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.

On the Hyperplane Sections Through two Given Points of an Algebraic Variety

1970 ◽  
Vol 22 (1) ◽  
pp. 128-133 ◽  
Author(s):  
Wei-Eihn Kuan
Keyword(s):  
Simple Proof ◽  
Hyperplane Section ◽  
Rational Points ◽  
Infinite Field ◽  
Important Special Case ◽  
Irreducible Curve ◽  
Irreducible Variety ◽  
Dimension 2 ◽  
Hyperplane Sections ◽  
Special Case

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

scholarly journals OCNEANU'S TRACE AND STARKEY'S RULE

2003 ◽  
Vol 12 (07) ◽  
pp. 899-904 ◽  
Author(s):  
MEINOLF GECK ◽  
NICOLAS JACON
Keyword(s):  
Simple Proof ◽  
Hecke Algebras ◽  
Jones Polynomial ◽  
Main Tool ◽  
Type A ◽  
Character Tables ◽  
Special Case ◽  
Knots And Links

We give a new simple proof for the weights of Ocneanu's trace on Iwahori–Hecke algebras of type A. This trace is used in the construction of the HOMFLYPT-polynomial of knots and links (which includes the famous Jones polynomial as a special case). Our main tool is Starkey's rule concerning the character tables of Iwahori–Hecke algebras of type A.

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.

A simple proof of the Mazur-Orlicz summability theorem

1981 ◽  
Vol 89 (3) ◽  
pp. 391-392 ◽  
Author(s):  
J. A. Fridy
Keyword(s):  
Functional Analysis ◽  
Simple Proof ◽  
Direct Proof ◽  
Topological Properties ◽  
Unbounded Sequence ◽  
Important Theorem ◽  
Slowly Oscillating Sequences

In 1933(1), Mazur and Orlicz stated that, if a conservative (i.e. convergence preserving) matrix sums a bounded nonconvergent sequence, then it must sum an unbounded sequence. Their proof of this result (2) was one of the early applications of functional analysis to summability theory, and it is based on rather deep topological properties of F K-spaces. In (3) Zeller obtained a proof of this important theorem as a consequence of his study of the summability of slowly oscillating sequences. The purpose of this note is to give a simple direct proof of this theorem using only the well-known Silverman-Töplitz conditions for regularity. In order to reduce the details of the argument, we state and prove the result for regular matrices rather than conservative matrices.

scholarly journals SKEW-PRODUCT DYNAMICAL SYSTEMS, ELLIS GROUPS AND TOPOLOGICAL CENTRE

2009 ◽  
Vol 79 (1) ◽  
pp. 129-145 ◽  
Author(s):  
A. JABBARI ◽  
H. R. E. VISHKI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Type Systems ◽  
General Construction ◽  
Skew Product ◽  
Topological Properties ◽  
Natural Extensions ◽  
Special Case ◽  
Topological Centre

AbstractIn this paper, a general construction of a skew-product dynamical system, for which the skew-product dynamical system studied by Hahn is a special case, is given. Then the ergodic and topological properties (of a special type) of our newly defined systems (called Milnes-type systems) are investigated. It is shown that the Milnes-type systems are actually natural extensions of dynamical systems corresponding to some special distal functions. Finally, the topological centre of Ellis groups of any skew-product dynamical system is calculated.

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.

Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products

2001 ◽  
Vol 21 (2) ◽  
pp. 563-603 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Hausdorff Dimension ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Skew Products ◽  
Wandering Domain ◽  
Upper Estimate ◽  
Interior Points

We consider dynamics of sub-hyperbolic and semi-hyperbolic semigroups of rational functions on the Riemann sphere and will show some no wandering domain theorems. The Julia set of a rational semigroup in general may have non-empty interior points. We give a sufficient condition that the Julia set has no interior points. From some information about forward and backward dynamics of the semigroup, we consider when the area of the Julia set is equal to zero or an upper estimate of the Hausdorff dimension of the Julia set.

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