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Week 10 Worksheet
- Prove that $T(x) = 3x \bmod 1$ is measure-preserving and ergodic and mixing for Lebesgue measure on $[0,1)$.
- Define $T : [0,1)^2 \to [0,1)^2$ by
\[
T(x,y) = (x + \alpha \bmod 1 , y + \beta \bmod 1)
\]
for all $x,y \in [0,1)$. Take for granted that
\[
\psi_{j,k}(x,y) = e^{2 \pi i (jx+ky)}
\]
is an orthonormal basis of $\Lp{2}(X,\B,\mu)$. Prove $T$ is ergodic.
- Prove that irrational rotations on $[0,1)$ are not mixing.
- Fix a linear map $B$ from $\Lp{2}(X,\B,\mu)$ to $\Lp{2}(X,\B,\mu)$. Prove that $B$ is continuous with respect to the norm $\| \cdot\|_\mathsf{2}$ if and only if $B$ is bounded in the sense that there is $C \g 0$ with
\[
\| B x \|_\mathsf{2} \le C \| x \|_\mathsf{2}
\]
for all $x$ in $\Lp{2}(X,\B,\mu)$.
- Fix a measure preserving map $T$ of a probability space $(X,\B,\mu)$. Suppose $f$ in $\Lp{2}(X,\B,\mu)$ is an eigenfunction of $T$ with eigenvalue $\eta$.
- Prove that
\[
\langle f, f \rangle = \langle Tf, Tf \rangle = |\eta|^2 \langle f, f \rangle
\]
and conclude $|\eta| = 1$.
- Prove that eigenfunctions with different eigenvalues are orthogonal.
- Prove that $|f|$ is a $T$-invariant function.
- Prove, assuming $T$ is ergodic, that $|f|$ is constant almost everywhere.
- Prove, assuming $T$ is ergodic, that any other eigenfuction $g$ of $T$ is equal almost everywhere to a constant multiple of $f$.