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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 bijectionbetween $F_{p^n}$ and the fixed points of $g^n_p$ that naturally relates someof the structure in both worlds. Also we extend our result to other families ofcontinuous functions that goes up and down $p$ times, in particular toChebyshev polynomials, where we get a better understanding of its fixed points.A generalization for other piece-wise linear functions that are not necessarilycontinuous is also provided.

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