On some qualitative properties of monotone linear extensions of dynamical systems

We study monotone linear extensions of dynamical systems. The problem of the existence of invariant manifolds and exponential separation is investigated for linear extensions that preserve the order structure. We also study the relationship between the monotonicity of linear extensions and the existence (weak regularity, quasiregularity) of bounded solutions of inhomogeneous linear extensions.     

On chaotic extensions of dynamical systems

In this paper, inspired by some results in linear dynamics, we will show that every dynamical system (X,f), where f is a continuous self-map on a separable metric space X, can be extended to a chaotic (in the sense of Devaney) dynamical system in an isometric way.     

On extensions of Hoeffding’s inequality for panel data

Hoeffding’s inequality provides a probability bound for the deviation between the average of n independent bounded random variables and its mean. This paper introduces two inequalities that extend Hoeffding’s inequality to panel data, which consists of several mutually independent sequences of dependent data with strong mixing or with a dependence structure being even more general than strong mixing. One is denoted as the Bosq’s Extension which is an extension of Bosq’s inequality (Bosq, 1993) to panel data and the other one is called the Triplex Extension, which extends the Triplex inequality (Jiang, 2009) to panel data. The Bosq’s Extension provides a tighter upper probability bound, while the Triplex Extension is more relaxed in assumption allowing unboundedness and more general dependence than strong mixing. We also apply these two inequalities to establish the convergence rate of empirical risk minimization for high dimensional panel data with variable selection.     

On equicontinuous factors of linear extensions of minimal dynamical systems

The concept of the equicontinuous factor of the linear extension of a minimal transformation group is introduced and investigated. It is shown that a subset of motions, bounded and distal with respect to the extension, forms a maximal equicontinuous subsplitting of the linear extension. As a consequence, any distal linear extension has a nontrivial equicontinuous invariant subsplitting. The linear extensions without exponential dichotomy possess similar subsplittings if the Favard condition is satisfied. The same statement holds for linear extensions with the property of recurrent motions additivity provided that at least one nonzero motion of this sort exists.Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 45, No. 2, pp. 233–238, February, 1993.     

Symbolic extensions and smooth dynamical systems

Let f:XX be a homeomorphism of the compact metric space X. A symbolic extension of (f,X) is a subshift on a finite alphabet (g,Y) which has f as a topological factor. We show that a generic C¹ non-hyperbolic (i.e., non-Anosov) area preserving diffeomorphism of a compact surface has no symbolic extensions. For r>1, we exhibit a residual subset of an open set of C^r diffeomorphisms of a compact surface such that if , then any possible symbolic extension has topological entropy strictly larger than that of f. These results complement the known fact that any C diffeomorphism has symbolic extensions with the same entropy. We also show that C^r generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two.     

On local perturbations of linear extensions of dynamical systems on a torus

We investigate the problem of preservation of regularity of linear extensions of dynamic systems on a torus under perturbations.     

Eigenvalue extensions of Bohr’s inequality

We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohr’s inequality due to Vasi? and Ke?ki?.     

Operator extensions of Hua’s inequality

We give an extension of Hua’s inequality in pre-Hilbert C^∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C^∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is equivalent to operator convexity of given continuous real function.     

Geometrical extensions of Wythoff’s game

In 1905 Bouton gave the complete theory of a two-player combinatorial game: the game of Nim. Two years later, Wythoff defined his game as “a modification” of the game of Nim. In this paper, we give the sets of the losing positions of geometrical extensions of Wythoff’s game, where allowed moves are considered according to a set of vectors (v₁,…,v_n). When n=3, we present algorithms and algebraic characterizations to determine the losing positions of such games. In the last part, we investigate a bounded version of Wythoff’s game, and give a polynomial way to decide whether a game position is losing or not.     

On some classes of regular linear extensions of dynamical systems on a torus

We select some classes of linear extensions of dynamical systems on a torus for which weak regularity implies regularity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1479–1485, November, 1994.This work was supported by the Ukrainian State Committee on Science and Technology.     

On extensions of Nelson’s logic satisfying Dummett’s axiom

The class of extensions is completely described of the logic obtained by adding Dummett’s axiom to the paraconsistent Nelson logic. Moreover, we prove that every extension of this logic is finitely axiomatizable and decidable and, given a formula, it is possible to determine which extension is axiomatized by this formula.     

Double extensions of dynamical systems and constructing mixing filtrations

Let T: X→X be an automorphism (a measurable invertible measure-preserving transformation) of a probability space (X, F, μ) and let two μ-symmetric Markov generators A_u and A_s acting on the space L₂=L₂ (X, F, μ) be “eigenfunctions” of the automorphism T with eigenvaluesθ _u > 1 andθ _s < 1, respectively. We construct an extension of the automorphism T having increasing and decreasing filtrations by means of a transformation on the path space of these processes. Under additional conditions, we give an estimate of the maximal correlation coefficient between the δ-fields chosen from these filtrations. Hyperbolic toral automorphisms are considered as an example. Applications to limit theorems are given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 61–72. Translated by M. I. Gordin.     

Metrizable extensions of dynamical systems from function algebras

Peters and Pennings introduced and studied in [9] and [11] a kind of extensions of dynamical systems induced by certain C^*-algebras, which they calltame extensions. In this paper we study the behaviour of metrizable tame extensions when some of the most important dynamical concepts related to the metric (such as sources, sinks, saddles, the shadowing property and distallity) are considered. We will confirm that these extensions can be troublesome in this context. We show, however, that sources and sinks are preserved under certain conditions. Entrata in Redazione il 24 febbraio 1997. Research partially supported by Generalitat Valenciana, (2223/94).     

Compact abelian group extensions of discrete dynamical systems

Summary This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.     

On the smoothness of bounded invariant manifolds of linear inhomogeneous extensions of dynamical systems

We establish sufficient conditions for the existence of smooth bounded invariant manifolds of dynamical systems.     

An aggregation framework based on coherent lower previsions: Application to Zadeh’s paradox and sensor networks

The problem of aggregating two or more sources of information containing knowledge about a common domain is considered. We propose an aggregation framework for the case where the available information is modelled by coherent lower previsions, corresponding to convex sets of probability mass functions. The consistency between aggregated beliefs and sources of information is discussed. A closed formula, which specializes our rule to a particular class of models, is also derived. Two applications consisting in a possible explanation of Zadeh’s paradox and an algorithm for estimation fusion in sensor networks are finally reported.