How to interpret a theorem stating that orbits are “uniform on average”
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I am reading through these notes: http://wwwf.imperial.ac.uk/~dcheragh/Teaching/2015-F-DS-MPE.pdf
There is a theorem (Theorem 1 under the section Distribution of Orbits) saying the following: 
Here $R_{alpha}^k (x)$ is the $k_{th}$ iteration of the rotation by $alpha$ map: $R_alpha(x) = x+alpha (text{mod} 1)$
Question:
There is a comment just above this theorem saying
The distribution of
the orbit is “uniform” on average.
Does this mean that the distributions of orbits on subsets of the circle are also uniform i.e. that if we fix some set $E$ on the circle and wait for the orbit to enter $E$, it has equal probability of landing anywhere in $E$?
probability ergodic-theory
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
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add a comment |
$begingroup$
I am reading through these notes: http://wwwf.imperial.ac.uk/~dcheragh/Teaching/2015-F-DS-MPE.pdf
There is a theorem (Theorem 1 under the section Distribution of Orbits) saying the following:
Here $R_{alpha}^k (x)$ is the $k_{th}$ iteration of the rotation by $alpha$ map: $R_alpha(x) = x+alpha (text{mod} 1)$
Question:
There is a comment just above this theorem saying
The distribution of
the orbit is “uniform” on average.
Does this mean that the distributions of orbits on subsets of the circle are also uniform i.e. that if we fix some set $E$ on the circle and wait for the orbit to enter $E$, it has equal probability of landing anywhere in $E$?
probability ergodic-theory
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
$endgroup$
add a comment |
$begingroup$
I am reading through these notes: http://wwwf.imperial.ac.uk/~dcheragh/Teaching/2015-F-DS-MPE.pdf
There is a theorem (Theorem 1 under the section Distribution of Orbits) saying the following:
Here $R_{alpha}^k (x)$ is the $k_{th}$ iteration of the rotation by $alpha$ map: $R_alpha(x) = x+alpha (text{mod} 1)$
Question:
There is a comment just above this theorem saying
The distribution of
the orbit is “uniform” on average.
Does this mean that the distributions of orbits on subsets of the circle are also uniform i.e. that if we fix some set $E$ on the circle and wait for the orbit to enter $E$, it has equal probability of landing anywhere in $E$?
probability ergodic-theory
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
$endgroup$
I am reading through these notes: http://wwwf.imperial.ac.uk/~dcheragh/Teaching/2015-F-DS-MPE.pdf
There is a theorem (Theorem 1 under the section Distribution of Orbits) saying the following:
Here $R_{alpha}^k (x)$ is the $k_{th}$ iteration of the rotation by $alpha$ map: $R_alpha(x) = x+alpha (text{mod} 1)$
Question:
There is a comment just above this theorem saying
The distribution of
the orbit is “uniform” on average.
Does this mean that the distributions of orbits on subsets of the circle are also uniform i.e. that if we fix some set $E$ on the circle and wait for the orbit to enter $E$, it has equal probability of landing anywhere in $E$?
probability ergodic-theory
probability ergodic-theory
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
asked Dec 5 '18 at 21:23
foshofosho
4,7991032
4,7991032
add a comment |
add a comment |
1 Answer
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If the set $E$ has positive measure, yes.
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
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add a comment |
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1 Answer
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1 Answer
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active
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active
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$begingroup$
If the set $E$ has positive measure, yes.
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
$endgroup$
add a comment |
$begingroup$
If the set $E$ has positive measure, yes.
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
$endgroup$
add a comment |
$begingroup$
If the set $E$ has positive measure, yes.
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
$endgroup$
If the set $E$ has positive measure, yes.
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
answered Dec 5 '18 at 21:47
kodlukodlu
3,390716
3,390716
add a comment |
add a comment |
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