scholarly journals Twist sets for maps of the circle

1984 ◽  
Vol 4 (3) ◽  
pp. 391-404 ◽  
Author(s):  
Michał Misiurewicz
Keyword(s):  
Rotation Number ◽  
Closed Set ◽  
Twist Map ◽  
Rotation Interval ◽  
Continuous Map ◽  
The One

AbstractLet f be a continuous map of degree one of the circle onto itself. We prove that for every number a from the rotation interval of f there exists an invariant closed set A consisting of points with rotation number a and such that f restricted to A preserves the order. This result is analogous to the one in the case of a twist map of an annulus.

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 Boundaries of instability zones for symplectic twist maps

2013 ◽  
Vol 13 (1) ◽  
pp. 19-41 ◽  
Author(s):  
M.-C. Arnaud
Keyword(s):  
Rotation Number ◽  
Irrational Rotation ◽  
Invariant Curve ◽  
Partial Answer ◽  
Twist Map ◽  
Instability Zone ◽  
Counter Example ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
The One

AbstractVery few things are known about the curves that are at the boundary of the instability zones of symplectic twist maps. It is known that in general they have an irrational rotation number and that they cannot be KAM curves. We address the following questions. Can they be very smooth? Can they be non-${C}^{1} $?Can they have a Diophantine or a Liouville rotation number? We give a partial answer for${C}^{1} $and${C}^{2} $twist maps.In Theorem 1, we construct a${C}^{2} $symplectic twist map$f$of the annulus that has an essential invariant curve$\Gamma $such that$\bullet $ $\Gamma $is not differentiable;$\bullet $the dynamics of${f}_{\vert \Gamma } $is conjugated to the one of a Denjoy counter-example;$\bullet $ $\Gamma $is at the boundary of an instability zone for$f$.Using the Hayashi connecting lemma, we prove in Theroem 2 that any symplectic twist map restricted to an essential invariant curve can be embedded as the dynamics along a boundary of an instability zone for some${C}^{1} $symplectic twist map.

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.

scholarly journals ROTATION SETS FOR ORBITS OF DEGREE ONE CIRCLE MAPS

2002 ◽  
Vol 12 (02) ◽  
pp. 429-437
Author(s):  
LLUÍS ALSEDÀ ◽  
FRANCESC MAÑOSAS ◽  
MOIRA CHAS
Keyword(s):  
Circle Map ◽  
Circle Maps ◽  
Rotation Interval ◽  
The One ◽  
Rotation Set ◽  
Dynamical Information

Let F be the lifting of a circle map of degree one. In [Bamón et al., 1984] a notion of F-rotation interval of a point [Formula: see text] was given. In this paper we define and study a new notion of a rotation set of point which preserves more of the dynamical information contained in the sequences [Formula: see text] than the one preserved from [Bamón et al., 1984]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the Main Theorem of [Bamón et al., 1984] in our settings.

OBSERVED ROTATION NUMBERS IN FAMILIES OF CIRCLE MAPS

2001 ◽  
Vol 11 (01) ◽  
pp. 73-89 ◽  
Author(s):  
MICHAEL A. SAUM ◽  
TODD R. YOUNG
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Narrow Band ◽  
Initial Conditions ◽  
Positive Probability ◽  
Circle Maps ◽  
Numerical Evidence ◽  
Rotation Interval ◽  
Large Numbers ◽  
Rotation Numbers

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers.

scholarly journals The speed interval: a rotation algorithm for endomorphisms of the circle

1988 ◽  
Vol 8 (1) ◽  
pp. 17-33 ◽  
Author(s):  
Jan Barkmeijer
Keyword(s):  
Rotation Interval ◽  
Continuous Map ◽  
Rotation Set

AbstractLetfbe a continuous map of the circle into itself of degree one. We introduce the notion of rotation algorithms. One of these algorithms associates eachz∈S1with an interval, the so-called speed intervalS(z,f), which is contained in the rotation interval ρ(f) off. In contrast with the rotation set ρ(z,f), the intervalS(z,f) sometimes allows us to ascertain that ρ(f) is non-degenerate, by using only finitely many elements of {fn(z) |n≥ 0}. We further show that all choices for ρ(z,f) andS(z,f) occur, for certainz∈S1provided that ρ(z,f) ⊂S(z,f) ⊂ ρ(f).

scholarly journals Large semigroups of cellular automata

2011 ◽  
Vol 32 (6) ◽  
pp. 1991-2010 ◽  
Author(s):  
YAIR HARTMAN
Keyword(s):  
Cellular Automata ◽  
Field Structure ◽  
The Other ◽  
Entire Space ◽  
Dimensional Torus ◽  
Semigroup Action ◽  
Closed Set ◽  
One Dimensional ◽  
Shift Space ◽  
The One

AbstractIn this article, we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to ‘largeness’: first, a semigroup has the ID (infinite is dense) property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space; the second property is maximal commutativity (MC). We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus, and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring ℤ/sℤ). It will be shown that these two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of a prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible), the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one-sided shift space a semigroup acting on a two-sided shift space, and vice versa, in a way that preserves the ID and the MC properties.

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

2001 ◽  
Vol 64 (1) ◽  
pp. 125-143 ◽  
Author(s):  
MEIRONG ZHANG
Keyword(s):  
Dirichlet Problem ◽  
Rotation Number ◽  
Periodic Potential ◽  
Schrödinger Operators ◽  
Negative Integer ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Periodic Potentials ◽  
The One

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

Invariant circles and depinning transition

2016 ◽  
Vol 38 (2) ◽  
pp. 761-787 ◽  
Author(s):  
WEN-XIN QIN ◽  
YA-NAN WANG
Keyword(s):  
Rotation Number ◽  
Critical Value ◽  
Invariant Circle ◽  
Twist Map ◽  
Depinning Transition ◽  
Invariant Circles ◽  
Irrational Rotation Number ◽  
Rotational Invariant ◽  
Image Position ◽  
Area Preserving

We associate the existence or non-existence of rotational invariant circles of an area-preserving twist map on the cylinder with a physically motivated quantity, the depinning force, which is a critical value in the depinning transition. Assume that $H:\mathbb{R}^{2}\mapsto \mathbb{R}$ is a $C^{2}$ generating function of an exact area-preserving twist map $\bar{\unicode[STIX]{x1D711}}$ and consider the tilted Frenkel–Kontorova (FK) model: $$\begin{eqnarray}{\dot{x}}_{n}=-D_{1}H(x_{n},x_{n+1})-D_{2}H(x_{n-1},x_{n})+F,\quad n\in \mathbb{Z},\end{eqnarray}$$ where $F\geq 0$ is the driving force. The depinning force is the critical value $F_{d}(\unicode[STIX]{x1D714})$ depending on the mean spacing $\unicode[STIX]{x1D714}$ of particles, above which the tilted FK model is sliding, and below which the particles are pinned. We prove that there exists an invariant circle with irrational rotation number $\unicode[STIX]{x1D714}$ for $\bar{\unicode[STIX]{x1D711}}$ if and only if $F_{d}(\unicode[STIX]{x1D714})=0$. For rational $\unicode[STIX]{x1D714}$, $F_{d}(\unicode[STIX]{x1D714})=0$ is equivalent to the existence of an invariant circle on which $\bar{\unicode[STIX]{x1D711}}$ is topologically conjugate to the rational rotation with rotation number $\unicode[STIX]{x1D714}$. Such conclusions were claimed much earlier by Aubry et al. We also show that the depinning force $F_{d}(\unicode[STIX]{x1D714})$ is continuous at irrational $\unicode[STIX]{x1D714}$.

A new perspective on the Riesz potential

2017 ◽  
Vol 6 (3) ◽  
pp. 317-326 ◽  
Author(s):  
Jie Xiao
Keyword(s):  
Radon Measure ◽  
Riesz Potential ◽  
The Other ◽  
Lebesgue Spaces ◽  
Content Type ◽  
Continuous Map ◽  
Other Hand ◽  
Associate Space ◽  
New Perspective ◽  
The One

AbstractThis paper offers a new perspective to look at the Riesz potential. On the one hand, it is shown that not only \mathfrak{L}^{q,qp^{-1}(n-\alpha p)}\cap\mathfrak{L}^{p,\kappa-\alpha p} contains {I_{\alpha}L^{p,\kappa}} under the conditions {1<p<\infty}, {1\leq q<\infty}, q(\kappa/p-\alpha)\leq\kappa\leq n, {0<\alpha<\min\{n,1+\kappa/p\}}, but also {\mathfrak{L}^{q,\lambda}} exists as an associate space under the condition {-q<\lambda<n}, where {I_{\alpha}L^{p,\kappa}} and {\mathfrak{L}^{q,\lambda}} are the Morrey–Sobolev and Campanato spaces on {\mathbb{R}^{n}} respectively. On the other hand, a nonnegative Radon measure μ is completely characterized to produce a continuous map {I_{\alpha}:L_{p,1}\to L^{q}_{\mu}} under the condition {1<p<\min\{q,{n}/{\alpha}\}} or {1<q\leq p<\min\{{q(n-\alpha p)}/({n-\alpha q(q-1)^{-1}}),{n}/{\alpha}\}}, where {L_{p,1}} and {L^{q}_{\mu}} are the {(p,1)}-Lorentz and {(q,\mu)}-Lebesgue spaces on {\mathbb{R}^{n}} respectively.

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