Bialgebraicity and dynamical Ax-Schanuel

Böttcher coordinates conjugate the dynamics of a polynomial near infinity to a power map. Whenever the Julia set of the polynomial is neither a circle nor an interval, the Böttcher coordinate is known to be transcendental. This naturally leads to the problem of determining which algebraic structures, if any, are preserved under the Böttcher coordinate. We refer to these as bialgebraic sets. It is also interesting to study the functional transcendence properties of Böttcher coordinates, in analogy with the theory developed for the exponential function, particularly in connection with the Ax–Schanuel theorem. In this talk, I will present a classification of bialgebraic sets under mild topological assumptions on the Julia set, and I will formulate a dynamical Ax–Schanuel conjecture, explaining how it can be proved when the Julia set is disconnected.

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information