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It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's Topics in ergodic theory p14

Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant, except the identity?

It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's Topics in ergodic theory p14

Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant?

It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's Topics in ergodic theory p14

Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant, except the identity?

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Po C.
Po C.
  • 487
  • 3
  • 14

I knowIt is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's Topics in ergodic theory p14

Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant?

I know that, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse.

Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant?

It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's Topics in ergodic theory p14

Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant?

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Po C.
Po C.
  • 487
  • 3
  • 14

Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?

I know that, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse.

Given a probability space $(X,\mathcal{B},\mu)$, can we always find a measure-preserving transformation $T:X \rightarrow X$ such that $\mu$ is $T$-invariant?