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C¹-Stably Positively Expansive Maps

Kazuhiro Sakai (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

The notion of C¹-stably positively expansive differentiable maps on closed C manifolds is introduced, and it is proved that a differentiable map f is C¹-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the ε-time dependent stability is shown. As a result, every expanding map is ε-time dependent stable.

Convergence Rates of the POD–Greedy Method

Bernard Haasdonk (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy...

Diffeomorphisms with weak shadowing

Kazuhiro Sakai (2001)

Fundamenta Mathematicae

The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a C closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.

Exponential limit shadowing

S. A. Ahmadi, M. R. Molaei (2013)

Annales Polonici Mathematici

We introduce the notion of exponential limit shadowing and show that it is a persistent property near a hyperbolic set of a dynamical system. We show that Ω-stability implies the exponential limit shadowing property.

On almost specification and average shadowing properties

Marcin Kulczycki, Dominik Kwietniak, Piotr Oprocha (2014)

Fundamenta Mathematicae

We study relations between the almost specification property, the asymptotic average shadowing property and the average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples showing that compactness is a necessary condition for these implications to hold. As a consequence, we also obtain a proof that limit shadowing in...

On the relationship between hyperbolic and cone-hyperbolic structures in metric spaces

Marcin Mazur (2013)

Annales Polonici Mathematici

We give necessary and sufficient conditions for topological hyperbolicity of a homeomorphism of a metric space, restricted to a given compact invariant set. These conditions are related to the existence of an appropriate finite covering of this set and a corresponding cone-hyperbolic graph-directed iterated function system.

On the ω-limit sets of tent maps

Andrew D. Barwell, Gareth Davies, Chris Good (2012)

Fundamenta Mathematicae

For a continuous map f on a compact metric space (X,d), a set D ⊂ X is internally chain transitive if for every x,y ∈ D and every δ > 0 there is a sequence of points ⟨x = x₀,x₁,...,xₙ = y⟩ such that d ( f ( x i ) , x i + 1 ) < δ for 0 ≤ i< n. In this paper, we prove that for tent maps with periodic critical point, every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed,...

Partial hyperbolicity and homoclinic tangencies

Sylvain Crovisier, Martin Sambarino, Dawei Yang (2015)

Journal of the European Mathematical Society

We show that any diffeomorphism of a compact manifold can be C 1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

Sylvain Crovisier (2006)

Publications Mathématiques de l'IHÉS

We prove that the chain-transitive sets of C1-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology...

Properties of dynamical systems with the asymptotic average shadowing property

Marcin Kulczycki, Piotr Oprocha (2011)

Fundamenta Mathematicae

This article investigates under what conditions nontransitivity can coexist with the asymptotic average shadowing property. We show that there is a large class of maps satisfying both conditions simultaneously and that it is possible to find such examples even among maps on a compact interval. We also study the limit shadowing property and its relation to the asymptotic average shadowing property.

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...

Shadowing in actions of some Abelian groups

Sergei Yu. Pilyugin, Sergei B. Tikhomirov (2003)

Fundamenta Mathematicae

We study shadowing properties of continuous actions of the groups p and p × p . Necessary and sufficient conditions are given under which a linear action of p on m has a Lipschitz shadowing property.

Shadowing in multi-dimensional shift spaces

Piotr Oprocha (2008)

Colloquium Mathematicae

We show that the class of expansive d actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction ν d . Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.

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