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Selection Theorem for Systems with Inheritance

A. N. Gorban (2010)

Mathematical Modelling of Natural Phenomena

The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of “natural”...

Semigroup actions on tori and stationary measures on projective spaces

Yves Guivarc'h, Roman Urban (2005)

Studia Mathematica

Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on ℝ d is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on ℝ d at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space ℙ d - 1 . In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits...

Sequence entropy pairs and complexity pairs for a measure

Wen Huang, Alejandro Maass, Xiangdong Ye (2004)

Annales de l’institut Fourier

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy n -tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...

Shadowing and expansivity in subspaces

Andrew D. Barwell, Chris Good, Piotr Oprocha (2012)

Fundamenta Mathematicae

We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.

Shadowing and internal chain transitivity

Jonathan Meddaugh, Brian E. Raines (2013)

Fundamenta Mathematicae

The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with...

Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces

Marian Mrozek (1994)

Fundamenta Mathematicae

We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.

Shape index in metric spaces

Francisco R. Ruiz del Portal, José M. Salazar (2003)

Fundamenta Mathematicae

We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of q-adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map in a locally...

Some cohomological aspects of the Banach fixed point principle

Ludvík Janoš (2011)

Mathematica Bohemica

Let T : X → X be a continuous selfmap of a compact metrizable space X . We prove the equivalence of the following two statements: (1) The mapping T is a Banach contraction relative to some compatible metric on X . (2) There is a countable point separating family ℱ ⊂ 𝒞 ( X ) of non-negative functions f ∈ 𝒞 ( X ) such that for every f ∈ ℱ there is g ∈ 𝒞 ( X ) with f = g - g ∘ T .

Some constructions of strictly ergodic non-regular Toeplitz flows

A. Iwanik, Y. Lacroix (1994)

Studia Mathematica

We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.

Some dynamical properties of S-unimodal maps

Tomasz Nowicki (1993)

Fundamenta Mathematicae

We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.

Some families of pseudo-processes

J. Kłapyta (1994)

Annales Polonici Mathematici

We introduce several types of notions of dis persive, completely unstable, Poisson unstable and Lagrange uns table pseudo-processes. We try to answer the question of how many (in the sense of Baire category) pseudo-processes with each of these properties can be defined on the space ℝ m . The connections are discussed between several types of pseudo-processes and their limit sets, prolongations and prolongational limit sets. We also present examples of applications of the above results to pseudo-processes...

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