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

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of R T Directional ergodicity and weak mixing The correct definitions Studying W M T 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T Given a free, measure preserving, ergodic Z d action on a Lebesgue probability space T = ( X , µ, { T � n } � n ∈ Z d ) 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T Given a free, measure preserving, ergodic Z d action on a Lebesgue probability space T = ( X , µ, { T � n } � n ∈ Z d ) studying the sub-dynamics of T : which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional v ∈ S 1 ⊂ R 2 . entropy of a Z d action for all directions � Ay¸ se S ¸ahin DePaul University Directional recurrence, ergodicity, and weak mixing

Motivation for directional dynamics Defining directional properties Directional recurrence Examples of R T Directional ergodicity and weak mixing The correct definitions Studying W M T Given a free, measure preserving, ergodic Z d action on a Lebesgue probability space T = ( X , µ, { T � n } � n ∈ Z d ) studying the sub-dynamics of T : which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional v ∈ S 1 ⊂ R 2 . If � entropy of a Z d action for all directions � 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T Given a free, measure preserving, ergodic Z d action on a Lebesgue probability space T = ( X , µ, { T � n } � n ∈ Z d ) studying the sub-dynamics of T : which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional v ∈ S 1 ⊂ R 2 . If � entropy of a Z d action for all directions � 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T Given a free, measure preserving, ergodic Z d action on a Lebesgue probability space T = ( X , µ, { T � n } � n ∈ Z d ) studying the sub-dynamics of T : which properties of T are inherited by subgroup actions? Milnor (’86) expanded this notion and defined the directional v ∈ S 1 ⊂ R 2 . If � entropy of a Z d action for all directions � 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T There are two approaches to defining directional properties: ◮ intrinsically in the Z d 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T There are two approaches to defining directional properties: ◮ intrinsically in the Z d action: Study the property by analyzing the behavior of T at rational approximants of a direction. ◮ By embedding the discrete action in an R d 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T There are two approaches to defining directional properties: ◮ intrinsically in the Z d action: Study the property by analyzing the behavior of T at rational approximants of a direction. ◮ By embedding the discrete action in an R d action T . Then v ∈ S 1 there is an R action in direction � for any � v , for any v ∈ S 1 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 R T Directional ergodicity and weak mixing The correct definitions Studying W M T There are two approaches to defining directional properties: ◮ intrinsically in the Z d action: Study the property by analyzing the behavior of T at rational approximants of a direction. ◮ By embedding the discrete action in an R d action T . Then v ∈ S 1 there is an R action in direction � for any � v , for any v ∈ S 1 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