Coefficients Averaging for Functional Operators Generated by Irrational Rotation

2007 ◽  
pp. 27-41
Author(s):  
A. B. Antonevich
Keyword(s):  
Irrational Rotation

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,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" 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>

Skew products over the irrational rotation

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

A Necessary Condition of Nonminimal Aubry–Mather Sets for Monotone Recurrence Relations

2014 ◽  
Vol 24 (01) ◽  
pp. 1450012 ◽  
Author(s):  
Ya-Nan Wang ◽  
Wen-Xin Qin
Keyword(s):  
Rotation Number ◽  
Recurrence Relations ◽  
Irrational Rotation ◽  
Necessary Condition ◽  
Irrational Rotation Number

In this paper, we show that a necessary condition for nonminimal Aubry–Mather sets of monotone recurrence relations is that the set of all Birkhoff minimizers with some irrational rotation number does not constitute a foliation, i.e. the gaps of the minimal Aubry–Mather set are not filled up with Birkhoff minimizers.

scholarly journals On the structure of the family of Cherry fields on the torus

1985 ◽  
Vol 5 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Colin Boyd
Keyword(s):  
Vector Field ◽  
Hausdorff Dimension ◽  
Rotation Number ◽  
Vector Fields ◽  
Cantor Set ◽  
Irrational Rotation ◽  
Codimension One ◽  
The Family ◽  
Irrational Rotation Number

AbstractA class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure ofhas irrational rotation number}, where ƒ is a certain map of the circle andRtis rotation throught. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.

Inductive Limit Toral Automorphisms of Irrational Rotation Algebras

2001 ◽  
Vol 44 (3) ◽  
pp. 335-336
Author(s):  
P. J. Stacey
Keyword(s):  
Inductive Limit ◽  
Continuous Functions ◽  
Irrational Rotation ◽  
Space Of Continuous Functions ◽  
Toral Automorphism ◽  
Matrix Algebras ◽  
Toral Automorphisms

AbstractIrrational rotation C*-algebras have an inductive limit decomposition in terms of matrix algebras over the space of continuous functions on the circle and this decomposition can be chosen to be invariant under the flip automorphism. It is shown that the flip is essentially the only toral automorphism with this property.

A topological invariant for volume preserving diffeomorphisms

1995 ◽  
Vol 15 (3) ◽  
pp. 535-541 ◽  
Author(s):  
Jean-Marc Gambaudo ◽  
Elisabeth Pécou
Keyword(s):  
Simple Proof ◽  
Normal Bundle ◽  
Topological Invariant ◽  
Irrational Rotation ◽  
Volume Preserving ◽  
The Way

AbstractFor a smooth diffeomorphism f in ℝn+2, which possesses an invariant n-torus , such that the restriction f is topologically conjugate to an irrational rotation, we define a number which represents the way the normal bundle to the torus asymptotically wraps around . We prove that this number is a topological invariant among volume-preserving maps. This result can be seen as a generalization of a theorem by Naishul, for which we give a simple proof.

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

scholarly journals Sets of bounded discrepancy for multi-dimensional irrational rotation

2015 ◽  
Vol 25 (1) ◽  
pp. 87-133 ◽  
Author(s):  
Sigrid Grepstad ◽  
Nir Lev
Keyword(s):  
Irrational Rotation

scholarly journals The automorphism group of the irrational rotation C*-algebra

1993 ◽  
Vol 155 (1) ◽  
pp. 3-26 ◽  
Author(s):  
George A. Elliott ◽  
Mikael Rørdam
Keyword(s):  
Automorphism Group ◽  
Irrational Rotation ◽  
C Algebra

scholarly journals Resonance between Cantor sets

2009 ◽  
Vol 29 (1) ◽  
pp. 201-221 ◽  
Author(s):  
YUVAL PERES ◽  
PABLO SHMERKIN
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Unit Interval ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Scaling Parameter ◽  
Iterated Function ◽  
Self Similar ◽  
Image Position ◽  
Geometric Resonance

AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

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