Explore the vast range of titles available.

Let f : X → X be a hereditarily locally connected continuum homeomorphism and denote respectively by P ( f ), AP ( f ) and Ω ( f ) , the sets of periodic points, almost periodic points and the non-wandering points of f . We show that any ω -limit set (resp. α -limit set) is minimal. Moreover, we show that Ω ( f ) = A P ( f ) . We also prove that if P ( f ) = ∅ , then there exists a unique minimal set. On the other hand, if P ( f ) ≠ ∅ then we prove that any infinite minimal set has the adding machine structure and the absence of Li-Yorke pairs. Consequently, we partially solve the positive entropy conjecture which remains open even in the case of hereditarily locally connected continuum.

This article aims to study parallelizable random dynamical systems by examining them through the terms of dissipation and stochastic Lyapunov functions. It is demonstrated that any random variable that is not a random fixed point admits a tube, and every non-wandering point is within one. The Lyapunov function is employed to characterize the asymptotic stability of compact and closed random sets. The section of a random dynamical system is used to define the parallelizable random dynamical system, and it is proven that a random dynamical system is parallelizable if and only if it admits a section. Furthermore, the principle of Lyapunov used this characterization to study the parallelizability of random dynamical systems. The concept of symmetry is defined, and then its impact on the behavior of stochastic dynamic systems, particularly the Lorenz system, is discussed. In addition, by using an appropriate stochastic Lyapunov function, we have shown that the random Lorenz system is parallelizable.

The problem of constructing reachable and null-controllable sets for stationary linear discrete-time systems with a summary constraint on the scalar control is considered. For the case of quadratic constraints and a diagonalizable matrix of the system, these sets are built explicitly in the form of ellipsoids. In the general case, the limit reachable and null-controllable sets are represented as fixed points of a contraction mapping in the metric space of compact sets. On the basis of the method of simple iteration, a convergent procedure for constructing their external estimates with an indication of the a priori approximation error is proposed. Examples are given.

In this paper, we consider the natural action of SL ( 3 , H ) on the quaternionic projective space P H 2 . Under this action, we investigate limit sets for cyclic subgroups of SL ( 3 , H ) . We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc’h, respectively.

Let be a dendrite, : → be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that -limit set ) of any point ∈ has the next properties: In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that The aim of this note is to show that the above results – do not hold for monotone maps on dendroids.

The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.

We consider semilinear parabolic equations of the form

For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

In this paper some qualitative and geometric aspects of nonsmooth vector fields theory are discussed. A Poincaré–Bendixson Theorem for a class of nonsmooth systems is presented. In addition, a minimal set in planar Filippov systems not predicted in classical Poincaré–Bendixson theory and whose interior is non-empty is exhibited. The concepts of limit sets, recurrence, and minimal sets for nonsmooth systems are defined and compared with the classical ones. Moreover some differences between them are pointed out.