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Rational weak mixing in infinite measure spaces Jon. Aaronson (TAU) July 2011
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Rational weak mixing in infinite measure spaces Jon. Aaronson (TAU) ETDS Workshop, Warwick July 2011 Preprint link: http://arxiv.org/abs/1105.3541 Jon. Aaronson (TAU) Rational weak mixing. х>0у 0<?/<1. 1. Hopf’s example Разобьём её целочисленными абсциссами х = 1, 2, . . . на бесконечное числа квадратов, T R X B занумеровав X R 0,их, 1 как0, 1показано 2 Z наX рисунке. where B isПреобразова­ the ние Т будет произведением Т"Т' двух Baker’s transformation on each сохраняющих box 0, 12 площадь n and преобразоR is . \ 1_ — 2 1 3 -*5 ваний, где Т' есть преобразование, переводящее каждый квадрат в самого* себя, рассмотренное в § 12 и применённое к каждому из этих квадра­ тов, а Т" переводит прямоугольники, получающиеся разбиением квадра1 тов линией у = ~ , один в другой так, к а к это показано на рисунке. *) W i e n e r . [1]. Предположение об обращении в нуль функции q (?) вне некото­ рого конечного интервала несущественно. 12* х>0у 0<?/<1. 1. Hopf’s example Разобьём её целочисленными абсциссами х = 1, 2, . . . на бесконечное числа квадратов, T R X B занумеровав X R 0,их, 1 как0, 1показано 2 Z наX рисунке. where B isПреобразова­ the ние Т будет произведением Т"Т' двух Baker’s transformation on each сохраняющих box 0, 12 площадь n and преобразоR is . \ 1_ 1 — 2 3 -*5 ваний, где Т' есть преобразование, переводящее каждый квадрат в самого* T preserves Lebesgue measure m & is ratio mixing: себя, рассмотренное в § 12 и применённое к каждому из этих квадра­ тов, а Т" ½ переводит прямоугольники, получающиеся разбиением квадраπn1 n A 9 Tвдругой B так, mкAакmэто B показано на рисунке. (Ð) тов линией у = ~m , один n ª 2 Ð bounded with m∂Aв нульmфункции ∂B q0.(?) вне некото­ *) W i e n¦e r .A, [1]. BПредположение об обращении рого конечного интервала несущественно. 12* х>0у 0<?/<1. 1. Hopf’s example Разобьём её целочисленными абсциссами х = 1, 2, . . . на бесконечное числа квадратов, T R X B занумеровав X R 0,их, 1 как0, 1показано 2 Z наX рисунке. where B isПреобразова­ the ние Т будет произведением Т"Т' двух Baker’s transformation on each сохраняющих box 0, 12 площадь n and преобразоR is . \ 1_ 1 — 2 3 -*5 ваний, где Т' есть преобразование, переводящее каждый квадрат в самого* T preserves Lebesgue measure m & is ratio mixing: себя, рассмотренное в § 12 и применённое к каждому из этих квадра­ тов, а Т" ½ переводит прямоугольники, получающиеся разбиением квадраπn1 n A 9 Tвдругой B так, mкAакmэто B показано на рисунке. (Ð) тов линией у = ~m , один n ª 2 Ð bounded with m∂Aв нульmфункции ∂B q0.(?) вне некото­ *) W i e n¦e r .A, [1]. BПредположение об обращении рого конечного интервала несущественно. 12* Y Hopf asked whether: (Ð) ¦ A, B bounded measurable sets ? х>0у 0<?/<1. 1. Hopf’s example Разобьём её целочисленными абсциссами х = 1, 2, . . . на бесконечное числа квадратов, T R X B занумеровав X R 0,их, 1 как0, 1показано 2 Z наX рисунке. where B isПреобразова­ the ние Т будет произведением Т"Т' двух Baker’s transformation on each сохраняющих box 0, 12 площадь n and преобразоR is . \ 1_ 1 — 2 3 -*5 ваний, где Т' есть преобразование, переводящее каждый квадрат в самого* T preserves Lebesgue measure m & is ratio mixing: себя, рассмотренное в § 12 и применённое к каждому из этих квадра­ тов, а Т" ½ переводит прямоугольники, получающиеся разбиением квадраπn1 n A 9 Tвдругой B так, mкAакmэто B показано на рисунке. (Ð) тов линией у = ~m , один n ª 2 Ð bounded with m∂Aв нульmфункции ∂B q0.(?) вне некото­ *) W i e n¦e r .A, [1]. BПредположение об обращении рого конечного интервала несущественно. 12* Y Hopf asked whether: (Ð) Implies ergodicity ¦ A, B bounded measurable sets ? х>0у 0<?/<1. 1. Hopf’s example Разобьём её целочисленными абсциссами х = 1, 2, . . . на бесконечное числа квадратов, T R X B занумеровав X R 0,их, 1 как0, 1показано 2 Z наX рисунке. where B isПреобразова­ the ние Т будет произведением Т"Т' двух Baker’s transformation on each сохраняющих box 0, 12 площадь n and преобразоR is . \ 1_ 1 — 2 3 -*5 ваний, где Т' есть преобразование, переводящее каждый квадрат в самого* T preserves Lebesgue measure m & is ratio mixing: себя, рассмотренное в § 12 и применённое к каждому из этих квадра­ тов, а Т" ½ переводит прямоугольники, получающиеся разбиением квадраπn1 n A 9 Tвдругой B так, mкAакmэто B показано на рисунке. (Ð) тов линией у = ~m , один n ª 2 Ð bounded with m∂Aв нульmфункции ∂B q0.(?) вне некото­ *) W i e n¦e r .A, [1]. BПредположение об обращении рого конечного интервала несущественно. 12* Y Hopf asked whether: (Ð) ¦ A, B bounded measurable sets ? Implies ergodicity ((Ð) does not!) х>0у 0<?/<1. 1. Hopf’s example Разобьём её целочисленными абсциссами х = 1, 2, . . . на бесконечное числа квадратов, T R X B занумеровав X R 0,их, 1 как0, 1показано 2 Z наX рисунке. where B isПреобразова­ the ние Т будет произведением Т"Т' двух Baker’s transformation on each сохраняющих box 0, 12 площадь n and преобразоR is . \ 1_ 1 — 2 3 -*5 ваний, где Т' есть преобразование, переводящее каждый квадрат в самого* T preserves Lebesgue measure m & is ratio mixing: себя, рассмотренное в § 12 и применённое к каждому из этих квадра­ тов, а Т" ½ переводит прямоугольники, получающиеся разбиением квадраπn1 n A 9 Tвдругой B так, mкAакmэто B показано на рисунке. (Ð) тов линией у = ~m , один n ª 2 Ð bounded with m∂Aв нульmфункции ∂B q0.(?) вне некото­ *) W i e n¦e r .A, [1]. BПредположение об обращении рого конечного интервала несущественно. 12* Y Hopf asked whether: (Ð) ¦ A, B bounded measurable sets ? Implies ergodicity ((Ð) does not!) & isomorphism invariance of º n. х>0у 0<?/<1. 1. Hopf’s example Разобьём её целочисленными абсциссами х = 1, 2, . . . на бесконечное числа квадратов, T R X B занумеровав X R 0,их, 1 как0, 1показано 2 Z наX рисунке. where B isПреобразова­ the ние Т будет произведением Т"Т' двух Baker’s transformation on each сохраняющих box 0, 12 площадь n and преобразоR is . \ 1_ 1 — 2 3 -*5 ваний, где Т' есть преобразование, переводящее каждый квадрат в самого* T preserves Lebesgue measure m & is ratio mixing: себя, рассмотренное в § 12 и применённое к каждому из этих квадра­ тов, а Т" ½ переводит прямоугольники, получающиеся разбиением квадраπn1 n A 9 Tвдругой B так, mкAакmэто B показано на рисунке. (Ð) тов линией у = ~m , один n ª 2 Ð bounded with m∂Aв нульmфункции ∂B q0.(?) вне некото­ *) W i e n¦e r .A, [1]. BПредположение об обращении рого конечного интервала несущественно. 12* Y Hopf asked whether: (Ð) ¦ A, B bounded measurable sets ? Implies ergodicity ((Ð) does not!) & isomorphism invariance of Y Summed (Ð) OK ¦ º A, B bdd. meas. via Markov property. n. 2. Weakly wandering sets Y Weakly wandering set for MPT X , B , m, T : W > B s.t. § n1 @ n2 @ . . . with T nk W k > N disjoint. 2. Weakly wandering sets Y Weakly wandering set for MPT X , B , m, T : W > B s.t. § n1 @ n2 @ . . . with T nk W k > N disjoint. , ¦ ([HK]) X , A > F , A 0, § B, m, T MPT w.o. a.c.i.p. Ô W > B A weakly wandering, mA W @ . 2. Weakly wandering sets Y Weakly wandering set for MPT X , B , m, T : W > B s.t. § n1 @ n2 @ . . . with T nk W k > N disjoint. , ¦ ([HK]) X , A > F , A 0, / Ô X , 0 § B, m, T MPT w.o. a.c.i.p. Ô W > B A weakly wandering, mA W @ . B, m, T MPT, un A 0 & W > F weakly wandering limn ª m W 9 T un nW x mW 2 . So no improved Ð. 2. Weakly wandering sets Y Weakly wandering set for MPT X , B , m, T : W > B s.t. § n1 @ n2 @ . . . with T nk W k > N disjoint. , ¦ ([HK]) X , A > F , A 0, / Ô X , 0 § B, m, T MPT w.o. a.c.i.p. Ô W > B A weakly wandering, mA W @ . B, m, T MPT, un A 0 & W > F weakly wandering limn ª m W 9 T un ½ x mW 2 . So no improved Ð πn density m A 9 T n B mAmB n ª 2 A, B bounded, measurable. For Hopf ex., ¦ nW Ð. (K) 2. Weakly wandering sets Y Weakly wandering set for MPT X , B , m, T : W > B s.t. § n1 @ n2 @ . . . with T nk W k > N disjoint. , ¦ ([HK]) X , A > F , A 0, / Ô X , 0 § B, m, T MPT w.o. a.c.i.p. Ô W > B A weakly wandering, mA W @ . B, m, T MPT, un A 0 & W > F weakly wandering limn ª m W 9 T un nW x mW 2 . So no improved ½ ¦ Ð πn density m A 9 T n B mAmB n ª 2 A, B bounded, measurable. For Hopf ex., Question about Hopf example Ð. (K) 2. Weakly wandering sets Y Weakly wandering set for MPT X , B , m, T : W > B s.t. § n1 @ n2 @ . . . with T nk W k > N disjoint. , ¦ ([HK]) X , A > F , A 0, / Ô X , 0 § B, m, T MPT w.o. a.c.i.p. Ô W > B A weakly wandering, mA W @ . B, m, T MPT, un A 0 & W > F weakly wandering limn ª m W 9 T un nW x mW 2 . So no improved ½ Ð πn density m A 9 T n B mAmB n ª 2 A, B bounded, measurable. For Hopf ex., ¦ Question about Hopf example set of finite measure? § Ð. (K) ? exhaustive weakly wandering 3. Rational weak mixing MPT X , B , m, T is rationally weakly mixing (RWM) if § F > F s.t. 3. Rational weak mixing MPT X , B , m, T is rationally weakly mixing (RWM) if § F > F s.t. m A 9 T n B un F where u F u F density Ъ n un F nC0 mAmB ¦ A, B > B 9 F mF 9T n F nC0 m F 2 & (Í) 3. Rational weak mixing MPT X , B , m, T is rationally weakly mixing (RWM) if § F > F s.t. m A 9 T n B un F where u F u F density Ъ n un F nC0 sn Ъ u density n mAmB A, B > B 9 F mF 9T n F nC0 m F 2 L means where K ` N is u-small i.e. ¦ sn n Ð & ª, n¶K L (Í) 3. Rational weak mixing MPT X , B , m, T is rationally weakly mixing (RWM) if § F > F s.t. m A 9 T n B un F where u F u F density Ъ n un F nC0 sn Ъ u density n mAmB A, B > B 9 F mF 9T n F nC0 m F 2 L means where K ` N is u-small i.e. ¦ sn Pk >K 9 1,n uk Pk > 1,n uk n Ð ª, n¶K Ъ n & 0. L (Í) 3. Rational weak mixing MPT X , B , m, T is rationally weakly mixing (RWM) if § F > F s.t. m A 9 T n B un F where u F u F density Ъ n un F nC0 sn Ъ u density n ¦ A, B > B 9 F mF 9T n F nC0 m F 2 L means where K ` N is u-small i.e. Y mAmB sn Pk >K 9 1,n uk Pk > 1,n uk n Ð ª, n¶K Ъ n & L 0. For un γ-regularly varying with γ > 1, 0, K ` N is u-small it has density zero. (Í) 3. Rational weak mixing MPT X , B , m, T is rationally weakly mixing (RWM) if § F > F s.t. m A 9 T n B un F where u F u F density Ъ n un F nC0 sn Ъ u density n ¦ A, B > B 9 F mF 9T n F nC0 m F 2 L means where K ` N is u-small i.e. Y mAmB sn Pk >K 9 1,n uk Pk > 1,n uk n Ð ª, n¶K Ъ n & L 0. For un γ-regularly varying with γ > 1, 0, K ` N is u-small it has density zero. So RWM of Hopf’s example & box in R T Ô (K). (Í) 4. Consequences g Suppose that X , B , m, T is a RWM, MPT, then 4. Consequences g Suppose that X , B , m, T is a RWM, MPT, then ¶1 T is weakly rationally ergodic (WRE) i.e. 1 n 1 m A 9 T k B mAmB n ª an F k 0 Q where an F Ð P u n 1 k 0 k F . ¦ § F > F s.t. A, B > B 9 F () 4. Consequences g Suppose that X , B , m, T is a RWM, MPT, then ¶1 T is weakly rationally ergodic (WRE) i.e. 1 n 1 m A 9 T k B mAmB n ª an F k 0 Q Ð where an F R T F P u n 1 k 0 k F . ¦ § F > F s.t. A, B > B 9 F () Moreover > F () holds F > F (Í) holds. 4. Consequences g Suppose that X , B , m, T is a RWM, MPT, then ¶1 T is weakly rationally ergodic (WRE) i.e. 1 n 1 m A 9 T k B mAmB n ª an F k 0 Q Ð where an F R T F P u n 1 k 0 k F . F > F s.t. A, B > B 9 F () Moreover > F () holds ¶1’Isomorphism: ¦ § T T RWM Ô F > F (Í) holds. T un T un Ъ u density n 1. 4. Consequences g Suppose that X , B , m, T is a RWM, MPT, then ¶1 T is weakly rationally ergodic (WRE) i.e. 1 n 1 m A 9 T k B mAmB n ª an F k 0 Q Ð where an F R T F P u n 1 k 0 k F . ¶2 A, B > B 9 F () Moreover > F () holds ¶1’Isomorphism: ¦ F > F s.t. § T T RWM F Ô T is weakly mixing (WM): T > F (Í) holds. T un T un Ъ u density n S is ergodic ¦ 1. EPPT S. 4. Consequences g Suppose that X , B , m, T is a RWM, MPT, then ¶1 T is weakly rationally ergodic (WRE) i.e. 1 n 1 m A 9 T k B mAmB n ª an F k 0 Q Ð where an F R T F P u n 1 k 0 k F . T T RWM T is weakly mixing (WM): T ¶3 T S is RWM ¦ WM, PPT S. F Ô ¶2 A, B > B 9 F () Moreover > F () holds ¶1’Isomorphism: ¦ F > F s.t. § > F (Í) holds. T un T un Ъ u density n S is ergodic ¦ 1. EPPT S. 5. Sufficient condition: Krickeberg mixing Given X , B , m, T a MPT & α ` B a countable generating partition, say that T is α-ratio mixing (α-RM) if 5. Sufficient condition: Krickeberg mixing Given X , B , m, T a MPT & α ` B a countable generating partition, say that T is α-ratio mixing (α-RM) if § ρn A 0 n C 1 such that ρ n m A 9 T n B Ð ª mAmB n ¦ A, B > Cα . (Ð) where Cα T N a1 , . . . , aN k j 1 jk aj N > N, k > Z, a1 , . . . , aN > α. 5. Sufficient condition: Krickeberg mixing Given X , B , m, T a MPT & α ` B a countable generating partition, say that T is α-ratio mixing (α-RM) if § ρn A 0 n C 1 such that ρ n m A 9 T n B Ð ª mAmB n ¦ A, B > Cα . (Ð) where Cα T N a1 , . . . , aN k jk aj N > N, k > Z, a1 , . . . , aN > α. j 1 ¶4 Suppose X , B , m, T WRE, MPT & partition α ` R T and Ω > Cα such that m A 9 T n B un where u Ъ u density n § a countable generating mAmB u Ω, then X , B , m, T is RWM. ¦ A, B > Cα 6. Markov Chains Irreducible, recurrent Markov shifts with the strong ratio limit property (SRLP) are α-RM with α state occupied at time 0. 6. Markov Chains Irreducible, recurrent Markov shifts with the strong ratio n n1 limit property (SRLP) (ps,s ps,s ¦ states s) are α-RM with α state occupied at time 0. 6. Markov Chains Irreducible, recurrent Markov shifts with the strong ratio n n1 limit property (SRLP) (ps,s ps,s ¦ states s) are α-RM with α state occupied at time 0. e.g. Hopf’s example isom. to the symm RW on Z with reflecting barrier at 0. 6. Markov Chains Irreducible, recurrent Markov shifts with the strong ratio n n1 limit property (SRLP) (ps,s ps,s ¦ states s) are α-RM with α state occupied at time 0. e.g. Hopf’s example isom. to the symm RW on Z with reflecting barrier at 0. Transition matrix has SRLP: ¼ n p0,0 2 πn . 6. Markov Chains Irreducible, recurrent Markov shifts with the strong ratio n n1 limit property (SRLP) (ps,s ps,s ¦ states s) are α-RM with α state occupied at time 0. e.g. Hopf’s example isom. to the symm RW on Z with reflecting barrier at 0. Transition matrix has SRLP: ¼ n p0,0 2 πn . ¶5 The irreducible, recurrent Markov shift is RWM iff for some and hence all states s, the renewal sequence u Qu n S k k 0 Qu n ps,s nC0 n uk 1 S o k k 0 as n . ª is smooth: 6. Markov Chains Irreducible, recurrent Markov shifts with the strong ratio n n1 limit property (SRLP) (ps,s ps,s ¦ states s) are α-RM with α state occupied at time 0. e.g. Hopf’s example isom. to the symm RW on Z with reflecting barrier at 0. Transition matrix has SRLP: ¼ n p0,0 2 πn . ¶5 The irreducible, recurrent Markov shift is RWM iff for some and hence all states s, the renewal sequence u Qu n S k is smooth: n uk 1 S k 0 , Qu n ps,s nC0 o k as n . ª k 0 Smoothness of ap. rec. u u0 , u1 , . . . with lifetime dist. 1 2 f > P N if e.g. § N C 1, ª 1BnBt n fn ) & n N V n2 @ ª (V t º n k 1 f k, ª o n. P P P 7. Modes of convergence For u > W weights u > `ª Z au n Qu Ð n k k 1 ª , 7. Modes of convergence For u > W weights u > `ª Z au n Qu Ð n k ª k 1 sn Ъ u density n L if § K ` N u-small such that sn n Ð ª, n¶K L; , 7. Modes of convergence For u > W weights u > `ª Z au n Qu Ð n k ª k 1 sn Ъ u density n sn L if § Ð n ª u s. K ` N u-small such that sn Cesaro L if 1 au n Qu n k Ssk k 0 LS n Ð ª 0. n Ð ª, n¶K L; , 7. Modes of convergence For u > W weights u > `ª Z au n Qu Ð n k ª k 1 sn Ъ u density n sn L if § Ð n ª u s. K ` N u-small such that sn Cesaro L if Evidently 1 au n Ð n ª u s. Qu n k Ssk k 0 Cesaro LS n Ð ª 0. n Ô unÐ ª density . Ð ª, n¶K L; , 7. Modes of convergence For u > W weights u > `ª Z au n Qu Ð n k ª k 1 sn Ъ u density n sn L if § K ` N u-small such that sn Ð n ª u s. Cesaro L if Q n 1 uk xk au n k 0 lim n ª, n¶K0 Qu n k Ssk Cesaro > W, x Ъ n xn C L L& Ô Ð ª, n¶K Ð ª 0. density . > RN bdd. below: K0 ` N, u-small s.t. xn Ð n ª u s. L; n Ô unÐ ª x1 , x2 , . . . § LS k 0 Ð n ª u s. Evidently ¶6 Smoothing Lemma: u 1 au n n Cesaro L. , 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that X Sk Ъ L2 P n E f ¦ f > L2 P . (MET) 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that (i) u > W smooth X Sk Ъ L2 P n Ô MET. E f ¦ f > L2 P . (MET) 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that (i) u > W smooth X Sk Ъ L2 P n E f ¦ f > L2 P . (MET) Ô MET. e.g. uF for F > R T , T RWM. 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that (i) u > W smooth X Sk Ъ L2 P n E f ¦ f > L2 P . (MET) Ô MET. e.g. uF for F > R T , T RWM. (ii) For T WRE & WM, u F is MET ¦ F > R T . 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that (i) u > W smooth X Sk Ъ L2 P n E f f > L2 P . (MET) Ô MET. e.g. uF for F > R T , T RWM. (ii) For T WRE & WM, u F is MET (iii) u > W MET ¦ u smooth. ¦ F > R T . 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that (i) u > W smooth X Sk Ъ L2 P n E f f > L2 P . (MET) Ô MET. e.g. uF for F > R T , T RWM. (ii) For T WRE & WM, u F is MET (iii) u > W MET ¦ ¦ F > R T . u smooth. Qn. for renewal sequences. 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that (i) u > W smooth X Sk Ъ L2 P n E f f > L2 P . ¦ F > R T . u smooth. Qn. for renewal sequences. (iv) u > W MET, Ω, A, P, S weakly mixing PPT. Ô (MET) Ô MET. e.g. uF for F > R T , T RWM. (ii) For T WRE & WM, u F is MET (iii) u > W MET ¦ P A 9 S n B Ъ u density n P AP B ¦ A, B > A. 8. Mean ergodic theorem with weights A weight u > W is (good for the) mean ergodic theorem (MET) if ¦ ergodic PPT Ω, A, P, S : 1 n 1 uk f au n k 0 Q Known that (i) u > W smooth X Ъ L2 P Sk n E f f > L2 P . (MET) Ô MET. e.g. uF for F > R T , T RWM. (ii) For T WRE & WM, u F is MET (iii) u > W MET ¦ ¦ F > R T . u smooth. Qn. for renewal sequences. (iv) u > W MET, Ω, A, P, S weakly mixing PPT. Ô P A 9 S n B Ъ u density n Ô T Use these to prove T RWM RWM ¦ weakly mixing PPT S. P AP B S ergodic ¦ ¦ A, B > A. ergodic PPT S & 9. Proof sketch of ¶4 ¶4 X , B , m, T WRE, MPT & α ` R T & Ω > Cα such that m A 9 T n B un u Ωdensity Ъ n § cble gen. ptn mAmB ¦ A, B > Cα (o) then X , B , m, T is RWM. Proof that Ω satisfies (Í). §1 WLOG X αZ , T shift & Cα clopen base Polish top. on X . 9. Proof sketch of ¶4 ¶4 X , B , m, T WRE, MPT & α ` R T & Ω > Cα such that m A 9 T n B un u Ωdensity Ъ n § cble gen. ptn mAmB ¦ A, B > Cα (o) then X , B , m, T is RWM. Proof that Ω satisfies (Í). §1 WLOG X §2 (o) ¦ αZ , T A, B > Uα shift & finite Cα clopen base Polish top. on X . unions of cylinders. 9. Proof sketch of ¶4 ¶4 X , B , m, T WRE, MPT & α ` R T & Ω > Cα such that m A 9 T n B un § u Ωdensity Ъ cble gen. ptn mAmB n ¦ A, B > Cα (o) then X , B , m, T is RWM. Proof that Ω satisfies (Í). §1 WLOG X §2 (o) ¦ αZ , T A, B > Uα §3 u density lim k ª compact. shift & finite mA9T k B uk Cα clopen base Polish top. on X . unions of cylinders. B mAmB ¦ A, B ` X 9. Proof sketch of ¶4 ¶4 X , B , m, T WRE, MPT & α ` R T & Ω > Cα such that m A 9 T n B un § u Ωdensity Ъ cble gen. ptn mAmB n ¦ A, B > Cα (o) then X , B , m, T is RWM. Proof that Ω satisfies (Í). §1 WLOG X §2 (o) ¦ αZ , T A, B > Uα §3 u density lim k ª shift & finite mA9T k B uk Cα clopen base Polish top. on X . unions of cylinders. B mAmB compact. §4 (o) holds ¦ A, B ` Ω compact. ¦ A, B ` X 9. Proof sketch of ¶4 ¶4 X , B , m, T WRE, MPT & α ` R T & Ω > Cα such that m A 9 T n B un § u Ωdensity Ъ cble gen. ptn mAmB n ¦ A, B > Cα (o) then X , B , m, T is RWM. Proof that Ω satisfies (Í). §1 WLOG X §2 (o) ¦ αZ , T A, B > Uα §3 u density lim k ª shift & finite mA9T k B uk Cα clopen base Polish top. on X . unions of cylinders. B mAmB ¦ A, B ` X compact. §4 (o) holds ¦ A, B ` Ω compact. §5 u density lim k ª mA9T k B uk C mAmB ¦ A, B > B 9 Ω. 9. Proof sketch of ¶4 ¶4 X , B , m, T WRE, MPT & α ` R T & Ω > Cα such that m A 9 T n B un § u Ωdensity Ъ cble gen. ptn mAmB n ¦ A, B > Cα (o) then X , B , m, T is RWM. Proof that Ω satisfies (Í). §1 WLOG X §2 (o) ¦ αZ , T A, B > Uα §3 u density lim k ª shift & finite mA9T k B uk Cα clopen base Polish top. on X . unions of cylinders. B mAmB ¦ A, B ` X compact. §4 (o) holds ¦ A, B ` Ω compact. mA9T k B uk §5 u density lim k ª §6 (o) holds ¦ A, B > B 9 Ω. C mAmB 2 ¦ A, B > B 9 Ω. 10. Examples: LLT sets. X , B, m, T WRE MPT, an an T γ-reg. var. (γ > 0, 1). 10. Examples: LLT sets. X , B, m, T WRE MPT, an an T γ-reg. var. (γ > 0, 1). Ω > R T , mΩ 1 LLT set if § cble, ptn β ` B Ω s.t. ϕΩ1 n > σ β ¦ n C 1 & s.t. ¦ A, B > Cβ TΩ , a1 nmA 9 TΩn B 9 ϕn uniformly in x > E ¦ kn n Ð ª, kn n E ` 0, ª compact fZγ x mAmB x 10. Examples: LLT sets. X , B, m, T WRE MPT, an an T γ-reg. var. (γ > 0, 1). Ω > R T , mΩ 1 LLT set if § cble, ptn β ` B Ω s.t. ϕΩ1 n > σ β ¦ n C 1 & s.t. ¦ A, B > Cβ TΩ , a1 nmA 9 TΩn B 9 ϕn uniformly in x > E where f E Zγγ fZγ 1. ¦ kn n Ð ª, kn n fZγ x mAmB x E ` 0, ª compact p.d.f. of +-ive, γ-stable RV Zγ , a normalized s.t. 10. Examples: LLT sets. X , B, m, T WRE MPT, an an T γ-reg. var. (γ > 0, 1). Ω > R T , mΩ 1 LLT set if § cble, ptn β ` B Ω s.t. ϕΩ1 n > σ β ¦ n C 1 & s.t. ¦ A, B > Cβ TΩ , a1 nmA 9 TΩn B 9 ϕn uniformly in x > E where f E Zγγ fZγ 1. ¦ kn n Ð ª, kn n fZγ x mAmB x E ` 0, ª compact p.d.f. of +-ive, γ-stable RV Zγ , a normalized s.t. E.G. Natural extensions of towers over Gibbs Markov fibred systems or AFU maps Ω, S, α with α-meas. height fn. and γ-rv return sequences. 10. Examples: LLT sets. X , B, m, T WRE MPT, an an T γ-reg. var. (γ > 0, 1). Ω > R T , mΩ 1 LLT set if § cble, ptn β ` B Ω s.t. ϕΩ1 n > σ β ¦ n C 1 & s.t. ¦ A, B > Cβ TΩ , a1 nmA 9 TΩn B 9 ϕn uniformly in x > E where f E Zγγ fZγ 1. ¦ kn n Ð ª, kn n fZγ x mAmB x E ` 0, ª compact p.d.f. of +-ive, γ-stable RV Zγ , a normalized s.t. E.G. Natural extensions of towers over Gibbs Markov fibred systems or AFU maps Ω, S, α with α-meas. height fn. and γ-rv return sequences. Refs. A & Denker; A, Denker, Sarig & Zweimüller & . . . 10. Examples: LLT sets. X , B, m, T WRE MPT, an an T γ-reg. var. (γ > 0, 1). Ω > R T , mΩ 1 LLT set if § cble, ptn β ` B Ω s.t. ϕΩ1 n > σ β ¦ n C 1 & s.t. ¦ A, B > Cβ TΩ , a1 nmA 9 TΩn B 9 ϕn uniformly in x > E where f E Zγγ fZγ 1. ¦ kn n Ð ª, kn n fZγ x mAmB x E ` 0, ª compact p.d.f. of +-ive, γ-stable RV Zγ , a normalized s.t. E.G. Natural extensions of towers over Gibbs Markov fibred systems or AFU maps Ω, S, α with α-meas. height fn. and γ-rv return sequences. Refs. A & Denker; A, Denker, Sarig & Zweimüller & . . . , In this situation, X , B , m, T is RWM. 11. Proof of , (as in [GL]) 11. Proof of , (as in [GL]) For un γan n , lim n ª mA9T n B un C mAmB ¦ A, B > Cβ TΩ . 11. Proof of , (as in [GL]) For un Proof γan n , lim n ª mA9T n B un C mAmB ¦ Fix A, B > Cβ TΩ , 0 @ c @ d @ ª & set xk,n A, B > Cβ TΩ . n , B k then 11. Proof of , (as in [GL]) For un Proof γan n , lim n ª mA9T n B un C mAmB ¦ Fix A, B > Cβ TΩ , 0 @ c @ d @ ª & set xk,n mA 9 T n B Qm A C Q Q n 9 TΩk B 9 ϕk k 1 1Bk Bn, xk,n > c,d 1Bk Bn, xk,n > c,d A, B > Cβ TΩ . n , B k then n mA 9 TΩk B 9 ϕk f xk,n mAmB B k xk,n B k 11. Proof of , (as in [GL]) For un Proof γan n , lim ª n mA9T n B un C mAmB ¦ Fix A, B > Cβ TΩ , 0 @ c @ d @ ª & set xk,n mA 9 T n B Qm A C Q Q n 9 TΩk B 9 ϕk k 1 1Bk Bn, xk,n > c,d 1Bk Bn, xk,n > c,d Q 1Bk Bn, xk,n > c,d f xk,n B k γan n γan n then xk,n B k f xk,n mAmB B k 1Bk Bn, xk,n >c,d c,d n , B k n mA 9 TΩk B 9 ϕk Q S A, B > Cβ TΩ . f x dx xγ xk,n xk 1,n γ xk,n f xk,n γan E1 c,d Zγ Zγγ . n 2 11. Proof of , (as in [GL]) For un Proof γan n , lim ª n mA9T n B un C mAmB ¦ Fix A, B > Cβ TΩ , 0 @ c @ d @ ª & set xk,n mA 9 T n B Qm A C Q Q n 9 TΩk B 9 ϕk k 1 1Bk Bn, xk,n > c,d 1Bk Bn, xk,n > c,d Q 1Bk Bn, xk,n > c,d f xk,n B k Above ex’s H-K mix. γan n γan n ¦ then xk,n B k f xk,n mAmB B k 1Bk Bn, xk,n >c,d c,d n , B k n mA 9 TΩk B 9 ϕk Q S A, B > Cβ TΩ . f x dx xγ xk,n xk 1,n γ xk,n f xk,n γan E1 c,d Zγ Zγγ . n 2 γ > 1 , 1. Refs: Thaler; Melbourne & Terhesiu. 12. Questions arising Spectral and rational weak mixing 12. Questions arising Spectral and rational weak mixing WM RWM 12. Questions arising Spectral and rational weak mixing WM § RWM squashable, WM MPTs , not WRE or RWM. 12. Questions arising Spectral and rational weak mixing WM § RWM squashable, WM MPTs , not WRE or RWM. Question Does weak rational ergodicity and spectral weak mixing imply rational weak mixing? 12. Questions arising Spectral and rational weak mixing WM § RWM squashable, WM MPTs , not WRE or RWM. Question Does weak rational ergodicity and spectral weak mixing imply rational weak mixing? L2 spectrum and rational weak mixing 12. Questions arising Spectral and rational weak mixing WM § RWM squashable, WM MPTs , not WRE or RWM. Question Does weak rational ergodicity and spectral weak mixing imply rational weak mixing? L2 spectrum and rational weak mixing All RWM examples here of form T K-automorphism and S WM PPT S where T is an infinite 12. Questions arising Spectral and rational weak mixing WM § RWM squashable, WM MPTs , not WRE or RWM. Question Does weak rational ergodicity and spectral weak mixing imply rational weak mixing? L2 spectrum and rational weak mixing All RWM examples here of form T K-automorphism and S WM PPT S where T is an infinite Koopman operators have countable Lebesgue spectrum. 12. Questions arising Spectral and rational weak mixing WM § RWM squashable, WM MPTs , not WRE or RWM. Question Does weak rational ergodicity and spectral weak mixing imply rational weak mixing? L2 spectrum and rational weak mixing All RWM examples here of form T K-automorphism and S WM PPT S where T is an infinite Koopman operators have countable Lebesgue spectrum. Sample question Does the Koopman operator of a rationally weakly mixing measure preserving transformation necessarily have countable Lebesgue spectrum? 12. Questions arising Spectral and rational weak mixing WM § RWM squashable, WM MPTs , not WRE or RWM. Question Does weak rational ergodicity and spectral weak mixing imply rational weak mixing? L2 spectrum and rational weak mixing All RWM examples here of form T K-automorphism and S WM PPT S where T is an infinite Koopman operators have countable Lebesgue spectrum. Sample question Does the Koopman operator of a rationally weakly mixing measure preserving transformation necessarily have countable Lebesgue spectrum? Thank you for listening.
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