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Algebraic dynamics from topological and holomorphic dynamics

Banff International Research Station for Mathematical Innovation and Discovery

Let f:S^2 —> S^2 be a postcritically finite branched covering from the 2-sphere to itself with postcritical set P. Thurston studied the dynamics of f using an induced holomorphic self-map T(f) of the Teichmuller space of complex structures on (S^2, P). Koch found that this holomorphic dynamical system on Teichmuller space descends to algebraic dynamical systems: 1. T(f) always descends to a multivalued self map H(f) of the moduli space M_{0,P} of markings of the Riemann sphere by the finite set P 2. When P contains a point x at which f is fully ramified, under certain combinatorial conditions on f, the inverse of T(f) descends to a rational self-map M(f) of projective space CP^n. When, in addition, x is a fixed point of f, i.e. f is a `topological polynomial’, the induced self-map M(f) is regular. The dynamics of H(f) and M(f) may be studied via numerical invariants called dynamical degrees: the k-th dynamical degree of an algebraic dynamical system measures the asymptotic growth rate, under iteration, of the degrees of k-dimensional subvarieties. I will introduce the dynamical systems T(f), H(f) and M(f), and dynamical degrees. I will then discuss why it is useful to study H(f) (resp. M(f)) simultaneously on several compactifications of M_{0,P}. We find that the dynamical degrees of H(f) (resp. M(f)) are algebraic integers whose properties are constrained by the dynamics of f on the finite set P. In particular, when M(f) exists, then the more f resembles a topological polynomial, the more M(f) : CP^n - - -> CP^n behaves like a regular map.

Mathematics; Dynamical systems and ergodic theory; Algebraic geometry; Dynamical systems

video/mp4

Author affiliation: Harvard University

2018-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65871

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/