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Reference on time continuous measure-preserving systems
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In Ergodic Theory a measure-preserving dynamical system $(X, varphi)$ is a pair of a probility space $X = (X, Sigma, mu)$ and a measurable map $T:X to X$ such that $mu$ is $T$-invariant, i.e., $mu(A) = mu(T^-1(A))$ for each $A in Sigma$. This is the starting point to a great theory. Many results in this discrete setting are covered in various references.
But I am interested in the following setting: Let $X = (X, Sigma, mu)$ be a probility space and $(T_s)_s in mathbb R$ a family of measurable mappings that $mu$ is $T_s$-invariant for each $s in mathbb R$. I know there is a more general theory that deals with this continuous setting but I am unable to find good references. So I would appreciate good sources for this kinds of questions.
functional-analysis measure-theory reference-request operator-theory ergodic-theory
asked Aug 6 at 9:39
Yaddle
2,906827
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In Ergodic Theory a measure-preserving dynamical system $(X, varphi)$ is a pair of a probility space $X = (X, Sigma, mu)$ and a measurable map $T:X to X$ such that $mu$ is $T$-invariant, i.e., $mu(A) = mu(T^-1(A))$ for each $A in Sigma$. This is the starting point to a great theory. Many results in this discrete setting are covered in various references.
But I am interested in the following setting: Let $X = (X, Sigma, mu)$ be a probility space and $(T_s)_s in mathbb R$ a family of measurable mappings that $mu$ is $T_s$-invariant for each $s in mathbb R$. I know there is a more general theory that deals with this continuous setting but I am unable to find good references. So I would appreciate good sources for this kinds of questions.
functional-analysis measure-theory reference-request operator-theory ergodic-theory
asked Aug 6 at 9:39
Yaddle
2,906827
1
I mean naturally, one would need to impose the condition that $T_t+s = T_t circ T_s$, in which case yes, there are results.
– Shalop
Aug 9 at 22:56
@Shalop Yes, I forgot to write that down. I am looking for a reference on semigroup results :)
– Yaddle
Aug 9 at 23:06
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In Ergodic Theory a measure-preserving dynamical system $(X, varphi)$ is a pair of a probility space $X = (X, Sigma, mu)$ and a measurable map $T:X to X$ such that $mu$ is $T$-invariant, i.e., $mu(A) = mu(T^-1(A))$ for each $A in Sigma$. This is the starting point to a great theory. Many results in this discrete setting are covered in various references.
But I am interested in the following setting: Let $X = (X, Sigma, mu)$ be a probility space and $(T_s)_s in mathbb R$ a family of measurable mappings that $mu$ is $T_s$-invariant for each $s in mathbb R$. I know there is a more general theory that deals with this continuous setting but I am unable to find good references. So I would appreciate good sources for this kinds of questions.
functional-analysis measure-theory reference-request operator-theory ergodic-theory
asked Aug 6 at 9:39
Yaddle
2,906827
In Ergodic Theory a measure-preserving dynamical system $(X, varphi)$ is a pair of a probility space $X = (X, Sigma, mu)$ and a measurable map $T:X to X$ such that $mu$ is $T$-invariant, i.e., $mu(A) = mu(T^-1(A))$ for each $A in Sigma$. This is the starting point to a great theory. Many results in this discrete setting are covered in various references.
But I am interested in the following setting: Let $X = (X, Sigma, mu)$ be a probility space and $(T_s)_s in mathbb R$ a family of measurable mappings that $mu$ is $T_s$-invariant for each $s in mathbb R$. I know there is a more general theory that deals with this continuous setting but I am unable to find good references. So I would appreciate good sources for this kinds of questions.
functional-analysis measure-theory reference-request operator-theory ergodic-theory
asked Aug 6 at 9:39
Yaddle
2,906827
asked Aug 6 at 9:39
Yaddle
2,906827
asked Aug 6 at 9:39
Yaddle
2,906827
asked Aug 6 at 9:39
Yaddle
2,906827
2,906827
1
I mean naturally, one would need to impose the condition that $T_t+s = T_t circ T_s$, in which case yes, there are results.
– Shalop
Aug 9 at 22:56
@Shalop Yes, I forgot to write that down. I am looking for a reference on semigroup results :)
– Yaddle
Aug 9 at 23:06
add a comment |Â
1
I mean naturally, one would need to impose the condition that $T_t+s = T_t circ T_s$, in which case yes, there are results.
– Shalop
Aug 9 at 22:56
@Shalop Yes, I forgot to write that down. I am looking for a reference on semigroup results :)
– Yaddle
Aug 9 at 23:06
1
1
I mean naturally, one would need to impose the condition that $T_t+s = T_t circ T_s$, in which case yes, there are results.
– Shalop
Aug 9 at 22:56
I mean naturally, one would need to impose the condition that $T_t+s = T_t circ T_s$, in which case yes, there are results.
– Shalop
Aug 9 at 22:56
@Shalop Yes, I forgot to write that down. I am looking for a reference on semigroup results :)
– Yaddle
Aug 9 at 23:06
@Shalop Yes, I forgot to write that down. I am looking for a reference on semigroup results :)
– Yaddle
Aug 9 at 23:06
add a comment |Â
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1
I mean naturally, one would need to impose the condition that $T_t+s = T_t circ T_s$, in which case yes, there are results.
– Shalop
Aug 9 at 22:56
@Shalop Yes, I forgot to write that down. I am looking for a reference on semigroup results :)
– Yaddle
Aug 9 at 23:06