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On foundations of the Conley index theory

Roman Srzednicki (1999)

Banach Center Publications

The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical...

On gaps in Rényi β -expansions of unity for β > 1 an algebraic number

Jean-Louis Verger-Gaugry (2006)

Annales de l’institut Fourier

Let β > 1 be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi β -expansion   d β ( 1 ) of unity which controls the set ℤ β of β -integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in d β ( 1 ) are shown to exhibit a “gappiness” asymptotically bounded above by   log ( M ( β ) ) / log ( β ) , where   M ( β )   is the Mahler measure of   β . The proof of this result provides in a natural way a new classification of algebraic numbers > 1 with classes called Q...

On gradient-like random dynamical systems

Aya Hmissi, Farida Hmissi, Mohamed Hmissi (2012)

ESAIM: Proceedings

This paper deals with some characterizations of gradient-like continuous random dynamical systems (RDS). More precisely, we establish an equivalence with the existence of random continuous section or with the existence of continuous and strict Liapunov function. However and contrary to the deterministic case, parallelizable RDS appear as a particular case of gradient-like RDS.The obtained results are generalizations of well-known analogous theorems in the framework of deterministic dynamical systems....

On heredity of strongly proximal actions

C. Robinson Edward Raja (2003)

Archivum Mathematicum

We prove that action of a semigroup T on compact metric space X by continuous selfmaps is strongly proximal if and only if T action on 𝒫 ( X ) is strongly proximal. As a consequence we prove that affine actions on certain compact convex subsets of finite-dimensional vector spaces are strongly proximal if and only if the action is proximal.

On local aspects of topological weak mixing in dimension one and beyond

Piotr Oprocha, Guohua Zhang (2011)

Studia Mathematica

We introduce the concept of weakly mixing sets of order n and show that, in contrast to weak mixing of maps, a weakly mixing set of order n does not have to be weakly mixing of order n + 1. Strictly speaking, we construct a minimal invertible dynamical system which contains a non-trivial weakly mixing set of order 2, whereas it does not contain any non-trivial weakly mixing set of order 3. In dimension one this difference is not that much visible, since we prove that every continuous...

On Lyapunov stability in hypoplasticity

Kovtunenko, Victor A., Krejčí, Pavel, Bauer, Erich, Siváková, Lenka, Zubkova, Anna V. (2017)

Proceedings of Equadiff 14

We investigate the Lyapunov stability implying asymptotic behavior of a nonlinear ODE system describing stress paths for a particular hypoplastic constitutive model of the Kolymbas type under proportional, arbitrarily large monotonic coaxial deformations. The attractive stress path is found analytically, and the asymptotic convergence to the attractor depending on the direction of proportional strain paths and material parameters of the model is proved rigorously with the help of a Lyapunov function....

On maximizing measures of homeomorphisms on compact manifolds

Fábio Armando Tal, Salvador Addas-Zanata (2008)

Fundamenta Mathematicae

We prove that given a compact n-dimensional connected Riemannian manifold X and a continuous function g: X → ℝ, there exists a dense subset of the space of homeomorphisms of X such that for all T in this subset, the integral ∫ X g d μ , considered as a function on the space of all T-invariant Borel probability measures μ, attains its maximum on a measure supported on a periodic orbit.

On multiplicatively dependent linear numeration systems, and periodic points

Christiane Frougny (2002)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

On multiplicatively dependent linear numeration systems, and periodic points

Christiane Frougny (2010)

RAIRO - Theoretical Informatics and Applications

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

On Pawlak's problem concerning entropy of almost continuous functions

Tomasz Natkaniec, Piotr Szuca (2010)

Colloquium Mathematicae

We prove that if f: → is Darboux and has a point of prime period different from 2 i , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

On perturbation of continuous maps

Maria Carbinatto (1999)

Banach Center Publications

In [1], the concept of singular isolating neighborhoods for a continuous family of continuous maps was presented. The work was based on Conley's result for a continuous family of continuous flows (cf. [2]). In this note, we study a particular family of continuous maps to illustrate the results in [1].

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