Seminars & Colloquia Calendar

Algebra Seminar

Algebraic dynamics from topological and holomorphic dynamics

Location:  Hill 425
Date & time: Wednesday, 28 February 2018 at 2:00PM - 3:00PM

Let $$f:S^2 \rightarrow S^2$$ be an orientation-preserving branched covering from the 2-sphere to itself whose postcritical set
$$P := f^n(x) | x$$ is a critical point of $$f$$ and $$n>0$$ is finite.  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 $$P^n$$.  When, in addition, $$x$$ is a fixed point of $$f$$, i.e. $$f$$ is a
`topological polynomial' $$\rightarrow$$, 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 $$M(f): P^n \rightarrow P^n$$ behaves like a regular map.

Special Note to All Travelers

Directions: map and driving directions. If you need information on public transportation, you may want to check the New Jersey Transit page.

Unfortunately, cancellations do occur from time to time. Feel free to call our department: 848-445-6969 before embarking on your journey. Thank you.