### Article

## Cantor Type Basic Sets of Surface A-endomorphisms

The paper is devoted to an investigation of the genus of an orientable closed surface M2

which admits A-endomorphisms whose nonwandering set contains a one-dimensional strictly

invariant contracting repeller Λr with a uniquely defined unstable bundle and with an admissible

boundary of finite type. First, we prove that, if M2 is a torus or a sphere, then M2 admits

such an endomorphism. We also show that, if Ω is a basic set with a uniquely defined unstable

bundle of the endomorphism f : M2 → M2 of a closed orientable surface M2 and f is not a

diffeomorphism, then Ω cannot be a Cantor type expanding attractor. At last, we prove that,

if f : M2 → M2 is an A-endomorphism whose nonwandering set consists of a finite number

of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of

Cantor type Ωr with a uniquely defined unstable bundle and such that the lamination consisting

of stable manifolds of Ωr is regular, then M2 is a two-dimensional torus T2 or a two-dimensional

sphere S2.

In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called a bone, and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i. e., "big." It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.

We investigated conformal foliations $(M,F)$ of codimension $q\geq 3$ and proved a criterion for them to be Riemannian. In particular, the application of this criterion allowed us to proof the existence of an attractor that is a minimal set for each non-Riemannian conformal foliation. Moreover, if foliated manifold is compact then non-Riemannian conformal foliation $(M,F)$ is $(Conf(S^q),S^q)$-foliation with finitely many minimal sets. They are all attractors, and each leaf of the foliation belongs to the basin of at least one of them. The specificity of the proper conformal foliations is indicated. Special attention is given to complete conformal foliations.

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.

Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations

are developed.

An attractor, in complex systems theory, is any state that is more easily or more often entered or acquired than departed or lost; attractor states therefore accumulate more members than non-attractors, other things being equal. In the context of language evolution, linguistic attractors include sounds, forms, and grammatical structures that are prone to be selected when sociolinguistics and language contact make it possible for speakers to choose between competing forms. The reasons why an element is an attractor are linguistic (auditory salience, ease of processing, paradigm structure, etc.), but the factors that make selection possible and propagate selected items through the speech community are non-linguistic. This paper uses the consonants in personal pronouns to show what makes for an attractor and how selection and diffusion work, then presents a survey of several language families and areas showing that the derivational morphology of pairs of verbs like fear and frighten, or Turkish korkmak 'fear, be afraid' and korkutmak 'frighten, scare', or Finnish istua 'sit' and istutta 'seat (someone)', or Spanish sentarse 'sit down' and sentar 'seat (someone)' is susceptible to selection. Specifically, the Turkish and Finnish pattern, where 'seat' is derived from 'sit' by addition of a suffix-is an attractor and a favored target of selection. This selection occurs chiefly in sociolinguistic contexts of what is defined here as linguistic symbiosis, where languages mingle in speech, which in turn is favored by certain demographic, sociocultural, and environmental factors here termed frontier conditions. Evidence is surveyed from northern Eurasia, the Caucasus, North and Central America, and the Pacific and from both modern and ancient languages to raise the hypothesis that frontier conditions and symbiosis favor causativization.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.