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мне для экзамена нужно, однако типичные образчики \exercise[20 points] Let $A:\;T^n \arrow T^n$ be given by a matrix $A\in GL(n, \Z)$. Prove that $A$ preserves the usual measure. Prove that $A$ is mixing if and only if none of the eigenvalues of $A$ is a root of unity. \ez \exercise[20 points] Consider the map $T(x)=4x(1-x)$ from the interval $[0,1]$ to itself, Prove or disprove that $T$ is uniquely ergodic. \ez \exercise[20 points] Let $f:\; M \arrow M$ be an isometry of a compact metric space. Assume that $\mu$ is a finite, uniquely ergodic measure on $M$. Prove that support of $\mu$ is $M$. \ez \exercise[20 points] Let $f:\; M \arrow M$ be an isometry of a non-compact metric space. Assume that $\mu$ is a finite, uniquely ergodic measure on $M$. Prove that support of $\mu$ is $M$, or find a counterexample \ez Добавить комментарий: |
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