Math Seminars
Fedor Nazarov
Michigan State University
A remark on the product of 2x2 matrices
Let H(t) be the 2 x 2 matrix (1,t;0,1).
Let R_k be an arbitrary sequence of real matrices of determinant
1. Finally, let
P_n(t)=R_n H(t) R_{n-1} H(t) ... R_1 H(t).
Define
B={t\in R : sup_n ||P_n(t)||<+\infty}.
We show that the set B is always of finite Lebesgue measure
but it is not necessarily essentially bounded and can contain
an arbitrary given countable set. This answers a question raised
by Leonid Polterovich.