A reconstruction theorem for complex polynomials

Abstract

Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.]. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. Here we show that the result from [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.] also holds for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is 0. In [The reconstruction theorem for endomorphisms, Bull. Braz. Math. Soc. (N.S.) 33(2) (2002) 231–262.] Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the (2m + 1) th image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.