Week 10 Worksheet

  1. Prove that T(x)=3xmod1 is measure-preserving and ergodic and mixing for Lebesgue measure on [0,1).
  2. Define T:[0,1)2[0,1)2 by T(x,y)=(x+αmod1,y+βmod1) for all x,y[0,1). Take for granted that ψj,k(x,y)=e2πi(jx+ky) is an orthonormal basis of L2(X,B,μ). Prove T is ergodic.
  3. Prove that irrational rotations on [0,1) are not mixing.
  4. Fix a linear map B from L2(X,B,μ) to L2(X,B,μ). Prove that B is continuous with respect to the norm 2 if and only if B is bounded in the sense that there is C>0 with Bx2Cx2 for all x in L2(X,B,μ).
  5. Fix a measure preserving map T of a probability space (X,B,μ). Suppose f in L2(X,B,μ) is an eigenfunction of T with eigenvalue η.
    1. Prove that f,f=Tf,Tf=|η|2f,f and conclude |η|=1.
    2. Prove that eigenfunctions with different eigenvalues are orthogonal.
    3. Prove that |f| is a T-invariant function.
    4. Prove, assuming T is ergodic, that |f| is constant almost everywhere.
    5. Prove, assuming T is ergodic, that any other eigenfuction g of T is equal almost everywhere to a constant multiple of f.