Mixing
Dynamical system with $f(x)leq f(Tx) leq f(T^2 x) leq …$
$begingroup$
Let $(X,mathcal{F},mu,T)$ be a measure preserving system and $fin L^1(X,mathcal{F},mu)$.
Problem: show that if $f(x)leq f(Tx)$ $forall xin X$ then $f(x)=f(Tx)$ $mu$-a.e.
It seems to be useful to have it about $C:=lbrace x: f(x)= crbrace$, then by measure preservingness we get $mulbrace x: f(x)= crbrace = mu(C)=mu(T^{-1}C) = mulbrace x: f(Tx)= crbrace$. Or maybe inequalities are more useful.
For the rest I have really no idea what to do. Is there anyone who can give a solution? Thanks in advance.
ergodic-theory
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
$endgroup$
add a comment |
$begingroup$
Let $(X,mathcal{F},mu,T)$ be a measure preserving system and $fin L^1(X,mathcal{F},mu)$.
Problem: show that if $f(x)leq f(Tx)$ $forall xin X$ then $f(x)=f(Tx)$ $mu$-a.e.
It seems to be useful to have it about $C:=lbrace x: f(x)= crbrace$, then by measure preservingness we get $mulbrace x: f(x)= crbrace = mu(C)=mu(T^{-1}C) = mulbrace x: f(Tx)= crbrace$. Or maybe inequalities are more useful.
For the rest I have really no idea what to do. Is there anyone who can give a solution? Thanks in advance.
ergodic-theory
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
$endgroup$
3
$begingroup$
Consider the function $g(x):=f(Tx)-f(x)$. Note that $g$ is non-negative but has integral $0$.
$endgroup$
– Blackbird
Jan 19 at 0:37
add a comment |
$begingroup$
Let $(X,mathcal{F},mu,T)$ be a measure preserving system and $fin L^1(X,mathcal{F},mu)$.
Problem: show that if $f(x)leq f(Tx)$ $forall xin X$ then $f(x)=f(Tx)$ $mu$-a.e.
It seems to be useful to have it about $C:=lbrace x: f(x)= crbrace$, then by measure preservingness we get $mulbrace x: f(x)= crbrace = mu(C)=mu(T^{-1}C) = mulbrace x: f(Tx)= crbrace$. Or maybe inequalities are more useful.
For the rest I have really no idea what to do. Is there anyone who can give a solution? Thanks in advance.
ergodic-theory
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
$endgroup$
Let $(X,mathcal{F},mu,T)$ be a measure preserving system and $fin L^1(X,mathcal{F},mu)$.
Problem: show that if $f(x)leq f(Tx)$ $forall xin X$ then $f(x)=f(Tx)$ $mu$-a.e.
It seems to be useful to have it about $C:=lbrace x: f(x)= crbrace$, then by measure preservingness we get $mulbrace x: f(x)= crbrace = mu(C)=mu(T^{-1}C) = mulbrace x: f(Tx)= crbrace$. Or maybe inequalities are more useful.
For the rest I have really no idea what to do. Is there anyone who can give a solution? Thanks in advance.
ergodic-theory
ergodic-theory
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
edited Jan 18 at 20:32
Rocco van Vreumingen
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
asked Jan 18 at 16:55
Rocco van VreumingenRocco van Vreumingen
928
928
3
$begingroup$
Consider the function $g(x):=f(Tx)-f(x)$. Note that $g$ is non-negative but has integral $0$.
$endgroup$
– Blackbird
Jan 19 at 0:37
add a comment |
3
$begingroup$
Consider the function $g(x):=f(Tx)-f(x)$. Note that $g$ is non-negative but has integral $0$.
$endgroup$
– Blackbird
Jan 19 at 0:37
3
3
$begingroup$
Consider the function $g(x):=f(Tx)-f(x)$. Note that $g$ is non-negative but has integral $0$.
$endgroup$
– Blackbird
Jan 19 at 0:37
$begingroup$
Consider the function $g(x):=f(Tx)-f(x)$. Note that $g$ is non-negative but has integral $0$.
$endgroup$
– Blackbird
Jan 19 at 0:37
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3078496%2fdynamical-system-with-fx-leq-ftx-leq-ft2-x-leq%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3078496%2fdynamical-system-with-fx-leq-ftx-leq-ft2-x-leq%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
3
$begingroup$
Consider the function $g(x):=f(Tx)-f(x)$. Note that $g$ is non-negative but has integral $0$.
$endgroup$
– Blackbird
Jan 19 at 0:37