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There exists a measure-preserving transformation with any given (nonnegative) entropy

Notation for an event related to some measure-preserving transformationDefinition of entropy of an ergodic measureExistence of a measure-preserving mapping between two given measure spaces?Does any measure preserving system have an invertible extension?What does “there exists a subset $S_0 subset S$ of full $mu$-measure” mean?Show that frac(1/x) is a measure-preserving transformationshow that there is a set $ A $ with positive measure such that $ T(A) cap A= phi $Entropy of a Measure Preserving TransformationDifferential entropy vs Kolmogorov-Sinai “partition trick”measure preserving transformation inequality

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Let $(X,mathscr{B},mu,T)$ be a measure-preserving system and let $xi$ be a partition of $X$ with finite entropy. Then the entropy of $T$ with respect to $xi$ is
$$h_mu(T,xi)=lim_{nto infty}frac{1}{n}H_mu(bigvee_{i=0}^{n-1}T^{-i}xi)=sup_{nto infty}frac{1}{n}H_mu(bigvee_{i=0}^{n-1}T^{-i}xi).$$
The entropy of $T$ is
$$h_mu(T)=sup_{xi:H_mu(xi)<infty}h_mu(T,xi).$$

Let $h$ be a nonnegative number. I wonder if there always exists a measure-preserving transformation with $h$ as its entropy.

measure-theory information-theory

share|cite|improve this question

edited Mar 9 at 17:07

No One

asked Mar 9 at 16:31

No OneNo One

2,0661519

$endgroup$

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  $begingroup$
  Hint (and it's hard to think of an hint that's not a complete giveaway): look at the definition of $H_mu$.
  $endgroup$
  – kimchi lover
  Mar 9 at 17:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
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  @kimchilover $H_mu$ is defined on partitions... It is not obvious to me how looking at its definition gives me an approach. Note that I hope this proposition is true for a general measure space.
  $endgroup$
  – No One
  Mar 9 at 20:32
  

  
  
  
  

add a comment | 

1

1

$begingroup$

Let $(X,mathscr{B},mu,T)$ be a measure-preserving system and let $xi$ be a partition of $X$ with finite entropy. Then the entropy of $T$ with respect to $xi$ is
$$h_mu(T,xi)=lim_{nto infty}frac{1}{n}H_mu(bigvee_{i=0}^{n-1}T^{-i}xi)=sup_{nto infty}frac{1}{n}H_mu(bigvee_{i=0}^{n-1}T^{-i}xi).$$
The entropy of $T$ is
$$h_mu(T)=sup_{xi:H_mu(xi)<infty}h_mu(T,xi).$$

Let $h$ be a nonnegative number. I wonder if there always exists a measure-preserving transformation with $h$ as its entropy.

measure-theory information-theory

share|cite|improve this question

edited Mar 9 at 17:07

No One

asked Mar 9 at 16:31

No OneNo One

2,0661519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hint (and it's hard to think of an hint that's not a complete giveaway): look at the definition of $H_mu$.
  $endgroup$
  – kimchi lover
  Mar 9 at 17:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @kimchilover $H_mu$ is defined on partitions... It is not obvious to me how looking at its definition gives me an approach. Note that I hope this proposition is true for a general measure space.
  $endgroup$
  – No One
  Mar 9 at 20:32
  

  
  
  
  

add a comment | 

$begingroup$

Let $(X,mathscr{B},mu,T)$ be a measure-preserving system and let $xi$ be a partition of $X$ with finite entropy. Then the entropy of $T$ with respect to $xi$ is
$$h_mu(T,xi)=lim_{nto infty}frac{1}{n}H_mu(bigvee_{i=0}^{n-1}T^{-i}xi)=sup_{nto infty}frac{1}{n}H_mu(bigvee_{i=0}^{n-1}T^{-i}xi).$$
The entropy of $T$ is
$$h_mu(T)=sup_{xi:H_mu(xi)<infty}h_mu(T,xi).$$

Let $h$ be a nonnegative number. I wonder if there always exists a measure-preserving transformation with $h$ as its entropy.

measure-theory information-theory

share|cite|improve this question

edited Mar 9 at 17:07

No One

asked Mar 9 at 16:31

No OneNo One

2,0661519

$endgroup$

share|cite|improve this question

edited Mar 9 at 17:07

No One

asked Mar 9 at 16:31

No OneNo One

2,0661519

share|cite|improve this question

edited Mar 9 at 17:07

No One

asked Mar 9 at 16:31

No OneNo One

2,0661519

asked Mar 9 at 16:31

No OneNo One

2,0661519

asked Mar 9 at 16:31

No OneNo One

2,0661519

1 Answer
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For each $h>0$ there is a Bernoulli shift $B(h)$ with entropy $h$. (Details and further info in the wikipedia article. The hard-math part here is finding a $k$ such that $h<log k$ and then finding probabilities $p_1,ldots,p_k$ such that $h=-sum p_ilog p_i$.)

(The construction is basically to let the coordinates be independent identically distributed copies of a random variable which takes the value $i$ with probability $p_i$, where $T$ is the shift. Because of independence, the $H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ is just the entropy of an $n$ tuple whose coordinates are independent, when $xi$ is the partition induced by a single coordinate. Because of the multiplicative property of independence, and the way logarithms work, all the $ H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ terms evaluate to $nh$, so for that $xi$, $H_mu(T,xi)=h$. And so on. Billingsley's book Ergodic Theory and Information has details.)

You can pick a single measure space, say $S=([0,1],mathcal B, lambda)$ with Lebesgue measure and, for any given $h$, find a measure-theoretic isomorphism between $B(h)$ and $S$, and use it to make a $T$ such that $([0,1],mathcal B, lambda,T)$ does what you want. (Conjugate the shift on $B(h)$ by the isomorphism.)

I don't quite understand what your comment "I hope this proposition is true for a general measure space" means, but maybe this is good enough for you.

I am also puzzled by how you came across this definition of a shift's entropy without also coming across examples of ergodic processes such as the Bernoulli shift, which are to information theory and ergodic theory as triangles are to Euclidean geometry. Most textbooks talk about the Kolmogorov Sinai definition and then about the Ornstein theorem, by which point your question should be obvious.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

kimchi loverkimchi lover

11.1k31229

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For the Bernoulli shift case, even if we have found $h=-sum p_ilog p_i$, I am still not sure why taking limits and supremum keeps this $h$ unchanged
  $endgroup$
  – No One
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have edited my answer.
  $endgroup$
  – kimchi lover
  2 days ago
  

  
  
  
  

add a comment | 

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0

$begingroup$

For each $h>0$ there is a Bernoulli shift $B(h)$ with entropy $h$. (Details and further info in the wikipedia article. The hard-math part here is finding a $k$ such that $h<log k$ and then finding probabilities $p_1,ldots,p_k$ such that $h=-sum p_ilog p_i$.)

(The construction is basically to let the coordinates be independent identically distributed copies of a random variable which takes the value $i$ with probability $p_i$, where $T$ is the shift. Because of independence, the $H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ is just the entropy of an $n$ tuple whose coordinates are independent, when $xi$ is the partition induced by a single coordinate. Because of the multiplicative property of independence, and the way logarithms work, all the $ H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ terms evaluate to $nh$, so for that $xi$, $H_mu(T,xi)=h$. And so on. Billingsley's book Ergodic Theory and Information has details.)

You can pick a single measure space, say $S=([0,1],mathcal B, lambda)$ with Lebesgue measure and, for any given $h$, find a measure-theoretic isomorphism between $B(h)$ and $S$, and use it to make a $T$ such that $([0,1],mathcal B, lambda,T)$ does what you want. (Conjugate the shift on $B(h)$ by the isomorphism.)

I don't quite understand what your comment "I hope this proposition is true for a general measure space" means, but maybe this is good enough for you.

I am also puzzled by how you came across this definition of a shift's entropy without also coming across examples of ergodic processes such as the Bernoulli shift, which are to information theory and ergodic theory as triangles are to Euclidean geometry. Most textbooks talk about the Kolmogorov Sinai definition and then about the Ornstein theorem, by which point your question should be obvious.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

kimchi loverkimchi lover

11.1k31229

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For the Bernoulli shift case, even if we have found $h=-sum p_ilog p_i$, I am still not sure why taking limits and supremum keeps this $h$ unchanged
  $endgroup$
  – No One
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have edited my answer.
  $endgroup$
  – kimchi lover
  2 days ago
  

  
  
  
  

add a comment | 

0

$begingroup$

For each $h>0$ there is a Bernoulli shift $B(h)$ with entropy $h$. (Details and further info in the wikipedia article. The hard-math part here is finding a $k$ such that $h<log k$ and then finding probabilities $p_1,ldots,p_k$ such that $h=-sum p_ilog p_i$.)

(The construction is basically to let the coordinates be independent identically distributed copies of a random variable which takes the value $i$ with probability $p_i$, where $T$ is the shift. Because of independence, the $H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ is just the entropy of an $n$ tuple whose coordinates are independent, when $xi$ is the partition induced by a single coordinate. Because of the multiplicative property of independence, and the way logarithms work, all the $ H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ terms evaluate to $nh$, so for that $xi$, $H_mu(T,xi)=h$. And so on. Billingsley's book Ergodic Theory and Information has details.)

You can pick a single measure space, say $S=([0,1],mathcal B, lambda)$ with Lebesgue measure and, for any given $h$, find a measure-theoretic isomorphism between $B(h)$ and $S$, and use it to make a $T$ such that $([0,1],mathcal B, lambda,T)$ does what you want. (Conjugate the shift on $B(h)$ by the isomorphism.)

I don't quite understand what your comment "I hope this proposition is true for a general measure space" means, but maybe this is good enough for you.

I am also puzzled by how you came across this definition of a shift's entropy without also coming across examples of ergodic processes such as the Bernoulli shift, which are to information theory and ergodic theory as triangles are to Euclidean geometry. Most textbooks talk about the Kolmogorov Sinai definition and then about the Ornstein theorem, by which point your question should be obvious.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

kimchi loverkimchi lover

11.1k31229

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For the Bernoulli shift case, even if we have found $h=-sum p_ilog p_i$, I am still not sure why taking limits and supremum keeps this $h$ unchanged
  $endgroup$
  – No One
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have edited my answer.
  $endgroup$
  – kimchi lover
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

For each $h>0$ there is a Bernoulli shift $B(h)$ with entropy $h$. (Details and further info in the wikipedia article. The hard-math part here is finding a $k$ such that $h<log k$ and then finding probabilities $p_1,ldots,p_k$ such that $h=-sum p_ilog p_i$.)

(The construction is basically to let the coordinates be independent identically distributed copies of a random variable which takes the value $i$ with probability $p_i$, where $T$ is the shift. Because of independence, the $H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ is just the entropy of an $n$ tuple whose coordinates are independent, when $xi$ is the partition induced by a single coordinate. Because of the multiplicative property of independence, and the way logarithms work, all the $ H_mu(bigvee_{i=1}^{n-1} T^{-i}xi)$ terms evaluate to $nh$, so for that $xi$, $H_mu(T,xi)=h$. And so on. Billingsley's book Ergodic Theory and Information has details.)

You can pick a single measure space, say $S=([0,1],mathcal B, lambda)$ with Lebesgue measure and, for any given $h$, find a measure-theoretic isomorphism between $B(h)$ and $S$, and use it to make a $T$ such that $([0,1],mathcal B, lambda,T)$ does what you want. (Conjugate the shift on $B(h)$ by the isomorphism.)

I don't quite understand what your comment "I hope this proposition is true for a general measure space" means, but maybe this is good enough for you.

I am also puzzled by how you came across this definition of a shift's entropy without also coming across examples of ergodic processes such as the Bernoulli shift, which are to information theory and ergodic theory as triangles are to Euclidean geometry. Most textbooks talk about the Kolmogorov Sinai definition and then about the Ornstein theorem, by which point your question should be obvious.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

kimchi loverkimchi lover

11.1k31229

$endgroup$

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

kimchi loverkimchi lover

11.1k31229

answered 2 days ago

kimchi loverkimchi lover

11.1k31229

answered 2 days ago

kimchi loverkimchi lover

11.1k31229