Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 21.01.1961
Phone: +7 906-195-45-16
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, 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.

Main publications:

1. Kachurovskii A. G., “Rates of convergence in ergodic theorems”, Russian Math. Surveys, 51:4 (1996), 653–703            
2. Kachurovskii A. G., “General theories unifying ergodic averages and martingales”, Proc. Steklov Inst. Math., 256 (2007), 160–187        
3. 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        
4. Kachurovskii A. G., Podvigin I. V., “Fejer Sums for Periodic Measures and the von Neumann Ergodic Theorem”, Dokl. Math., 98:1 (2018), 344–347        
5. Kachurovskii A. G., Podvigin I. V., “Measuring the rate of convergence in the Birkhoff ergodic theorem”, Math. Notes, 106:1–2 (2019), 52–62          

Publications in Math-Net.Ru

Talks and lectures

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Organisations:

• Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk