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?



Algebraic dynamics in finite fields
There is a growing body of literature on dynamics of rational maps over finite fields. The following paper would seem to be relevant. 


* 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. 


* 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>


