# Methods of Functional Analysis

and Topology

Editors-in-Chief: Yu. M. Berezansky,
Yu. G. Kondratiev

ISSN: 1029-3531 (Print) 2415-7503 (Online)

Methods of Functional Analysis and Topology (MFAT), founded in 1995, is a peer-reviewed arXiv overlay journal publishing original articles and surveys on general methods and techniques of functional analysis and topology with a special emphasis on applications to modern mathematical physics.

MFAT is an open access journal, free for authors and free for readers.

MFAT is indexed in: MathSciNet, zbMATH, Web of Science (ESCI), Google Scholar.

## Latest Articles (September, 2016)

### Homeotopy groups of rooted tree like non-singular foliations on the plane

Methods Funct. Anal. Topology **22** (2016), no. 3, 283-294

Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $\pi_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}(F)$ is an analogue of a mapping class group for foliated homeomorphisms. We will describe the algebraic structure of $\pi_0 H^{+}(F)$ under an assumption that the corresponding space of leaves of $F$ has a structure similar to a rooted tree of finite diameter.

### Foliations with all non-closed leaves on non-compact surfaces

Sergiy Maksymenko, Eugene Polulyakh

Methods Funct. Anal. Topology **22** (2016), no. 3, 266-282

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.

### Actions of finite groups and smooth functions on surfaces

Methods Funct. Anal. Topology **22** (2016), no. 3, 210-219

Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and $\mathcal{E}_V$ be its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and $\mathcal{S}_V(f) = \{h\in\mathcal{S}(f)\,| h(V) = V\}.$ The group $\mathcal{S}_V(f)$ acts on the set $\pi_0\partial \mathcal{E}_V$ of connected components of the boundary of $\mathcal{E}_V.$ Therefore we have a homomorphism $\phi:\mathcal{S}(f)\to \mathrm{Aut}(\pi_0\partial \mathcal{E}_V)$. Let also $G = \phi(\mathcal{S}(f))$ be the image of $\mathcal{S}(f)$ in $\mathrm{Aut}(\pi_0\partial \mathcal{E}_V).$ Suppose that the inclusion $\partial \mathcal{E}_V\subset M\setminus V$ induces a bijection $\pi_0 \partial \mathcal{E}_V\to\pi_0(M\setminus V).$ Let $H$ be a subgroup of $G.$ We present a sufficient condition for existence of a section $s:H\to \mathcal{S}_V(f)$ of the homomorphism $\phi,$ so, the action of $H$ on $\partial \mathcal{E}_V$ lifts to the $H$-action on $M$ by $f$-preserving diffeomorphisms of $M$. This result holds for a larger class of smooth functions $f:M\to \mathbb{R}$ having the following property: for each critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple linear factors.

### Fractional statistical dynamics and fractional kinetics

José Luís da Silva, Anatoly N. Kochubei, Yuri Kondratiev

Methods Funct. Anal. Topology **22** (2016), no. 3, 197-209

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.