Reconstruction theorem for complex polynomials

Abstract

Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters \cite{FP}. 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 \cite{FP} 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 \cite{T2} 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 $2m+1$-st 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.

arXiv commentThis is the errata of the original manuscript. It contains a new proof of the main result in section 4. The results remain unchanged