Topological conjugacy of 𝑛-multiple Cartesian products of circle rough transformations

2021 ◽  
Vol 29 (6) ◽  
pp. 851-862
Author(s):  
Iuliana Golikova ◽  
◽  
Svetlana Zinina ◽  
◽  
Keyword(s):  
Conjugacy Class ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Topological Invariant ◽  
Topological Invariants ◽  
Topological Conjugacy ◽  
Cartesian Products ◽  
Torus Surface

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

Knotted periodic orbits in suspensions of annulus maps

1987 ◽  
Vol 411 (1841) ◽  
pp. 351-378 ◽  
Keyword(s):  
Periodic Orbits ◽  
Rotation Number ◽  
Existence And Uniqueness ◽  
Torus Knots ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Rotation Numbers ◽  
Higher Genus

We consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3-manifold T 2 x I. Using results of Birman & Williams ( Topology 22, 47‒82 (1983); Contemp. Math . 20, 1‒60 (1983)), we construct a knotholder or template that carries the set of periodic orbits of the flow. We define rotation numbers and show that any orbit of period q and rotation number p / q can be arranged as a positive braid on p strands. This yields existence and uniqueness results for families of resonant torus knots ( p -braids that are ( p , q )-torus knots of period q > p which correspond to order-preserving (Birkhoff-) periodic orbits of the diffeomorphism. We show that all other q -periodic p -braids have higher genus, and we establish bounds on the genera of such knots. We obtain existence and uniqueness results for a number of other, non-resonant, torus knots, including non-order-preserving ( q + s , q )-torus knots of rotation number 1.

scholarly journals Tackling tangledness of cosmic strings by knot polynomial topological invariants

2017 ◽  
Vol 32 (27) ◽  
pp. 1750164 ◽  
Author(s):  
Xinfei Li ◽  
Xin Liu ◽  
Yong-Chang Huang
Keyword(s):  
Cosmic Strings ◽  
Topological Defects ◽  
Topological Invariant ◽  
Topological Invariants ◽  
String Topology ◽  
Promising Candidate ◽  
Linking Numbers ◽  
Quintessence Field ◽  
Chern Simons ◽  
Knot Polynomial

Cosmic strings in the early universe have received revived interest in recent years. In this paper, we derive these structures as topological defects from singular distributions of the quintessence field of dark energy. Our emphasis is placed on the topological charge of tangled cosmic strings, which originates from the Hopf mapping and is a Chern–Simons action possessing strong inherent tie to knot topology. It is shown that the Kauffman bracket knot polynomial can be constructed in terms of this charge for unoriented knotted strings, serving as a topological invariant much stronger than the traditional Gauss linking numbers in characterizing string topology. Especially, we introduce a mathematical approach of breaking-reconnection which provides a promising candidate for studying physical reconnection processes within the complexity-reducing cascades of tangled cosmic strings.

scholarly journals Topological invariants of germs of real analytic functions

1997 ◽  
Vol 39 (1) ◽  
pp. 85-89 ◽  
Author(s):  
Piotr Dudziński
Keyword(s):  
Real Analytic Function ◽  
Topological Invariant ◽  
Topological Invariants ◽  
Isolated Singularity ◽  
Simple Consequence ◽  
Milnor Fibre ◽  
Real Analytic Functions ◽  
Nonisolated Singularity ◽  
Real Analytic ◽  
Weighted Homogeneous Polynomial

Let f: (ℝn, 0)→ (ℝ,0) be a germ of a real analytic function. Let L and F(f) denote the link of f and the Milnor fibre of fc respectively, i. e., L = {x ∈ Sn−1 | f(x) = 0}, , where 0 ≤ ξ ≪ r ≪ 1, . In [2] Szafraniec introduced the notion of an -germ as a generalization of a germ defined by a weighted homogeneous polynomial satisfying some condition concerning the relation between its degree and weights (definition 1). He also proved that if f is an -germ (presumably with nonisolated singularity) then the number χ(F(f)/d mod 2 is a topological invariant of f, where χ(F(f)) is the Euler characterististic of F(f), and gave the formula for χ(L)/2 mod 2 (it is a well-known fact that F(L) is an even number). As a simple consequence he got the fact that χ(F(f)mod 2 is a topological invariant for any f, which is a generalization of Wall's result [3] (he considered only germs with an isolated singularity).

scholarly journals Unfoldings of discrete dynamical systems

1984 ◽  
Vol 4 (3) ◽  
pp. 421-486 ◽  
Author(s):  
Joel W. Robbin
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conjugacy Class ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Topological Conjugacy ◽  
Universal Unfolding ◽  
Moser Iteration

AbstractA universal unfolding of a discrete dynamical system f0 is a manifold F of dynamical systems such that each system g sufficiently near f0 is topologically conjugate to an element f of F with the conjugacy φ and the element f depending continuously on f0. An infinitesimally universal unfolding of f0 is (roughly speaking) a manifold F transversal to the topological conjugacy class of f0. Using Nash-Moser iteration we show infinitesimally universal unfoldings are universal and (in part II) give a class of examples relating to moduli of stability introduced by Palis and De Melo.

scholarly journals Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

1985 ◽  
Vol 5 (4) ◽  
pp. 501-517 ◽  
Author(s):  
Lluís Alsedà ◽  
Jaume Llibre ◽  
Michał Misiurewicz ◽  
Carles Simó
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Rotation Interval ◽  
Continuous Map ◽  
Continuous Maps

AbstractLet f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

scholarly journals Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

2019 ◽  
Vol 30 (14) ◽  
pp. 1950073 ◽  
Author(s):  
Hong-Duc Nguyen ◽  
Tien-Son Phạm ◽  
Phi-Dũng Hoàng
Keyword(s):  
Complex Plane ◽  
Plane Curve ◽  
Complex Variables ◽  
Topological Invariant ◽  
Topological Invariants ◽  
Real Plane ◽  
Plane Curve Singularities ◽  
Łojasiewicz Exponents ◽  
Curve Singularities ◽  
Formula Computing

In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are not necessarily reduced. We first show that, for complex plane curve singularities, the set of polar quotients is a topological invariant. We next prove that the Łojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. For real plane curve singularities, we also give a formula computing the Łojasiewicz gradient exponent via real polar branches. As an application, we give effective estimates of the Łojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.

scholarly journals TOPOLOGICAL INVARIANTS AND ANYONIC PROPAGATORS

1999 ◽  
Vol 14 (28) ◽  
pp. 1933-1936 ◽  
Author(s):  
WELLINGTON DA CRUZ
Keyword(s):  
Hausdorff Dimension ◽  
Integral Representation ◽  
Particle Flux ◽  
Topological Invariant ◽  
Topological Invariants ◽  
Continuous Family ◽  
Path Integral Representation ◽  
Flux System ◽  
Fractal Properties ◽  
Genus Number

We obtain the Hausdorff dimension, h=2-2s, for particles with fractional spins in the interval, 0≤ s ≤0.5, such that the manifold is characterized by a topological invariant given by, [Formula: see text]. This object is related to fractal properties of the path swept out by fractional spin particles, the spin of these particles, and the genus (number of anyons) of the manifold. We prove that the anyonic propagator can be put into a path integral representation which gives us a continuous family of Lagrangians in a convenient gauge. The formulas for, h and [Formula: see text], were obtained taking into account the anyon model as a particle-flux system and by a qualitative inference of the topology.

scholarly journals Bifurcation of an attracting invariant circle: a Denjoy attractor

1983 ◽  
Vol 3 (1) ◽  
pp. 87-118 ◽  
Author(s):  
Glen R. Hall
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Periodic Points ◽  
Invariant Circle ◽  
Worst Case ◽  
Dense Orbits ◽  
Parameter Values ◽  
Two Parameter

AbstractWe construct an example of a C∞ diffeomorphism of an annulus into itself which has an attracting invariant circle such that the map restricted to this circle has no periodic points and no dense orbits. By studying two parameter families of maps of the plane which undergo Hopf bifurcation, particularly the set of parameter values for which the rotation number is irrational, we see that the above example can be considered as a ‘worst case’ of the loss of smoothness of an attracting invariant circle without periodic orbits.

scholarly journals Kneading sequences for toy models of Hénon maps

2020 ◽  
pp. 1-20
Author(s):  
ERMERSON ARAUJO
Keyword(s):  
Conjugacy Class ◽  
Topological Conjugacy ◽  
Two Dimensional ◽  
One Dimensional ◽  
Henon Maps ◽  
Hénon Maps ◽  
Critical Lines ◽  
Model Family

Abstract The purpose of this article is to study the relation between combinatorial equivalence and topological conjugacy, specifically how a certain type of combinatorial equivalence implies topological conjugacy. We introduce the concept of kneading sequences for a setting that is more general than one-dimensional dynamics: for the two-dimensional toy model family of Hénon maps introduced by Benedicks and Carleson, we define kneading sequences for their critical lines, and prove that these sequences are a complete invariant for a natural conjugacy class among the toy model family. We also establish a version of Singer’s theorem for the toy model family.

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