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.

REMARKS ON SKEW PRODUCTS OVER IRRATIONAL ROTATIONS

2005 ◽  
Vol 15 (11) ◽  
pp. 3675-3689 ◽  
Author(s):  
L. M. LERMAN
Keyword(s):  
Sufficient Conditions ◽  
Irrational Rotation ◽  
Invariant Sets ◽  
Asymptotically Stable ◽  
Skew Product ◽  
Almost Periodic ◽  
Invariant Curves ◽  
Skew Products ◽  
Irrational Rotation Number ◽  
Periodic Equation

We prove several results of the orbit behavior of skew product diffeomorphisms generated by quasi-periodic differential systems. The first diffeomorphism is derived from a periodic differential equation on the circle by means of a construction proposed by Z. Opial to get a scalar quasi-periodic equation with all its solutions bounded but without an almost periodic solution. We consider both possible cases for the irrational rotation number, transitive and singular (intransitive). The main result for a transitive case is that the related skew product diffeomorphism has a foliation into invariant curves with pure irrational rotation on each curve (being the same for each curve). For intransitive case, we get invariant sets of two types: a collection of continuous invariant curves and invariant sets being dimensionally inhomogeneous ones.Section 3 is devoted to perturbations of a skew product diffeomorphism over an irrational rotation being initially foliated into invariant curves. We prove an analog of Poincaré–Pontryagin theorem which sets conditions when a perturbation of a one-degree-of-freedom Hamiltonian system (given in an annulus and written down in action-angle variables) has limit cycles. Our theorem provides sufficient conditions when a perturbation of a foliated skew product diffeomorphism has isolated invariant curves (asymptotically stable or unstable).In Sec. 4 we present some results on the geometry of skew product diffeomorphisms derived by a quasi-periodic Riccati equation.

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 Smale endomorphisms over graph-directed Markov systems

2020 ◽  
pp. 1-34
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Equilibrium Measure ◽  
Parabolic Systems ◽  
Skew Product ◽  
Pointwise Dimension ◽  
Markov Systems ◽  
Large Classes ◽  
Equilibrium Measures ◽  
Skew Products ◽  
Natural Extensions ◽  
Induced Maps

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

scholarly journals Equilibrium states for a class of skew products

2019 ◽  
Vol 40 (11) ◽  
pp. 3030-3050
Author(s):  
MARIA CARVALHO ◽  
SEBASTIÁN A. PÉREZ
Keyword(s):  
Existence And Uniqueness ◽  
Global Analysis ◽  
American Mathematical Society ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Axiom A ◽  
Skew Products ◽  
Pure Mathematics ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We consider skew products on $M\times \mathbb{T}^{2}$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\unicode[STIX]{x1D6FA}$-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in$T^{4}$ (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.

scholarly journals Return- and hitting-time distributions of small sets in infinite measure preserving systems

2019 ◽  
Vol 40 (8) ◽  
pp. 2239-2273
Author(s):  
SIMON RECHBERGER ◽  
ROLAND ZWEIMÜLLER
Keyword(s):  
Sufficient Conditions ◽  
Finite Measure ◽  
Hitting Time ◽  
Independent Random Variables ◽  
Limit Laws ◽  
Interval Maps ◽  
Infinite Measure ◽  
Sufficient Conditions For Convergence ◽  
Set Up ◽  
The One

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\unicode[STIX]{x1D707}(E_{k})\rightarrow 0$ in recurrent ergodic dynamical systems preserving an infinite measure $\unicode[STIX]{x1D707}$. Some properties which are easy in finite measure situations break down in this null-recurrent set-up. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D6FC}\in (0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge if and only if the corresponding hitting-time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to ${\mathcal{E}}^{1/\unicode[STIX]{x1D6FC}}{\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$, where ${\mathcal{E}}$ and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ are independent random variables, with ${\mathcal{E}}$ exponentially distributed and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ following the one-sided stable law of order $\unicode[STIX]{x1D6FC}$ (and ${\mathcal{G}}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperbolic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\unicode[STIX]{x1D6FC}=0$ case.

scholarly journals Translation Invariant Spaces and Asymptotic Properties of Variational Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-36 ◽  
Author(s):  
Adina Luminiţa Sasu ◽  
Bogdan Sasu
Keyword(s):  
Control Systems ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Invariant Function ◽  
Function Spaces ◽  
Skew Product ◽  
Translation Invariant ◽  
Variational Equations ◽  
Skew Product Flows ◽  
New Perspective

We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew-product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew-product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

scholarly journals A counterexample to a positive entropy skew product generalization of the Pinsker conjecture

1985 ◽  
Vol 5 (3) ◽  
pp. 379-407
Author(s):  
Jonathan King
Keyword(s):  
Positive Entropy ◽  
Skew Product ◽  
Skew Products ◽  
Fibre Transformation

AbstractThe class of k-automorphisms is not contained in a certain class of skew products over a Bernoulli base. The non-identity fibre transformation in the skew is allowed to have positive or even infinite entropy. A difficulty presented by positive entropy is handled via an apparently new property of independent processes (lemma 7.24).

On Weak Lyapunov Exponent and Sensitive Dependence of Interval Maps

2014 ◽  
Vol 24 (10) ◽  
pp. 1450120
Author(s):  
Xinxing Wu ◽  
Guanrong Chen ◽  
Peiyong Zhu
Keyword(s):  
Lyapunov Exponent ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Interval Maps ◽  
System A ◽  
Sensitive Dependence

By using weak Lyapunov exponent, some sufficient conditions ensuring a system ([a, b], f) to be sensitively dependent on the initial value x0 ∈ [a, b] are obtained.

Topologically transitive skew-products of operators

2009 ◽  
Vol 30 (1) ◽  
pp. 33-49 ◽  
Author(s):  
FRÉDÉRIC BAYART ◽  
GEORGE COSTAKIS ◽  
DEMETRIS HADJILOUCAS
Keyword(s):  
Linear Operators ◽  
Skew Product ◽  
Weighted Shifts ◽  
Linear Dynamics ◽  
Topologically Transitive ◽  
Skew Products ◽  
Product Systems ◽  
Hypercyclic Vectors ◽  
Backward Shift

AbstractThe purpose of the present paper is to provide a link between skew-product systems and linear dynamics. In particular, we give a criterion for skew-products of linear operators to be topologically transitive. This is then applied to certain families of linear operators including scalar multiples of the backward shift, backward unilateral weighted shifts, composition, translation and differentiation operators. We also prove the existence of common hypercyclic vectors for certain families of skew-product systems.

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