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, available in the harmonic analysis 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.
It was proved (2019; with I.V. Podvigin) the existence of estimates of the rate of convergence in the Birkhoff theorem which hold a.e. (for the ergodic case); criteria for the maximum possible such a rate were obtained.
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:
Kachurovskii A. G., “Rates of convergence in ergodic theorems”, Russian Math. Surveys, 51:4 (1996), 653–703
Kachurovskii A. G., “General theories unifying ergodic averages and martingales”, Proc. Steklov Inst. Math., 256 (2007), 160–187
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
Kachurovskii A. G., Podvigin I. V., “Fejer Sums for Periodic Measures and the von Neumann Ergodic Theorem”, Dokl. Math., 98:1 (2018), 344–347
Kachurovskii A. G., Podvigin I. V., “Measuring the rate of convergence in the Birkhoff ergodic theorem”, Math. Notes, 106:1–2 (2019), 52–62
A. G. Kachurovskii, I. V. Podvigin, V. E. Todikov, A. J. Khakimbaev, “A spectral criterion for power-law convergence rate in the ergodic theorem for ${\Bbb Z}^d$ and ${\Bbb R}^d$ actions”, Sibirsk. Mat. Zh., 65:1 (2024), 92–114
A. G. Kachurovskii, I. V. Podvigin, A. J. Khakimbaev, “Uniform Convergence on Subspaces in von Neumann Ergodic
Theorem with Discrete Time”, Mat. Zametki, 113:5 (2023), 713–730; Math. Notes, 113:5 (2023), 680–693
A. G. Kachurovskii, I. V. Podvigin, V. E. Todikov, “Uniform convergence on subspaces in von Neumann's ergodic theorem with continuous time”, Sib. Èlektron. Mat. Izv., 20:1 (2023), 183–206
A. G. Kachurovskii, I. V. Podvigin, A. A. Svishchev, “Zero-One law for the rates of convergence in the Birkhoff ergodic theorem with continuous time”, Mat. Tr., 24:2 (2021), 65–80
A. G. Kachurovskii, M. N. Lapshtaev, A. J. Khakimbaev, “Von Neumann's ergodic theorem and Fejer sums for signed measures on the circle”, Sib. Èlektron. Mat. Izv., 17 (2020), 1313–1321
6.
A. G. Kachurovskii, I. V. Podvigin, A. A. Svishchev, “The maximum pointwise rate of convergence in Birkhoff's ergodic theorem”, Zap. Nauchn. Sem. POMI, 498 (2020), 18–25
A. G. Kachurovskii, I. V. Podvigin, “Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 106:1 (2019), 40–52; Math. Notes, 106:1 (2019), 52–62
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; Siberian Adv. Math., 28:1 (2018), 23–38
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; Trans. Moscow Math. Soc., 77 (2016), 1–53
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; Math. Notes, 94:4 (2013), 524–531
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; Math. Notes, 91:4 (2012), 582–587
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; Sb. Math., 202:8 (2011), 1105–1125
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; Siberian Math. J., 52:5 (2011), 824–835
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; Math. Notes, 87:5 (2010), 720–727
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; Sb. Math., 201:4 (2010), 493–500
A. G. Kachurovskii, “The entropy brick of an automorphism of a Lebesgue space”, Mat. Zametki, 80:6 (2006), 943–945; Math. Notes, 80:6 (2006), 885–887
1999
19.
A. G. Kachurovskii, “Convergence of averages in the ergodic theorem for groups $\mathbb Z^d$”, Zap. Nauchn. Sem. POMI, 256 (1999), 121–128; J. Math. Sci. (New York), 107:5 (2001), 4231–4236
A. G. Kachurovskii, “Spectral measures and convergence rates in the ergodic theorem”, Dokl. Akad. Nauk, 347:5 (1996), 593–596
22.
A. G. Kachurovskii, “The rate of convergence in ergodic theorems”, Uspekhi Mat. Nauk, 51:4(310) (1996), 73–124; Russian Math. Surveys, 51:4 (1996), 653–703
A. G. Kachurovskii, “Fluctuation of averages in the strong law of large numbers”, Mat. Zametki, 50:5 (1991), 151–153; Math. Notes, 50:5 (1991), 1202–1203
A. G. Kachurovskii, “Boundedness of the fluctuation of mean sequences in the ergodic Birkhoff–Khinchin theorem”, Dokl. Akad. Nauk SSSR, 315:3 (1990), 530–532; Dokl. Math., 42:3 (1991), 810–812
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