Volume 75, Nº 2 (2020)

Ano: 2020
Artigos: 8
URL: https://journals.rcsi.science/0042-1316/issue/view/7515

Solenoidal attractors of diffeomorphisms of annular sets

Glyzin S., Kolesov A., Rozov N.

Resumo

An arbitrary diffeomorphism $\Pi$ of an annular set of the form $K=B\times \mathbb{T}$ is considered, where $B$ is a ball in a Banach space and $\mathbb{T}$ is a (finite- or infinite-dimensional) torus. A system of effective sufficient conditions is proposed which ensure that $P$ has a global attractor $A=\bigcap_{n\geqslant 0}\Pi^n(K)$ that can be represented as a generalized solenoid, that is, the inverse limit $\mathbb{T}\xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots\xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots$, where $G$ is an expanding linear endomorphism of the torus $\mathbb{T}$. Furthermore, the restriction $\Pi|_{A}$ is topologically conjugate to a shift map of the solenoid.Bibliography: 25 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(2):3-60

3-60

(Rus)

(JATS XML)

Uniform attractors for measure-driven quintic wave equations

Savostianov A., Zelik S.

Resumo

This is a detailed study of damped quintic wave equations with non-regular and non-autonomous external forces which are measures in time. In the 3D case with periodic boundary conditions, uniform energy-to-Strichartz estimates are established for the solutions, the existence of uniform attractors in a weak or strong topology in the energy phase space is proved, and their additional regularity is studied along with the possibility of representing them as the union of all complete bounded trajectories.Bibliography: 45 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(2):61-132

61-132

(Rus)

(JATS XML)

In search of infinite-dimensional Kähler geometry

Sergeev A.

Resumo

This paper is devoted to a survey of recent results in the Kähler geometry of infinite-dimensional Kähler manifolds. Three particular classes of such manifolds are investigated: the loop spaces of compact Lie groups, Hilbert–Schmidt Grassmannians, and the universal Teichmüller space. These investigations have been prompted both by requirements in Kähler geometry itself and by connections with string theory, which are considered in the last section.Bibliography: 43 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(2):133-184

133-184

(Rus)

(JATS XML)

Alexandre Mikhailovich Vinogradov (obituary)

Astashov A., Astashova I., Bocharov A., Buchstaber V., Vassiliev V., Verbovetsky A., Vershik A., Veselov A., Vinogradov M., Vitagliano L., Vitolo R., Voronov T., Kac V., Kosmann-Schwarzbach Y., Krasil'shchik I., Krichever I., Krishchenko A., Lando S., Lychagin V., Marvan M., Maslov V., Mishchenko A., Novikov S., Rubtsov V., Samokhin A., Sosinskii A., Stasheff J., Fuchs D., Helemskii A., Khor'kova N., Chetverikov V., Schwarz A.

Uspekhi Matematicheskikh Nauk. 2020;75(2):185-190

185-190