Week 8 Worksheet

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1. Fix $p\in \mathbb{N}$. Define $T:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ by $T(x)=x+1modp$. Let $\mu$ be normalized counting measure on $\mathbb{Z}/p\mathbb{Z}$.
   1. Verify that $\mu$ is $T$ invariant.
   2. Verify that $\mu$ is ergodic for $T$.
2. Fix $r,s\in \mathbb{N}$. Put $X=\mathbb{Z}/r\mathbb{Z}$ and define $S:X\to X$ by $$S(x)=x+smod$$ for all $x\in X$. When is $S$ ergodic?
3. Define $T:[0,1)\to [0,1)$ by $T(x)=2xmod1$. Verify that Lebesgue measure restricted to $[0,1)$ is $T$ invariant.
4. Take $X={\mathbb{R}}^2/{\mathbb{Z}}^2$. Define $T:X\to X$ by $$T(x+{\mathbb{Z}}^2)=Ax+{\mathbb{Z}}^2$$ for all $x\in {\mathbb{R}}^2$ where $A=[\begin{array}{cc}2& 1\\ 1& 1\end{array}]$.
   1. Check for every rectangle $R\subset X$ that $T^{-1}(R)$ has the same area as $R$.
   2. Given an example of a point $x\in X$ that does not have dense orbit for $T$.
   3. Write down a measure on $X$ that is $T$ invariant.
5. Suppose $(X,\mathcal{B},\mu ,T)$ is ergodic. Fix a measurable function $f:X\to [0,\mathrm{\infty })$ with finite integral. Prove that if $f\circ T=f$ then $f$ is equal, on a set of full measure, to a constant function.