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 Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds

2020 ◽  
Author(s):  
Leandro Lichtenfelz ◽  
Gerard Misiołek ◽  
Stephen C Preston
Keyword(s):  
Euler Equations ◽  
Riemannian Geometry ◽  
Exponential Map ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Ideal Fluids ◽  
Volume Preserving ◽  
The Way

Abstract We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples, we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston’s eight model geometries.

Transverse regularity for maps of homology manifolds

1973 ◽  
Vol 74 (1) ◽  
pp. 29-38 ◽  
Author(s):  
N. Martin
Keyword(s):  
General Position ◽  
Normal Bundle ◽  
Homology Manifold ◽  
Homology Manifolds ◽  
The Way

Recall that in (2) we showed that it was possible to make transverse a homology manifold and PL-manifold inside a large dimensional homology manifold, subject to being able to do some general position inside the large manifold. In (1) we were able to relax the condition that one of the submanifolds be a PL-manifold to it being a homotopy manifold. The way in which the making transverse was achieved was via a system of h-cobordisms from the original situation to the transverse one. The problem we tackle here is that of making a map between homology manifolds transverse regular. Thus we ask: given a map f: M → N of homology manifolds with P a proper submanifold of N, is it possible to homotop f to a map g: M → N such that g−1(P) is a proper submanifold of M and g induces a map from the normal bundle of g−1(P) in M to the normal bundle of P in N?

STATISTICS OF PRIME DIVISORS IN FUNCTION FIELDS

2009 ◽  
Vol 05 (01) ◽  
pp. 141-152 ◽  
Author(s):  
ROBERT C. RHOADES
Keyword(s):  
Simple Proof ◽  
Function Field ◽  
Function Fields ◽  
Random Polynomial ◽  
Prime Divisors ◽  
Number Of Prime Divisors ◽  
Field Setting ◽  
The Way

We show that the prime divisors of a random polynomial in 𝔽q[t] are typically "Poisson distributed". This result is analogous to the result in ℤ of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple proof of the Erdös–Kac theorem in the function field setting. This approach gives stronger results about the moments of the sequence {ω(f)}f∈𝔽q[t] than was previously known, where ω(f) is the number of prime divisors of f.

scholarly journals TANGENT DIRAC STRUCTURES AND SUBMANIFOLDS

2005 ◽  
Vol 02 (05) ◽  
pp. 759-775 ◽  
Author(s):  
IZU VAISMAN
Keyword(s):  
Poisson Structure ◽  
Normal Bundle ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Isotropic Submanifold ◽  
Coisotropic Submanifold ◽  
Definition Of ◽  
Simple Definition ◽  
The Way

We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas once again prove a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure if and only if N is a Dirac submanifold. In the presymplectic case it is the isotropy of the normal bundle which characterizes the corresponding notion of a Dirac submanifold. On the way, we give a simple definition of the tangent Dirac structure, make new remarks about it and establish its characteristic, local formulas for various interesting classes of submanifolds of a Dirac manifold.

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.

scholarly journals Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings

2020 ◽  
Vol 14 (1) ◽  
pp. 150-168
Author(s):  
Miodrag Mateljevic ◽  
Marek Svetlik
Keyword(s):  
Simple Proof ◽  
Hyperbolic Geometry ◽  
Harmonic Functions ◽  
Hyperbolic Metric ◽  
Schwarz Lemma ◽  
Optimal Estimates ◽  
The Schwarz Lemma ◽  
The Way

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasiregular, shortly HQR) mappings with the strip codomain. Along the way, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (without the assumption that 0 is mapped to 0 by the corresponding map). Using the Schwarz-Pick lemma related to distortion for harmonic functions and the elementary properties of the hyperbolic geometry of the strip we get optimal estimates for modulus of HQR mappings.

A simple proof of Denjoy's theorem

1988 ◽  
Vol 103 (2) ◽  
pp. 299-303 ◽  
Author(s):  
R. S. MacKay
Keyword(s):  
Rotation Number ◽  
Simple Proof ◽  
Irrational Rotation ◽  
Topological Conjugacy ◽  
Uniform Rotation ◽  
Circle Maps ◽  
Irrational Rotation Number

AbstractA simple proof of Denjoy's theorem on topological conjugacy of circle maps with irrational rotation number to uniform rotation is presented.

The Lindelöf Tychonoff Theorem and choice principles

1991 ◽  
Vol 110 (1) ◽  
pp. 57-65 ◽  
Author(s):  
G. Schlitt
Keyword(s):  
Simple Proof ◽  
Frame Theory ◽  
Logical Strength ◽  
Axiom Of Countable Choice ◽  
Tychonoff Theorem ◽  
The Way

AbstractIt is an important result in frame theory that the coproduct of a family of regular Lindelöf frames is Lindelöf [3]. We show that this ‘Lindelöf Tychonoff Theorem’ or ‘LTT’ is independent of ZF and indeed lies close in logical strength to the Axiom of Countable Choice, quite unlike the case with the usual (frame) Tychonoff Theorem. Along the way we construct the regular Lindelöf coreflection and obtain a simple proof of the LTT as a corollary.

Self-sacrifice for in-group's history: A diachronic perspective

2018 ◽  
Vol 41 ◽  
Author(s):  
Maria Babińska ◽  
Michal Bilewicz
Keyword(s):  
Clinical Psychology ◽  
Historical Events ◽  
Emotional Events ◽  
Reciprocal Process ◽  
The Way

AbstractThe problem of extended fusion and identification can be approached from a diachronic perspective. Based on our own research, as well as findings from the fields of social, political, and clinical psychology, we argue that the way contemporary emotional events shape local fusion is similar to the way in which historical experiences shape extended fusion. We propose a reciprocal process in which historical events shape contemporary identities, whereas contemporary identities shape interpretations of past traumas.

What is the purpose of cognition?

2020 ◽  
Vol 43 ◽  
Author(s):  
Aba Szollosi ◽  
Ben R. Newell
Keyword(s):  
Infinite Number ◽  
Human Cognition ◽  
The Way

Abstract The purpose of human cognition depends on the problem people try to solve. Defining the purpose is difficult, because people seem capable of representing problems in an infinite number of ways. The way in which the function of cognition develops needs to be central to our theories.

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