In the current note we extend results by Marmi, Moussa and Yoccoz about cohomological equations for interval exchange transformations to irreducible linear involutions.
Consider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold Ws(O), one-dimensional unstable manifold Wu(O), and with the saddle value σ different from 1. We assume that Ws(O) and Wu(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when σ > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.
We study topological properties of automorphisms of 4-dimensional torus generatedby integer matrices being symplectic either with respect to the standard symplecticstructure in R4 or w.r.t. a nonstandard symplectic structure generated by an integer skew-symmetric nondegenerate matrix. Such symplectic matrix generates a partially hyperbolic automorphism of the torus, if its eigenvalues are a pair of reals outsidethe unit circle and a complex conjugate pair on the unit circle. The main classifying element is the topology of a foliation generated by unstable (stable) leaves ofthe automorphism. There are two different cases, transitive and decomposable ones.For the first case the foliation into unstable (stable) leaves is transitive, for the second case the foliation itself has a sub-foliation into 2-dimensional tori. For both cases the classification is given.
We introduce Smale–Vietoris diffeomorphisms that include the classical DE mappings with Smale solenoids. We describe the correspondence between basic sets of axiom A Smale–Vietoris diffeomorphisms and basic sets of non-singular axiom A-endomorphisms. For Smale–Vietoris diffeomorphisms of 3-manifolds, we prove the uniqueness of non-trivial solenoidal basic set. We construct a bifurcation between different types of solenoidal basic sets which can be considered as a destruction (or birth) of Smale solenoid.
The modern qualitative theory of dynamical systems is thoroughly intertwined with the fairly young science of topology. Many strange constructions of topology are found sooner or later in dynamics of discrete or continuous dynamical systems. In the present paper we show that the wild Fox-Artin arc emerges naturally in invariant sets of dynamical systems.
Numerous magnetic fragments in the interior of the Sun give rise to many interesting energy processes in the solar corona, for example solar flares and prominences. Magnetic charge topology explains these phenomena by appearance and disappearance of heteroclinic trajectories (separators) --- magnetic lines that belong to an intersection of stable and unstable invariant two-dimensional manifolds (fans) of different saddle singularities (nulls). Separators are the locations in the magnetic field configuration where the magnetic energy is transferred from one region (a connected component into which the fans divide the solar corona) to another. Many recent papers has gone into investigation of the configurations that arise in different concrete models. In the present paper we solve the problem of interrelation between existence of separators of any given magnetic field in the Solar corona and the type and the number of saddle singularities and charges. Following the classical definition we introduce the concept of equivalence of two magnetic fields and get a classification of such fields up to topological equivalence.