Russian Mathematical Surveys, 2025, Volume 80, Issue 1, Pages 155–160 DOI: https://doi.org/10.4213/rm10217e (Mi rm10217)

DOI: https://doi.org/10.4213/rm10217e

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On 26 October 2024 the outstanding researcher, member of the Russian Academy of Sciences Dmitry Treschev observed his 60th birthday. He made a lasting contribution to the dynamics of Hamiltonian systems, perturbation theory, Arnold diffusion, stability, integrability, chaos, and KAM theory. Treschev published more than 110 scientific articles and 3 monographs.

Dmitry Valerievich Treschev was born on October 26th 1964, in the town of Olenegorsk in Murmansk Oblast, in a family of a military officer. Soon after that his parents moved to Kyiv, where he started basic education. As a repeated winner of mathematical olympiads for schoolchildren, he enrolled in the Kolmogorov Boarding School under the auspices of Moscow State University, now known as AESC MSU.

In 1981 he started studying at the Faculty of Mechanics and Mathematics of Moscow State University (MSU). During his second year at the university, professors noticed Treschev’s extraordinary mathematical talents. Valery Vasilievich Kozlov became his thesis advisor. Their fruitful collaboration continues to this day.

In 1986 Dmitry Treschev graduated with honours, and started his postgraduate studies in MSU. In 1988 he defended his Ph.D. thesis “Geometric methods of investigation of periodic trajectories in Hamiltonian systems”. Four years later he defended his D.Sc. thesis “Qualitative methods in Hamiltonian systems close to integrable ones”.

In 1986 Treschev started teaching in AESC MSU, and in 1988 he began working in the Department of Theoretical Mechanics of the Faculty of Mechanics and Mathematics at MSU. In 1993 he became a leading researcher, in 1998 he received a professor position, and in 2006 was promoted to the head of the department, the office he holds to this day. Since 1995 he has also been working in Steklov Mathematical Institute of Russian Academy of Sciences, and he became its director in 2017.

Dmitry Treschev has had seven successful Ph.D. students, one of whom became a doctor of sciences. Treschev transmits his researcher enthusiasm to people around him. His students are motivated to solve ever more difficult problems and constantly improve their mathematical skills. He finds natural and regular approaches even to the most difficult problems. When he reports about his solutions, the listener gets a feeling of simplicity of the result. This is the case even when the problem in question has been challenging the scientific community for decades.

In 2003 Treschev was elected a corresponding member of the Russian Academy of Sciences and in 2016 became a full member. He is a member of the editorial boards of the journals Nonlinearity and Regular and Chaotic Dynamics, and also is the editor-in-chief of Matematicheskie Zametki.^1[x]1Translated into English as Mathematical Notes. In 1995 Dmitry Treschev was awarded the State Prize of the Russian Federation for Young Scientists, and in 2007 he became a Lyapunov Prize laureate for his series of works “Separatrix map and its application to problems in Hamiltonian mechanics”. He was an invited speaker at the International Congress of Mathematicians in Beijing in 2002.

Let us briefly mention some of Treschev’s areas of research and main scientific results.

Treschev invented the continuous averaging method. We will shortly describe this method. Suppose we have a dynamical system $\dot x=\hat v(x)$ and are looking for a change of variables to simplify it. Usually, to construct such a change one uses the method of successive approximations. Treschev proposed replacing the discrete chain of changes of variables by a continuous one, parametrized by a variable $\delta\in [0,+\infty)$. The family of changes is constructed as a shift along the solutions of a certain auxiliary system

• $\bullet$ the problem of the inclusion of a diffeomorphism in a flow [8];
• $\bullet$ the estimate of an unremovable exponentially small dependence on the fast phase in slow-fast systems [9], [12], [11];
• $\bullet$ the calculation of asymptotic behaviour for the exponentially small separatrix splitting in the Chirikov standard map, in a pendulum with rapidly oscillating suspension point, and in other systems [21], [13], [14];
• $\bullet$ new results in the theory of normal forms in neighbourhoods of equilibria [20].

In [12] Treschev considered a dynamical system

In [9] Treschev and his student A. Pronin considered a classical multi-frequency slow-fast system. Using the continuous averaging method, they showed that a non-integrable perturbation can be weakened to exponentially small terms. Moreover, they obtained sharp estimates of the exponents.

Another example of a dynamical system with exponentially small effects is a pendulum with rapidly oscillating suspension point [13]. In that work Treschev, using the continuous averaging method, was the first to calculate the asymptotics of an exponentially small splitting of separatrices. These results cannot be obtained using the method of the Poincaré–Melnikov integral.

Treschev developed the separatrix mapping method and, with its help, obtained new fundamental results in the perturbation theory of Hamiltonian systems. The separatrix map is an analogue of the Poincaré map, when instead of a periodic trajectory, a homoclinic trajectory to an equilibrium state, or, more generally, to an invariant set, is used. Such a map had been used by B. V. Chirikov at the physical level of rigour to analyze a small periodic perturbation of a mathematical pendulum.

Treschev discovered that for the correct choice of variables the Chirikov separatrix map is defined by explicit universal formulae, only slightly depending on the specific system. Using the formulae obtained, he was the first to estimate rigorously (from above and below alike) the size of the stochastic layer of a perturbed system in a neighbourhood of the separatrix [21], [13], [14].

Treschev discovered that when separatrices split, stable periodic solutions appear in the holes (lunes) formed by the split separatrices. In collaboration with Neistadt, V. V. Sidorenko, K. Simo, and A. A. Vasiliev, he showed that in the region of transitions through the separatrix, in systems with parameters slowly changing at a rate $\sim \varepsilon$ there are many (of the order of $1/\varepsilon$) stable periodic trajectories. Each of these trajectories is surrounded by a stability island, whose measure is bounded below by a quantity of order $\varepsilon$, so that the total measure of the stability islands is bounded below by a quantity independent of $\varepsilon$. The proof is based on the study of asymptotic formulae for the corresponding Poincaré return map.

Next, Treschev constructed a generalization of the separatrix map to multidimensional Hamiltonian systems that are perturbations of integrable systems possessing a hyperbolic invariant torus with doubled stable and unstable manifolds. It turned out that there also are universal formulae for such a map.

Using the formulae obtained, Treshchev was the first to establish the existence of Arnold diffusion for general a priori unstable Hamiltonian systems and obtained sharp lower bounds for the rate of diffusion. Previously, Arnold diffusion had been found only in very special cases, without estimates for the rate of diffusion.

Now we describe some results due to Treschev in KAM theory. In [10] he showed that resonant tori of Liouville-integrable Hamiltonian systems do not fully disintegrate under perturbations: some of their non-resonant subtori of lower dimension are preserved as a rule and become partially normally hyperbolic. In [6], together with Neishtadt and Treschev’s student A. G. Medvedev, he studied a family of Lagrangian tori appearing in the vicinity of a resonance of a nearly integrable Hamiltonian system. Such families are absent in the unperturbed system. Treschev with co-authors also showed that, generally speaking, in the case of a resonance of order 1 the relative measure of the set of these tori is large in the following sense: the measure of the remaining chaotic set has an order of $\sqrt\varepsilon$ . Thus, for small $\varepsilon>0$ with random initial conditions in the $\sqrt\varepsilon$-neighbourhood of the resonance the probability of the occurrence of a quasi-periodic motion turns out to be significantly higher than in a ‘chaotic’ set.

For Hamiltonian systems with one and a half degrees of freedom, V. I. Arnold posed the following question: what is the difference in frequencies on Kolmogorov tori adjacent to a stochastic layer? Treschev solved this problem in an analytic and a non-analytic formulation. In particular, he proved that in analytic systems this difference is of the order of $\varepsilon$. Despite the apparent simplicity of the answer, this result requires the use of the separatrix mapping technique and theorems on averaging up to exponentially small terms.

Treschev showed that in the potential interaction of a finite-dimensional Hamiltonian system with a linear infinite-dimensional system effective dissipation arises, as a rule, which leads to simple final dynamics.

He proposed a generalization of Kolmogorov–Sinai entropy to quantum systems [18], [19], [1]. The classical Kolmogorov–Sinai entropy is defined for an endomorphism of a measure space $(M,\mu)$. Treschev posed and solved the problem of the construction of entropy for general unitary operators in the Hilbert space $L^2(M,\mu)$. For the Koopman operator corresponding to an endomorphism of a measure space, this entropy coincides with Kolmogorov–Sinai entropy. The construction is based on the new concept of the $\mu$-norm of an operator in the space $L^2(M,\mu)$.

In joint works with Kozlov [3]–[5] Treschev found all integrable systems in the class of (classical or quantum) Hamiltonian systems with toric position space and potential in the form of a trigonometric polynomial, and obtained generalizations to the case of systems with exponential interaction (generalized Toda chains). He also developed non-equilibrium statistical mechanics within the framework of Gibbs’s ensemble theory.

Jointly with S. V. Bolotin, Treschev developed a general theory of the anti-integrable limit [2], which he applied successfully in the study of Arnold diffusion.

We congratulate sincerely Dmitry Valerievich Treschev on his 60th birthday and wish him good health, happiness, and further success in science.

Bibliography
1.	K. A. Afonin and D. V. Treschev, “Entropy of a unitary operator on $L^2(\mathbb{T}^n)$”, Sb. Math., 213:7 (2022), 925–980
2.	S. V. Bolotin and D. V. Treschev, “The anti-integrable limit”, Russian Math. Surveys, 70:6 (2015), 975–1030
3.	V. V. Kozlov and D. V. Treshchëv, “On the integrability of Hamiltonian systems with toral position space”, Sb. Math., 63:1 (1989), 121–139
4.	V. V. Kozlov and D. V. Treshchev, “Polynomial integrals of Hamiltonian systems with exponential interaction”, Math. USSR-Izv., 34:3 (1990), 555–574
5.	V. V. Kozlov and D. V. Treshchev, “Kovalevskaya numbers of generalized Toda chains”, Math. Notes, 46:5 (1989), 840–848
6.	A. G. Medvedev, A. I. Neishtadt, and D. V. Treschev, “Lagrangian tori near resonances of near-integrable Hamiltonian systems”, Nonlinearity, 28:7 (2015), 2105–2130
7.	A. I. Neishtadt, V. V. Sidorenko, and D. V. Treschev, “Stable periodic motions in the problem on passage through a separatrix”, Chaos, 7:1 (1997), 2–11
8.	A. V. Pronin and D. V. Treschev, “On the inclusion of analytic maps into analytic flows”, Regul. Chaotic Dyn., 2:2 (1997), 14–24
9.	A. V. Pronin and D. V. Treschev, “Continuous averaging in multi-frequency slow-fast systems”, Regul. Chaotic Dyn., 5:2 (2000), 157–170
10.	D. V. Treshchëv, “The mechanism of destruction of resonance tori of Hamiltonian systems”, Math. USSR-Sb., 68:1 (1991), 181–203
11.	D. V. Treschev, “An averaging method for Hamiltonian systems, exponentially close to integrable ones”, Chaos, 6:1 (1996), 6–14
12.	D. V. Treschev, “The method of continuous averaging in the problem of separation of fast and slow motions”, Regul. Chaotic Dyn., 2:3-4 (1997), 9–20 (Russian)
13.	D. V. Treschev, “Splitting of separatrices for a pendulum with rapidly oscillating suspension point”, Russ. J. Math. Phys., 5:1 (1997), 63–98
14.	D. Treschev, “Width of stochastic layers in near-integrable two-dimensional symplectic maps”, Phys. D, 116:1-2 (1998), 21–43
15.	D. Treschev, “Trajectories in a neighbourhood of asymptotic surfaces of a priori unstable Hamiltonian systems”, Nonlinearity, 15:6 (2002), 2033–2052
16.	D. Treschev, “Evolution of slow variables in a priori unstable Hamiltonian systems”, Nonlinearity, 17:5 (2004), 1803–1841
17.	D. Treschev, “Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems”, Nonlinearity, 25:9 (2012), 2717–2757
18.	D. V. Treschev, “$\mu$-norm of an operator”, Proc. Steklov Inst. Math., 310 (2020), 262–290
19.	D. Treschev, “$\mu$-norm and regularity”, J. Dynam. Differential Equations, 33:3 (2021), 1269–1295
20.	D. V. Treschev, “Normalization flow”, Regul. Chaotic Dyn., 28:4-5 (2023), 781–804
21.	D. Treschev and O. Zubelevich, Introduction to the perturbation theory of Hamiltonian systems, Springer Monogr. Math., Springer-Verlag, Berlin, 2010, x+211 pp.
22.	A. A. Vasiliev, A. I. Neishtadt, C. Simó, and D. V. Treschev, “Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems”, Proc. Steklov Inst. Math., 259 (2007), 236–247

Citation in format AMSBIB

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\by S.~V.~Bolotin, O.~\`E.~Zubelevich, V.~V.~Kozlov, S.~B.~Kuksin, A.~I.~Neishtadt
\paper Dmitry Valerievich Treschev (on his sixtieth birthday)
\jour Russian Math. Surveys
\yr 2025
\vol 80
\issue 1
\pages 155--160
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\crossref{https://doi.org/10.4213/rm10217e}
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