### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Brian Nakamura, Rutgers University, bnaka {at} math [dot] rutgers [dot] edu
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: The 2^n Conjecture and Related Questions

Speaker: Roger Nussbaum, Rutgers University

Date: Thursday, February 28, 2013, 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

The "sup norm" on R^n is defined by ||x||:=max{x_i: 1<=i<=n}, where x:=(x_1,x_2,...,x_n). If D is a subset of R^n, a map T:D--->R^n is called nonexpansive (with respect to the sup norm) if ||T(x)-T(y)||<=||x-y|| for all x and y in D. A point x in D is called a periodic point of T of period p if (T^j) (x) is defined for all positive j and (T^p)(x)=x, where p is minimal. (Here T^j denotes the jth iterate of T.) The 2^n conjecture asserts that p<=2^n, which, if true, would be an optimal upper bound. In this talk we shall explain why an analyst might be interested in this question and describe what results are known concerning the 2^n conjecture. Time permitting, we shall also discuss related questions for maps which are nonexpansive with respect to other "polyhedral norms" ||.||, where a norm is called polyhedral if {x in R^n: ||x||<=1} is a polyhedron.