# Seminars & Colloquia Calendar

## Algebraic dynamics from topological and holomorphic dynamics

#### Rohini Ramadas, Harvard

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.

R. Shapiro Organizer's Page

Eilidh McKemmie -Charles Weibel Organizer's Page

Narek Hovsepyan and Ewerton Rocha Vieira Organizer's page

Ziming Shi, Sagun Chanillo, Xiaojun Huang, Chi Li, Jian Song Seminar website Old seminar website

Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Robert Dougherty-Bliss and Doron Zeilberger --> homepage

Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Hong Chen Seminar webpage

Fanxin Wu and Nkhalo Malawo Organizer's website

James Holland; Organizer website

Organizers: Maxime Van de Moortel and Avy Soffer. Organizer's Page

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

Organizer: Luochen Zhao

Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song Organizer's Page

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Dennis Kriventsov Organizer's Webpage

Alex V. Kontorovich, Vlada Sedláček seminar website

Stephen D. Miller

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Fioralba Cakoni Seminar webpage

Organizer's webpage: Organizer's webpage

For information on the Statistical Mechanics Conference, visit HERE

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