اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدQuantitative multiple recurrence for two and three transformations
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We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2)$ with two commuting transformations such that for every $0<ell< 4$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2, T_{3})$ with three commuting transformations such that for every $ell>0$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}Acap T_{3}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $ell>0$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0.$$
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