Algebraic dynamics in finite fields What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field $\mathbb{F}_{2^n}$ and the map $x\rightarrow f(x)=x+1/x$. We may consider such a map as oriented graph with outdegrees 1 (except outdegree 0 for $x=0$) and indegrees usually 0 or 2 (except again $x=0$ with indegree 0). Each weak connected component contains exactly 1 cycle (except connected component of 0, which is a tree.) What is known about number of cycles of different lengths? About trees growing from these cycles?
 A: There is a growing body of literature on dynamics of rational maps over finite fields. The following paper would seem to be relevant. 


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*Ugolini, S., Graphs associated with the map $x\mapsto x+x^{-1}$ in finite fields of characteristic two, Theory and applications of finite fields, Contemp. Math. 579, 197-204, Amer. Math. Soc., 2012.


Here are a few more to get you started. 


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*Alberto de Faria, Joao and Hutz, Benjamin, Combinatorics of cycle lengths on Wehler K3 Surfaces over finite fields, New Zealand J. Math. 45 (2015), 19-31.

*Alden Gassert, T., Chebyshev action on finite fields, Discrete Math. 315 (2014), 83–94.

*Bach, Eric and Bridy, Andrew, On the number of distinct functional graphs of affine-linear transformations over finite fields, Linear Algebra Appl. 439 (2013), 1312-1320.

*Burnette, Charles and Schmutz, Eric, Periods of Iterated Rational Functions over a Finite Field, 2015, arXiv:1508.04193.

*Flynn, Ryan and Garton, Derek, Graph components and dynamics over finite fields, Int. J. Number Theory 10 (2014), 779-792.

*Roberts, John A. G. and Vivaldi, Franco, Signature of time-reversal symmetry in polynomial automorphisms over finite fields, Nonlinearity 18 (2005), 2171-2192.

*Roberts, John A. G. and Vivaldi, Franco, A combinatorial model for reversible rational maps over finite fields, Nonlinearity 22 (2009), 1965--1982.


And you can try searching for the phrase "finite field" in the arithmetic dynamics bibliography that I've compiled at http://www.math.brown.edu/~jhs/ArithDyn.bib
