# hw1-ERG - Ergodic Theory MTG6401 Ergodic HW Uno Prof. JLF...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Ergodic Theory MTG6401 Ergodic HW Uno Prof. JLF King 29Jan2010 ( Due Wedn., 16Sept, at the beginning of class. Please staple this sheet as the first page of your write-up. ) Notation. Let K be the unit circle, [ , 1 ) wrapped into a circle, with m () its arclength mea- sure. For R , let R be rotation by on K . For explanation of your solutions to each of these three problems, pictures are appropriate. H1: Here ( T : X, X , ) is a mpt, and A is an invariant set, i.e, T 1 ( A ) a.e = A . a Suppose T is bi-mpt. Construct a set E a.e = A so that E is exactly-invariant , i.e, T 1 ( E ) = E . b No longer assume that T is invertible. With B := T k =0 T- k ( A ), prove that B a.e = A and T 1 ( B ) B . Now use B to construct a set E a.e = B which is exactly-invariant. H2: Let R = R be an irrational rotation on K . Partition the circle into two half-open intervals, say A := 0 , 1 3 ) and B := 1 3 , 1 ) ....
View Full Document

## This note was uploaded on 01/26/2012 for the course MTG 6401 taught by Professor Staff during the Fall '09 term at University of Florida.

Ask a homework question - tutors are online