Skew products over the irrational rotation

1990 ◽  
Vol 69 (1) ◽  
pp. 65-74 ◽  
Author(s):  
D. A. Pask
Keyword(s):  
Irrational Rotation ◽  
Skew Products

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 -recurrence in skew products

2013 ◽  
Vol 34 (5) ◽  
pp. 1525-1537 ◽  
Author(s):  
JON CHAIKA ◽  
DAVID RALSTON
Keyword(s):  
Growth Rate ◽  
Lyapunov Exponents ◽  
Bounded Variation ◽  
Probability Space ◽  
Irrational Rotation ◽  
Infinite Measure ◽  
Skew Products

AbstractThe rate of recurrence to measurable subsets in a conservative, ergodic infinite-measure-preserving system is quantified by generic divergence or convergence of certain sums given by a function $\omega (n)$. In the context of skew products over transformations of a probability space, we relate this notion to the more frequently studied question of the growth rate of ergodic sums (including Lyapunov exponents). We study in particular skew products over an irrational rotation given by bounded variation $ \mathbb{Z} $-valued functions: first the generic situation is studied and recurrence quantified, and then certain specific skew products over rotations are shown to violate this generic rate of recurrence.

scholarly journals On torus homeomorphisms semiconjugate to irrational rotations

2014 ◽  
Vol 35 (7) ◽  
pp. 2114-2137 ◽  
Author(s):  
T. JÄGER ◽  
A. PASSEGGI
Keyword(s):  
Simple Proof ◽  
Irrational Rotation ◽  
Rotation Vector ◽  
Topological Properties ◽  
Empty Interior ◽  
Irrational Rotations ◽  
Skew Products ◽  
Invariant Foliation ◽  
Special Case

In the context of the Franks–Misiurewicz conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterize these maps by the existence of an invariant ‘foliation’ by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalizing a well-known result of Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

Topological entropy of piecewise bimonotone skew products

2009 ◽  
Vol 15 (1) ◽  
pp. 53-69
Author(s):  
Franz Hofbauer ◽  
Peter Maličký ◽  
L'ubomír Snoha
Keyword(s):  
Topological Entropy ◽  
Skew Products

scholarly journals Equicontinuous local dendrite maps

2021 ◽  
Vol 22 (1) ◽  
pp. 67
Author(s):  
Aymen Haj Salem ◽  
Hawete Hattab ◽  
Tarek Rejeiba
Keyword(s):  
Irrational Rotation

<p>Let X be a local dendrite, and f : X → X be a map. Denote by E(X) the set of endpoints of X. We show that if E(X) is countable, then the following are equivalent:</p><p>(1) f is equicontinuous;</p><p>(2) <img src="data:image/png;base64,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" alt="" /> f<sup>n</sup> (X) = R(f);</p><p>(3) f| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAA/CAYAAACW2FpzAAAE5ElEQVRoBe2af0zUZRzH4Y671FhMMtIRGhE4rc2to1JnKbOGLn8ABRTJmL8yXdPYUKkQRFpoiKekoiOSmZqbmhWIDih3MozgD8RIcKQSU0QXP3bI5Pb93l4N5eC48+783n2XWt/bvvs+v+55Xs/783zuefZ5zotH5OP1iHCigMptqf+jon1cPr2PnPRPSN+6H8OV21aiinTU/0B+djrpWwqp+NO6zqqZk6RMihqp+jKJxRsKOPbTUXavnUuIfyhv59XSLVyl5LN44tcXcab5Jp1tdRz5IoUdlUYnWPZVsoDuz9zGedNQ5+a2UpJ1fozRzSIyqZCmvqE6EGne/TlFbWbrQpdpWUDzco/TazOUcGEz00dqePbdb7kkDK/sLd7GV+dtCoc3scvJAlqZk85Ra4WMtexIiCa1IIeoCX5MfL+Q33sGxu5tIH91FgaJy1QWULp+o+DTZNalZ5GxdgWJiz5mb00X/cYVWkrYGDOFZ4LDmbNwAXPnLyWvustOMVcF8oAOjCIab9DebbVYrUYXe9ppvW5EtCqTkpQVVMrAUtsqoFIVc9VeUdSVQlLr5VXUdJWqA5tJfm82L4UFEeA3Au2o0YwLnsy0+cvYsKuEP7ql7UiWCckE2suF71KIDPFF5e2Ft9qXwBemEjFnAVHz3uA1XShPar3x8vJGExBOYq6Bdom8noOKLRxf/TL+am80T+lIyDxMTZv9tiN2NVORv2ZgMo8RvHA7NRLOJZ6Bmjv4ZZ0OX5WasRFplLbex/5trKcw6UV8VT4ERhXQdO/9wWLxwbdHoF2lH/Ccj4rRr2dRLUEdxFaOLA5Dq3qCmbmN97VbuQ8q1JGhG4HKLwJ9oxsbY3cZq0I1qALiOHTD9YJ1G9RUmUyYj5qg5SftjniD9nKaMNOSF8Eo78eJ3NN25wDjrLmboCJN2VPRqANZesL2JOpsuOF15hY9s7Qq/BOOuZysm6B9lH84HrVmBjmX3DC7hbfvJMsD1Whn6mlxYX03QW9TsmQcKu1sdlofmC0A9/s2GVgd4oNmajYXXczXY9Bd111I4QzaZGDN8wpov0RDplcUtVkyyhq1EeTeWcWZBnVRnGlQCtuE4ky2itwzrzjToCyKMw1KYZv4rztTHxWrJqAe+RaFHbZzl5A3VbN+sg/aWXr+cnFadFNREK/9yo8/N3FLApd9UzPttcWcargb9LWvHypxG3Soi38npYDKrfNDp6gjn3rAoGaulW0nLS2Nxa+OYVLKWRyFoh4wqGWB9McJgpksC6j5b6oP6tHvO8NVoZfm8iJ2bt2M/nAdnY7sZeFw+ZYR1Hgqi43FF8mPfJrwuFR2GtowiVfIjQjjo9OWy06Bc3uW805MNNHRjp+YxC0YLDd5dyYhG6jI5YoyGm5VkTIpkLhDA0EtoZ4MXQgryy2gLqVz0EA20Lv9i81bmDF+CcUDcTGxMZvpQYl8Pyw2KiIKAoKTR7QL38gKaubG1/MYG1XE3e1doGHTKwQvO4Hx2gGy9zT233xybu8K4uNiiY11/MQl5TgwfZUcXt9H+cqJzN5+eSBCLHJB/ybTFmWycf0OTne451HdZ79hU8Y6Yqf4ExCeQGrGJg7W2f9ISfp5Enu66RnGI2C82Yn91YKDpehBsSRQD8bx+KsKqMcS2nSgKGojiMfZh1jR4X/4+geRrDru1x/giQAAAABJRU5ErkJggg==" alt="" /> f<sup>n</sup> (X) is equicontinuous;</p><p>(4) f| <img src="data:image/png;base64,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" alt="" />f<sup>n</sup> (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ;</p><p>(5) ω(x, f) = Ω(x, f) for all x ∈ X.</p><p>This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].</p>

Sign in / Sign up

Export Citation Format

Share Document