4
$\begingroup$

Be $\beta > 1$ non-integer.

$T_{\beta}: [0,1)\rightarrow[0,1)$ with $T_{\beta}x = \beta x$ mod$(1) = \beta x-\lfloor\beta x\rfloor$.

Show with Knopp's Lemma that $T_{\beta}$ is ergodic with respect to $\lambda$ Lebesgue measure.
(If $T_{\beta}^{-1}A = A$, then $\lambda(A)=0$ or $1$).

$\underline{\textrm{Knopp's Lemma:}}$ $B$ Lebesgue set. $\mathscr{C}$ is class of subintervals of $[0,1)$ with

a) $\forall$ open subinterval of $[0,1)$ is at most a countable union of disjoint elements from $\mathscr{C}$

b) $\forall A\in\mathscr{C}$: $\lambda(A\cap B)\geq \gamma\lambda(A)$ with $\gamma>0$ independent of A.

Then $\lambda(B)=1$.

  • 0
    Are you really assuming $\beta$ to be *non*-integer? The Lebesgue measure does *not* look like an invariant measure in that case.2014-04-19

1 Answers 1