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1

$begingroup$

Let $f:M to M$ a measurable transformation and $mu$ a invariant probability and $D subset M $ a subset of positive measure.
Prove that almost every point of $D$ passes a positive fraction of the time in $D$:

$$limsup_n dfrac1n# 0 ≤ j ≤ n − 1 : f^j(x) ∈ D > 0$$

for $μ$-almost every point $x ∈ D$.

I would try to prove that the set of $x$'s for which the given limit is zero has zero measure, but I do not know how to do it. Anyone have an idea how to get started?

ergodic-theory

share|cite|improve this question

edited Mar 11 at 21:21

gt6989b

34.9k22557

asked Mar 11 at 21:09

Ricardo FreireRicardo Freire

547210

$endgroup$

add a comment | 

1

$begingroup$

Let $f:M to M$ a measurable transformation and $mu$ a invariant probability and $D subset M $ a subset of positive measure.
Prove that almost every point of $D$ passes a positive fraction of the time in $D$:

$$limsup_n dfrac1n# 0 ≤ j ≤ n − 1 : f^j(x) ∈ D > 0$$

for $μ$-almost every point $x ∈ D$.

I would try to prove that the set of $x$'s for which the given limit is zero has zero measure, but I do not know how to do it. Anyone have an idea how to get started?

ergodic-theory

share|cite|improve this question

edited Mar 11 at 21:21

gt6989b

34.9k22557

asked Mar 11 at 21:09

Ricardo FreireRicardo Freire

547210

$endgroup$

add a comment | 

$begingroup$

Let $f:M to M$ a measurable transformation and $mu$ a invariant probability and $D subset M $ a subset of positive measure.
Prove that almost every point of $D$ passes a positive fraction of the time in $D$:

$$limsup_n dfrac1n# 0 ≤ j ≤ n − 1 : f^j(x) ∈ D > 0$$

for $μ$-almost every point $x ∈ D$.

I would try to prove that the set of $x$'s for which the given limit is zero has zero measure, but I do not know how to do it. Anyone have an idea how to get started?

ergodic-theory

share|cite|improve this question

edited Mar 11 at 21:21

gt6989b

34.9k22557

asked Mar 11 at 21:09

Ricardo FreireRicardo Freire

547210

$endgroup$

share|cite|improve this question

edited Mar 11 at 21:21

gt6989b

34.9k22557

asked Mar 11 at 21:09

Ricardo FreireRicardo Freire

547210

share|cite|improve this question

edited Mar 11 at 21:21

gt6989b

34.9k22557

asked Mar 11 at 21:09

Ricardo FreireRicardo Freire

547210

edited Mar 11 at 21:21

gt6989b

34.9k22557

edited Mar 11 at 21:21

gt6989b

34.9k22557

asked Mar 11 at 21:09

Ricardo FreireRicardo Freire

547210

asked Mar 11 at 21:09

Ricardo FreireRicardo Freire

547210