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Kachurovskii, Alexander Grigoryevich

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Total publications: 17
Scientific articles: 17
Presentations: 5

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Kachurovskii, Alexander Grigoryevich
Doctor of physico-mathematical sciences (2000)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 21.01.1961
E-mail:
Keywords: ergodic theory, rates of convergence in ergodic theorems, Fejer sums, unifications of ergodic theorems and martingale convergence theorems
UDC: 517.987, 519.214, 519.216
MSC: 28D, 37A, 60F, 60G

Subject:

It was proved (1996) that a power rate of convergence in von Neumanns ergodic theorem is equivalent to the power (with the same exponent) singularity at zero point of a spectral measure of averaging function with respect to the dynamical system. I.e. it was shown that the estimates of convergence rates in this ergodic theorem are necessarily the spectral ones. Estimates of the rates of convergence were obtained (1996; since 2010 – with the students): in von Neumanns ergodic theorem – via the singularity at zero point of the spectral measure, and via the speed of decay of correlations (i.e., the Fourier coefficients of this measure); in Birkhoffs ergodic theorem – via the rate of convergence in von Neumanns theorem, and via a speed of decay of probabilities of large deviations. Asymptotically exact estimates of the rates of convergence are obtained in both these ergodic theorems: for certain well-known billiards, and Anosov systems.

It was shown (1998) that both ergodic averages and martingales can be viewed as particular degenerate cases of the one new general class of stochastic processes; convergences of this new general process a.e. (an extra condition of integrability of the supremum of module of the process was omitted by the student I.V. Podvigin in 2010) and in the norm, are proved – and both maximal and dominant estimates take place, too.

It turns out (2018), that the Fejer sums for measures on the circle and the norms of the deviations from the limit in the von Neumann ergodic theorem both are calculating, in fact, with the same formulas (by integrating of the Fejer kernels) --- and so, this ergodic theorem is a statement about the asymptotics of the growth of the Fejer sums at zero for the corresponding spectral measure. As a result, availablle in the literature, numerous estimates for the deviations of Fejer sums at a point allowed to obtain new estimates for the rate of convergence in this ergodic theorem.

Biography

Graduated from Faculty of Mathematics and Mechanics of Novosibirsk State University in 1983 (Department of Mathematical Analysis). Ph.D thesis was defended in 1987 at Sobolev Institute of Mathematics, Novosibirsk. D.Sci thesis was defended in 1999 at Steklov Mathematical Institute at St. Petersburg. Principal place of work since 1983 – Sobolev Institute of Mathematics (with an interruption in 1997–1999 for the doctorate at Steklov Mathematical Institute at St. Petersburg).

   
Main publications:
  1. Kachurovskii A. G., “Rates of convergence in ergodic theorems”, Russian Math. Surveys, 51:4 (1996), 653–703  crossref  mathscinet  zmath  adsnasa  isi  scopus
  2. Kachurovskii A. G., “General theories unifying ergodic averages and martingales”, Proc. Steklov Inst. Math., 256 (2007), 160–187  crossref  mathscinet  zmath  scopus
  3. Kachurovskii A. G., Sedalishchev V. V., “Constants in estimates for the rates of convergence in von Neumann’s and Birkhoff’s ergodic theorems”, Sb. Math., 202:8 (2011), 1105–1125  crossref  mathscinet  zmath  adsnasa  isi  scopus
  4. Kachurovskii A. G., Podvigin I. V., “Large deviations and the rate of convergence in the Birkhoff ergodic theorem”, Math. Notes, 94:4 (2013), 524–531  crossref  mathscinet  zmath  isi  scopus
  5. Kachurovskii A. G., Podvigin I. V., “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems”, Trans. Moscow Math. Soc., 77 (2016), 1–53  crossref  mathscinet  zmath  scopus

http://www.mathnet.ru/eng/person17305
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https://mathscinet.ams.org/mathscinet/MRAuthorID/238538
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Publications in Math-Net.Ru
2019
1. A. G. Kachurovskii, I. V. Podvigin, “Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 106:1 (2019),  40–52  mathnet  elib; Math. Notes, 106:1 (2019), 52–62  isi  scopus
2018
2. A. G. Kachurovskii, “The Fejer integrals and the von Neumann ergodic theorem with continuous time”, Zap. Nauchn. Sem. POMI, 474 (2018),  171–182  mathnet
2017
3. A. G. Kachurovskiĭ, I. V. Podvigin, “Large deviations of the ergodic averages: from Hölder continuity to continuity almost everywhere”, Mat. Tr., 20:1 (2017),  97–120  mathnet  elib; Siberian Adv. Math., 28:1 (2018), 23–38  scopus
2016
4. A. G. Kachurovskii, I. V. Podvigin, “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems”, Tr. Mosk. Mat. Obs., 77:1 (2016),  1–66  mathnet  elib; Trans. Moscow Math. Soc., 77 (2016), 1–53  scopus
2013
5. A. G. Kachurovskii, I. V. Podvigin, “Large Deviations and the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 94:4 (2013),  569–577  mathnet  mathscinet  zmath  elib; Math. Notes, 94:4 (2013), 524–531  isi  elib  scopus
2012
6. A. G. Kachurovskii, V. V. Sedalishchev, “On the Constants in the Estimates of the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 91:4 (2012),  624–628  mathnet  mathscinet  elib; Math. Notes, 91:4 (2012), 582–587  isi  elib  scopus
2011
7. A. G. Kachurovskii, V. V. Sedalishchev, “Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems”, Mat. Sb., 202:8 (2011),  21–40  mathnet  mathscinet  zmath  elib; Sb. Math., 202:8 (2011), 1105–1125  isi  scopus
8. N. A. Dzhulaĭ, A. G. Kachurovskiĭ, “Constants in the estimates of the rate of convergence in von Neumann's ergodic theorem with continuous time”, Sibirsk. Mat. Zh., 52:5 (2011),  1039–1052  mathnet  mathscinet; Siberian Math. J., 52:5 (2011), 824–835  isi  scopus
2010
9. A. G. Kachurovskii, V. V. Sedalishchev, “On the Constants in the Estimates of the Rate of Convergence in von Neumann's Ergodic Theorem”, Mat. Zametki, 87:5 (2010),  756–763  mathnet  mathscinet  elib; Math. Notes, 87:5 (2010), 720–727  isi  scopus
10. A. G. Kachurovskii, A. V. Reshetenko, “On the rate of convergence in von Neumann's ergodic theorem with continuous time”, Mat. Sb., 201:4 (2010),  25–32  mathnet  mathscinet  zmath  elib; Sb. Math., 201:4 (2010), 493–500  isi  elib  scopus
2007
11. A. G. Kachurovskii, “General Theories Unifying Ergodic Averages and Martingales”, Tr. Mat. Inst. Steklova, 256 (2007),  172–200  mathnet  mathscinet  zmath  elib; Proc. Steklov Inst. Math., 256 (2007), 160–187  elib  scopus
2006
12. A. G. Kachurovskii, “The entropy brick of an automorphism of a Lebesgue space”, Mat. Zametki, 80:6 (2006),  943–945  mathnet  mathscinet  zmath  elib; Math. Notes, 80:6 (2006), 885–887  isi  elib  scopus
1999
13. A. G. Kachurovskii, “Convergence of averages in the ergodic theorem for groups $\mathbb Z^d$”, Zap. Nauchn. Sem. POMI, 256 (1999),  121–128  mathnet  mathscinet  zmath  elib; J. Math. Sci. (New York), 107:5 (2001), 4231–4236
1998
14. A. G. Kachurovskii, “Martingale ergodic theorem”, Mat. Zametki, 64:2 (1998),  311–314  mathnet  mathscinet  zmath; Math. Notes, 64:2 (1998), 266–269  isi
1996
15. A. G. Kachurovskii, “The rate of convergence in ergodic theorems”, Uspekhi Mat. Nauk, 51:4(310) (1996),  73–124  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 51:4 (1996), 653–703  isi  scopus
1992
16. A. G. Kachurovskii, “Time fluctuations in the statistical ergodic theorem”, Mat. Zametki, 52:1 (1992),  146–148  mathnet  mathscinet  zmath; Math. Notes, 52:1 (1992), 744–745  isi
1991
17. A. G. Kachurovskii, “Fluctuation of averages in the strong law of large numbers”, Mat. Zametki, 50:5 (1991),  151–153  mathnet  mathscinet  zmath; Math. Notes, 50:5 (1991), 1202–1203  isi

Presentations in Math-Net.Ru
1. Суммы Фейера и эргодическая теорема фон Неймана
A. G. Kachurovskii
Seminar on Probability Theory and Mathematical Statistics
October 12, 2018 18:00
2. Суммы Фейера периодических мер и эргодическая теорема фон Неймана
A. G. Kachurovskii
Dynamical systems and differential equations
October 8, 2018 18:30
3. Rates of convergence in ergodic theorems for Anosov diffeomorphisms
Alexander Kachurovskii
International Conference “Anosov Systems and Modern Dynamics” dedicated to the 80th anniversary of Dmitry Anosov
December 23, 2016 15:40   
4. Deviations of Fejer sums and rates of convergence in the von Neumann ergodic theorem
A. G. Kachurovskii
International Conference "Geometric Analysis and Control Theory"
December 8, 2016 15:00   
5. Оценки скоростей сходимости в эргодических теоремах фон Неймана и Биркгофа
A. G. Kachurovskii
Seminar on Probability Theory and Mathematical Statistics
October 4, 2013 20:00

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