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Homogenization for nonlocal problems with smooth kernels
Properties of multicorrelation sequences and large returns under some ergodicity assumptions
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA |
We prove that given a measure preserving system $ (X,\mathcal{B},\mu,T_1,\dots, T_d) $ with commuting, ergodic transformations $ T_i $ such that $ T_iT_j^{-1} $ are ergodic for all $ i \neq j $, the multicorrelation sequence $ a(n) = \int_X f_0 \cdot T_1^nf_1 \cdot \dotso \cdot T_d^n f_d \ d\mu $ can be decomposed as $ a(n) = a_{ \rm{st}}(n)+a_{ \rm{er}}(n) $, where $ a_{ \rm{st}} $ is a uniform limit of $ d $-step nilsequences and $ a_{ \rm{er}} $ is a nullsequence (that is, $ \lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n = M}^{N-1} |a_{ \rm{er}}|^2 = 0 $). Under some additional ergodicity conditions on $ T_1,\dots,T_d $ we also establish a similar decomposition for polynomial multicorrelation sequences of the form $ a(n) = \int_X f_0 \cdot \prod_{i = 1}^dT_i^{p_{i,1}(n)}f_1\cdot\dotso \cdot \prod_{i = 1}^dT_i^{p_{i,k}(n)}f_k \ d\mu $, where each $ p_{i,k}: {\mathbb{Z}} \rightarrow {\mathbb{Z}} $ is a polynomial map. We also show, for $ d = 2 $, that if $ T_1, T_2, T_1T_2^{-1} $ are invertible and ergodic, we have large triple intersections: for all $ \varepsilon>0 $ and all $ A \in \mathcal{B} $, the set $ \{n \in {\mathbb{Z}} : \mu(A \cap T_1^{-n}A \cap T_2^{-n}A)>\mu(A)^3-\varepsilon\} $ is syndetic. Moreover, we show that if $ T_1, T_2, T_1T_2^{-1} $ are totally ergodic, and we denote by $ p_n $ the $ n $-th prime, the set $ \{n \in \mathbb{N} : \mu(A \cap T_1^{-(p_n-1)}A \cap T_2^{-(p_n-1)}A)>\mu(A)^3-\varepsilon\} $ has positive lower density.
References:
[1] |
V. Bergelson,
Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.
doi: 10.1017/S0143385700004090. |
[2] |
V. Bergelson, B. Host and B. Kra,
Multiple recurrence and nilsequences. With an appendix by Imre Rusza, Invent. Math., 160 (2005), 261-303.
doi: 10.1007/s00222-004-0428-6. |
[3] |
V. Bergelson and A. Leibman,
Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753.
doi: 10.1090/S0894-0347-96-00194-4. |
[4] |
V. Bergelson and A. Leibman, Cubic averages and large intersections, Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 5–19.
doi: 10.1090/conm/631/12592. |
[5] |
V. Bergelson, T. Tao and T. Ziegler,
Multiple recurrence and convergence results associated to $\Bbb {F}_p^\omega$-actions, J. Anal. Math., 127 (2015), 329-378.
doi: 10.1007/s11854-015-0033-1. |
[6] |
Q. Chu,
Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792.
doi: 10.1017/S0143385710000258. |
[7] |
S. Donoso, J. Moreira, A. N. Le and W. Sun, Optimal lower bounds for multiple recurrence, Ergodic Theory and Dynamical Systems, (2019), 1–29.
doi: 10.1017/etds.2019.72. |
[8] |
S. Donoso and W. Sun,
Quantitative multiple recurrence for two and three transformations, Israel J. Math., 226 (2018), 71-85.
doi: 10.1007/s11856-018-1690-4. |
[9] |
N. Frantzikinakis,
Multiple correlation sequences and nilsequences, Invent. Math., 202 (2015), 875-892.
doi: 10.1007/s00222-015-0579-7. |
[10] |
N. Frantzikinakis,
Multiple ergodic averages for three polynomials and applications, Trans. Amer. Math. Soc., 360 (2008), 5435-5475.
doi: 10.1090/S0002-9947-08-04591-1. |
[11] |
N. Frantzikinakis and B. Host,
Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam. Systems, 38 (2018), 81-142.
doi: 10.1017/etds.2016.19. |
[12] |
N. Frantzikinakis, B. Host and B. Kra,
The polynomial multidimensional Szemerédi theorem along shifted primes, Israel J. Math., 194 (2013), 331-348.
doi: 10.1007/s11856-012-0132-y. |
[13] |
N. Frantzikinakis and B. Kra,
Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems, 25 (2005), 799-809.
doi: 10.1017/S0143385704000616. |
[14] |
W. T. Gowers,
A new proof of Szemerédi's theorem, Geom. Funct. Anal., 11 (2001), 465-588.
doi: 10.1007/s00039-001-0332-9. |
[15] |
J. T. Griesmer, Ergodic Averages, Correlation Sequences, and Sumsets, Ph. D thesis, The Ohio State University, 2009. |
[16] |
B. Host,
Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49.
doi: 10.4064/sm195-1-3. |
[17] |
B. Host and B. Kra,
Nonconventional ergodic averages and nilmanifolds, Ann. of Math., 161 (2005), 397-488.
doi: 10.4007/annals.2005.161.397. |
[18] |
B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, 236, American Mathematical Society, Providence, RI, 2018.
doi: 10.1090/surv/236. |
[19] |
M. C. R. Johnson,
Convergence of polynomial ergodic averages of several variables for some commuting transformations, Illinois J. Math., 53 (2009), 865-882.
doi: 10.1215/ijm/1286212920. |
[20] |
A. Khintchine,
The method of spectral reduction in classical dynamics, Proceedings of the National Academy of Sciences, 19 (1933), 567-573.
doi: 10.1073/pnas.19.5.567. |
[21] |
B. O. Koopman and J. von Neumann,
Dynamical systems of continuous spectra, Proceedings of the National Academy of Sciences, 18 (1932), 255-263.
doi: 10.1073/pnas.18.3.255. |
[22] |
A. Koutsogiannis, A. Le, J. Moreira, and F. K. Richter, Structure of multicorrelation sequences with integer part polynomial iterates along primes, Proc. Amer. Math. Soc. 149 (2021), no. 1,209–216. |
[23] |
A. N. Le,
Nilsequences and multiple correlations along subsequences, Ergodic Theory Dynam. Systems, 40 (2020), 1634-1654.
doi: 10.1017/etds.2018.110. |
[24] |
A. Leibman,
Multiple polynomial correlation sequences and nilsequences, Ergodic Theory Dynam. Systems, 30 (2010), 841-854.
doi: 10.1017/S0143385709000303. |
[25] |
A. Leibman,
Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.
doi: 10.1017/etds.2013.36. |
[26] |
P. Walters, An introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982. |
show all references
References:
[1] |
V. Bergelson,
Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.
doi: 10.1017/S0143385700004090. |
[2] |
V. Bergelson, B. Host and B. Kra,
Multiple recurrence and nilsequences. With an appendix by Imre Rusza, Invent. Math., 160 (2005), 261-303.
doi: 10.1007/s00222-004-0428-6. |
[3] |
V. Bergelson and A. Leibman,
Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753.
doi: 10.1090/S0894-0347-96-00194-4. |
[4] |
V. Bergelson and A. Leibman, Cubic averages and large intersections, Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 5–19.
doi: 10.1090/conm/631/12592. |
[5] |
V. Bergelson, T. Tao and T. Ziegler,
Multiple recurrence and convergence results associated to $\Bbb {F}_p^\omega$-actions, J. Anal. Math., 127 (2015), 329-378.
doi: 10.1007/s11854-015-0033-1. |
[6] |
Q. Chu,
Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792.
doi: 10.1017/S0143385710000258. |
[7] |
S. Donoso, J. Moreira, A. N. Le and W. Sun, Optimal lower bounds for multiple recurrence, Ergodic Theory and Dynamical Systems, (2019), 1–29.
doi: 10.1017/etds.2019.72. |
[8] |
S. Donoso and W. Sun,
Quantitative multiple recurrence for two and three transformations, Israel J. Math., 226 (2018), 71-85.
doi: 10.1007/s11856-018-1690-4. |
[9] |
N. Frantzikinakis,
Multiple correlation sequences and nilsequences, Invent. Math., 202 (2015), 875-892.
doi: 10.1007/s00222-015-0579-7. |
[10] |
N. Frantzikinakis,
Multiple ergodic averages for three polynomials and applications, Trans. Amer. Math. Soc., 360 (2008), 5435-5475.
doi: 10.1090/S0002-9947-08-04591-1. |
[11] |
N. Frantzikinakis and B. Host,
Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam. Systems, 38 (2018), 81-142.
doi: 10.1017/etds.2016.19. |
[12] |
N. Frantzikinakis, B. Host and B. Kra,
The polynomial multidimensional Szemerédi theorem along shifted primes, Israel J. Math., 194 (2013), 331-348.
doi: 10.1007/s11856-012-0132-y. |
[13] |
N. Frantzikinakis and B. Kra,
Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems, 25 (2005), 799-809.
doi: 10.1017/S0143385704000616. |
[14] |
W. T. Gowers,
A new proof of Szemerédi's theorem, Geom. Funct. Anal., 11 (2001), 465-588.
doi: 10.1007/s00039-001-0332-9. |
[15] |
J. T. Griesmer, Ergodic Averages, Correlation Sequences, and Sumsets, Ph. D thesis, The Ohio State University, 2009. |
[16] |
B. Host,
Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49.
doi: 10.4064/sm195-1-3. |
[17] |
B. Host and B. Kra,
Nonconventional ergodic averages and nilmanifolds, Ann. of Math., 161 (2005), 397-488.
doi: 10.4007/annals.2005.161.397. |
[18] |
B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, 236, American Mathematical Society, Providence, RI, 2018.
doi: 10.1090/surv/236. |
[19] |
M. C. R. Johnson,
Convergence of polynomial ergodic averages of several variables for some commuting transformations, Illinois J. Math., 53 (2009), 865-882.
doi: 10.1215/ijm/1286212920. |
[20] |
A. Khintchine,
The method of spectral reduction in classical dynamics, Proceedings of the National Academy of Sciences, 19 (1933), 567-573.
doi: 10.1073/pnas.19.5.567. |
[21] |
B. O. Koopman and J. von Neumann,
Dynamical systems of continuous spectra, Proceedings of the National Academy of Sciences, 18 (1932), 255-263.
doi: 10.1073/pnas.18.3.255. |
[22] |
A. Koutsogiannis, A. Le, J. Moreira, and F. K. Richter, Structure of multicorrelation sequences with integer part polynomial iterates along primes, Proc. Amer. Math. Soc. 149 (2021), no. 1,209–216. |
[23] |
A. N. Le,
Nilsequences and multiple correlations along subsequences, Ergodic Theory Dynam. Systems, 40 (2020), 1634-1654.
doi: 10.1017/etds.2018.110. |
[24] |
A. Leibman,
Multiple polynomial correlation sequences and nilsequences, Ergodic Theory Dynam. Systems, 30 (2010), 841-854.
doi: 10.1017/S0143385709000303. |
[25] |
A. Leibman,
Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.
doi: 10.1017/etds.2013.36. |
[26] |
P. Walters, An introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982. |
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