Dynamical Varieties

Abstract

The locus of periodic cycles is of fundamental importance to any discrete time dynamical system. We introduce a family Cyc_p(F) of marked cycle varieties that parameterize the $p$-cycles of an algebraic family of dynamical systems. We describe a dynamically natural cell structure for marked cycle curves over the moduli space Per(1) of quadratic polynomials and over the moduli space Per(2) of quadratic rational maps with a 2-periodic critical point. We then analyze the combinatorics of the resulting cell structures, obtaining formulas for the number of cells of each dimension.