Directional recurrence, ergodicity, and weak mixing Ay se S ahin - - PowerPoint PPT Presentation

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Directional recurrence, ergodicity, and weak mixing

Ay¸ se S ¸ahin DePaul University June 8, 2013

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Given a free, measure preserving, ergodic Zd action on a Lebesgue probability space T = (X, µ, {T

n} n∈Zd)

studying the sub-dynamics of T: which properties of T are inherited by subgroup actions?

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Given a free, measure preserving, ergodic Zd action on a Lebesgue probability space T = (X, µ, {T

n} n∈Zd)

studying the sub-dynamics of T: which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional entropy of a Zd action for all directions v ∈ S1 ⊂ R2.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Given a free, measure preserving, ergodic Zd action on a Lebesgue probability space T = (X, µ, {T

n} n∈Zd)

studying the sub-dynamics of T: which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional entropy of a Zd action for all directions v ∈ S1 ⊂ R2. If v has rational slope then this is the usual entropy of T

• v. But a new

invariant when v is an irrational direction.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Given a free, measure preserving, ergodic Zd action on a Lebesgue probability space T = (X, µ, {T

n} n∈Zd)

studying the sub-dynamics of T: which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional entropy of a Zd action for all directions v ∈ S1 ⊂ R2. If v has rational slope then this is the usual entropy of T

• v. But a new

invariant when v is an irrational direction. Literature:

◮ Foundational work: Sinai (’85), Park (’94,’99).

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Given a free, measure preserving, ergodic Zd action on a Lebesgue probability space T = (X, µ, {T

n} n∈Zd)

studying the sub-dynamics of T: which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional entropy of a Zd action for all directions v ∈ S1 ⊂ R2. If v has rational slope then this is the usual entropy of T

• v. But a new

invariant when v is an irrational direction. Literature:

◮ Foundational work: Sinai (’85), Park (’94,’99). ◮ New areas: Expansive sub-dynamics Boyle and Lind (’97)

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

In this talk:

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

In this talk:

◮ Directional recurrence for infinite measure preserving actions

(joint with A.S.A. Johnson and D. Rudolph).

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

In this talk:

◮ Directional recurrence for infinite measure preserving actions

(joint with A.S.A. Johnson and D. Rudolph). Motivated by Feldman’s proof of the ratio ergodic theorem (90’s)

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

In this talk:

◮ Directional recurrence for infinite measure preserving actions

(joint with A.S.A. Johnson and D. Rudolph). Motivated by Feldman’s proof of the ratio ergodic theorem (90’s)

◮ Directional ergodicity and weak mixing (joint with E. A.

Robinson, Jr. and J. Rosenblatt)

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

In this talk:

◮ Directional recurrence for infinite measure preserving actions

(joint with A.S.A. Johnson and D. Rudolph). Motivated by Feldman’s proof of the ratio ergodic theorem (90’s)

◮ Directional ergodicity and weak mixing (joint with E. A.

Robinson, Jr. and J. Rosenblatt) Motivated by example due to Bergelson and Ward.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

There are two approaches to defining directional properties:

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

There are two approaches to defining directional properties:

◮ intrinsically in the Zd action:

Study the property by analyzing the behavior of T at rational approximants of a direction.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

There are two approaches to defining directional properties:

◮ intrinsically in the Zd action:

Study the property by analyzing the behavior of T at rational approximants of a direction.

◮ By embedding the discrete action in an Rd action T .

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

There are two approaches to defining directional properties:

◮ intrinsically in the Zd action:

Study the property by analyzing the behavior of T at rational approximants of a direction.

◮ By embedding the discrete action in an Rd action T . Then

for any v ∈ S1 there is an R action in direction v, for any

• v ∈ S1 given by

{T t

v}t∈R

and one can define the dynamics in the direction v of T in terms of T

v.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

There are two approaches to defining directional properties:

◮ intrinsically in the Zd action:

Study the property by analyzing the behavior of T at rational approximants of a direction.

◮ By embedding the discrete action in an Rd action T . Then

for any v ∈ S1 there is an R action in direction v, for any

• v ∈ S1 given by

{T t

v}t∈R

and one can define the dynamics in the direction v of T in terms of T

v.

A natural candidate for this continuous group action is the unit suspension of T.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Let’s restrict ourselves to the case of Z2 actions. The unit suspension of T is a free R2 action T = (X × [0, 1)2, µ × λ, {T

v} v∈R2) defined by

T

v(x, (r, s)) = (T (⌊v1+r⌋,⌊v2+s⌋)x, {v1 + r}, {v2 + s}).

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Let’s restrict ourselves to the case of Z2 actions. The unit suspension of T is a free R2 action T = (X × [0, 1)2, µ × λ, {T

v} v∈R2) defined by

T

v(x, (r, s)) = (T (⌊v1+r⌋,⌊v2+s⌋)x, {v1 + r}, {v2 + s}).

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Directional recurrence

An infinite measure preserving Z2 action T = (X, µ, {T

n} n∈Z2) is

recurrent in the direction v ∈ S1 if for any measurable set A with µA > 0 and any ǫ > 0 there exists t ∈ R so that t v ∈ Z2, lies in the ǫ-tunnel of v and µ

• T t

vA ∩ A

• > 0.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Directional recurrence

An infinite measure preserving Z2 action T = (X, µ, {T

n} n∈Z2) is

recurrent in the direction v ∈ S1 if for any measurable set A with µA > 0 and any ǫ > 0 there exists t ∈ R so that t v ∈ Z2, lies in the ǫ-tunnel of v and µ

• T t

vA ∩ A

• > 0.

Easy to see that if v is a rational direction, then the definition coincides with {T ⌊t

v⌋}t∈R being a recurrent Z action.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Theorem

An infinite measure preserving Z2 action T = (X, µ, {T

n} n∈Z2) is

recurrent in the direction v ∈ S1 if and only if T

v is recurrent as

an R action.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Theorem

An infinite measure preserving Z2 action T = (X, µ, {T

n} n∈Z2) is

recurrent in the direction v ∈ S1 if and only if T

v is recurrent as

an R action. Let RT denote the set of recurrent directions for T

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Theorem

An infinite measure preserving Z2 action T = (X, µ, {T

n} n∈Z2) is

recurrent in the direction v ∈ S1 if and only if T

v is recurrent as

an R action. Let RT denote the set of recurrent directions for T

Theorem

RT is a Gδ set.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Theorem

An infinite measure preserving Z2 action T = (X, µ, {T

n} n∈Z2) is

recurrent in the direction v ∈ S1 if and only if T

v is recurrent as

an R action. Let RT denote the set of recurrent directions for T

Theorem

RT is a Gδ set. The proof relies on an alternate characterization of recurrence: ǫ-sweeping out.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Definition

Let A ∈ B be such that 0 < µ(A) < ∞. A direction v is said to have the ǫ sweeping out property for A if for all α, 0 < α < 1

2,

there exist pairwise disjoint subsets A1, A2, . . . , Ak of A of positive measure and vectors v1, . . . , vk in the ǫ-tunnel of v so that the sets T

v1A1, . . . , T vkAk are a pairwise disjoint collection of subsets of A

satisfying:

• 1. µ(k

i=1 Ai) > α · µ(A), and

• 2. µ(k

i=1 T vi(Ai)) > α · µ(A)

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

If a direction v has the ǫ-sweeping out property for A for every ǫ > 0 we say it is uniform sweeping out for A.

Theorem

◮ If T is recurrent in direction

v ∈ S1 then v is uniform sweeping out for all measurable sets of positive, finite measure.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

If a direction v has the ǫ-sweeping out property for A for every ǫ > 0 we say it is uniform sweeping out for A.

Theorem

◮ If T is recurrent in direction

v ∈ S1 then v is uniform sweeping out for all measurable sets of positive, finite measure.

◮ If

v is uniform sweeping out for a countable collection of sets that generate the σ-algebra then T is recurrent in the direction v.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

If a direction v has the ǫ-sweeping out property for A for every ǫ > 0 we say it is uniform sweeping out for A.

Theorem

◮ If T is recurrent in direction

v ∈ S1 then v is uniform sweeping out for all measurable sets of positive, finite measure.

◮ If

v is uniform sweeping out for a countable collection of sets that generate the σ-algebra then T is recurrent in the direction v.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Can now prove the Gδ result by choosing a countable set of generators {Ai}, then RT: ∩∞

i=1 ∩∞ n=1 {

v ∈ S1 : v has the 1 n sweeping out property for Ai}.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Can now prove the Gδ result by choosing a countable set of generators {Ai}, then RT: ∩∞

i=1 ∩∞ n=1 {

v ∈ S1 : v has the 1 n sweeping out property for Ai}. Open set: comes from the fact that one incidence of recurrence in a direction v gives an incidence of recurrence for an interval of directions ( v − δ, v + δ) where δ depends on the magnitude of the vector t v achieving the recurrence.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Open Question: Can every Gδ set be realized as RT?

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Open Question: Can every Gδ set be realized as RT? What is known:

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Open Question: Can every Gδ set be realized as RT? What is known:

Theorem

There exists (X, µ, {T

n} n∈Z2), an infinite measure preserving,

recurrent, and ergodic Z2 action on a σ-finite Lebesgue space, with the property that RT = ∅.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Open Question: Can every Gδ set be realized as RT? What is known:

Theorem

There exists (X, µ, {T

n} n∈Z2), an infinite measure preserving,

recurrent, and ergodic Z2 action on a σ-finite Lebesgue space, with the property that RT = ∅.

Theorem

Let α1, . . . , αk ∈ [0, π) be irrational. There exists an infinite measure preserving, recurrent, and ergodic Z2 action (X, µ, {T

n} n∈Z2) on a σ-finite Lebesgue space with the property

that RT contains all rational directions but αi / ∈ RT for i = 1, . . . , k.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Proofs by construction, cutting and stacking ”machine” that produces a variety of examples.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Directional ergodicity and weak mixing:

Here we will start with the unit suspension approach. There are some mild complications that need to be dealt with before we can make directional definitions.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Directional ergodicity and weak mixing:

Here we will start with the unit suspension approach. There are some mild complications that need to be dealt with before we can make directional definitions. Recall:

◮ T is not ergodic in any rational direction.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

◮ T is not weak mixing as an R2 action and therefore T v is not

a weak mixing R action for any v ∈ S1.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

These facts prevent us from defining directional ergodicity or weak mixing directly from T . We need to make a small adjustment. Let H = {f : f (x, r, s) = G(r, s), G ∈ L2([0, 1)2)}⊥

Definition

T is ergodic in direction v if T

v is an H-ergodic R action.

T is weak mixing in direction v if T

v is an H-weak mixing R

action.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Equivalence of definitions (work in progress):

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Equivalence of definitions (work in progress): Directional ergodicity: unit suspension implies intrinsic. Converse? Weak mixing: equivalent (90%) In general, directional ergodicity is trickier so consider first directional weak mixing.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Let WMT denote the directions for which T is weak mixing. First consider the case when T is not weak mixing as a Z2 action. Then of course WMT contains no rational directions. In fact, it is

• empty. Suppose

α ∈ R2 and F ∈ L2(X) are such that F(T

nx) = e2πi α· nF(x)

for all n ∈ Z2. Then for any v ∈ R2 the function F(x, u) = e2πi

α· uF(x)

is an eigenfunction for T

v.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

A quick computation shows that F

• T

v(x,

u)

• = e2πi

v· αF(x,

u) So in particular, F is invariant under T

v when

v ⊥ α. So T not weak mixing implies both that WMT is empty and that there is a non-ergodic direction.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Now assume that T is weak mixing as a Z2 action.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Now assume that T is weak mixing as a Z2 action. There is an example (Bergelson, Ward) of a weak mixing Z2 action T with WMc

T ⊃ Z2.

Start with (S, Y , µ) weak mixing. Define X =

i∈Z Y and T by

T (n,m)(x) =

• T li(n,m)(Yi)

where the li range over all possible expressions of the form an + bm with a, b ∈ Z. So for every (n, m) there is some i for which an + bm = 0 and therefore Yi is invariant. A collection of similar examples can be constructed using the Gaussian measure construction.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

On the other hand, WMc

T is countable.

So what more could there be? Also work in progress.

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of RT Directional ergodicity and weak mixing The correct definitions Studying WMT

Suppose µ is a positive Borel probability measure on R2. If µ is supported on a line segment orthogonal to v through a point (p1, p2) then ˆ µ(t v) = e2πit

v· p for all t ∈ R.

Using the Gaussian measure construction can construct examples

• f different types of WMT (seems can avoid

v for any v ∈ S1) and also ET (?).

Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing