Boundary Conditions for Maps in Fibers and Topological Transitivity of Skew Products of Interval Maps

2015 ◽  
Vol 208 (1) ◽  
pp. 109-114 ◽  
Author(s):  
L. S. Efremova ◽  
A. S. Fil’chenkov
Keyword(s):  
Boundary Conditions ◽  
Topological Transitivity ◽  
Interval Maps ◽  
Skew Products

scholarly journals Small C1-smooth perturbations of skew products and the partial integrability property

2020 ◽  
Vol 5 (2) ◽  
pp. 317-328
Author(s):  
L.S. Efremova
Keyword(s):  
Sufficient Conditions ◽  
Skew Product ◽  
Interval Maps ◽  
Skew Products ◽  
Partial Integrability

AbstractIn this paper we investigate stability of the integrability property of skew products of interval maps under small C1-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.

scholarly journals Non-smooth saddle-node bifurcations I: existence of an SNA

2014 ◽  
Vol 36 (4) ◽  
pp. 1130-1155 ◽  
Author(s):  
GABRIEL FUHRMANN
Keyword(s):  
Lyapunov Exponent ◽  
General Result ◽  
Sufficient Conditions ◽  
Positive Lyapunov Exponent ◽  
Hyperbolic Attractor ◽  
Interval Maps ◽  
Skew Products ◽  
Saddle Node ◽  
Invariant Graph

We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as ‘SNA’. In contrast to former results in this direction, our conditions are${\mathcal{C}}^{2}$-open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

scholarly journals Skew products of interval maps over subshifts

2016 ◽  
Vol 22 (7) ◽  
pp. 941-958 ◽  
Author(s):  
Masoumeh Gharaei ◽  
Ale Jan Homburg
Keyword(s):  
Interval Maps ◽  
Skew Products

Dynamics of skew products of interval maps

2017 ◽  
Vol 72 (1) ◽  
pp. 101-178 ◽  
Author(s):  
L S Efremova
Keyword(s):  
Interval Maps ◽  
Skew Products

Stable topological transitivity properties of ℝn-extensions of hyperbolic transformations

2011 ◽  
Vol 32 (4) ◽  
pp. 1435-1443 ◽  
Author(s):  
A. MOSS ◽  
C. P. WALKDEN
Keyword(s):  
Dynamical Systems ◽  
Skew Product ◽  
Dense Orbit ◽  
Anosov Diffeomorphism ◽  
Topological Transitivity ◽  
Axiom A ◽  
Skew Products ◽  
Hyperbolic Flow ◽  
Hyperbolic Dynamical Systems ◽  
Dense Set

AbstractWe consider ℝn skew-products of a class of hyperbolic dynamical systems. It was proved by Niţică and Pollicott [Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269] that for an Anosov diffeomorphism ϕ of an infranilmanifold Λ there is (subject to avoiding natural obstructions) an open and dense set f:Λ→ℝN for which the skew-product ϕf(x,v)=(ϕ(x),v+f(x)) on Λ×ℝN has a dense orbit. We prove a similar result in the context of an Axiom A hyperbolic flow on an attractor.

APERIODICITY OF COCYCLES AND CONDITIONAL LOCAL LIMIT THEOREMS

2004 ◽  
Vol 04 (01) ◽  
pp. 31-62 ◽  
Author(s):  
J. AARONSON ◽  
M. DENKER ◽  
O. SARIG ◽  
R. ZWEIMÜLLER
Keyword(s):  
Large Class ◽  
Limit Theorems ◽  
Local Limit ◽  
Interval Maps ◽  
Skew Products ◽  
Transfer Operators ◽  
Stochastic Properties ◽  
Local Limit Theorems

We establish conditions for aperiodicity of cocycles (in the sense of [12]), obtaining, via a study of perturbations of transfer operators, conditional local limit theorems and exactness of skew-products. Our results apply to a large class of Markov and non-Markov interval maps, including beta transformations. This allows us to establish various stochastic properties of beta expansions.

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