-wise linear function $g_p:[0,1]\rightarrow[0,1]$ that \emph{goes up and down $p$ times} with slope $\pm 1/p$. We provide a bijection between $F_{p^n}$ and the fixed points of $g^n_p$ that naturally relates some of the structure in both worlds. Also we extend our result to other families of continuous functions that goes up and down $p$ times, in particular to Chebyshev polynomials, where we get a better understanding of its fixed points. A generalization for other piece-wise linear functions that are not necessarily continuous is also provided.