Attractors with irrational rotation number (original) (raw)

Index 1 fixed points of orientation reversing planar homeomorphisms

Topological Methods in Nonlinear Analysis, 2015

Let U ⊂ R 2 be an open subset, f : U → f (U) ⊂ R 2 be an orientation reversing homeomorphism and let 0 ∈ U be an isolated, as a periodic orbit, fixed point. The main theorem of this paper says that if the fixed point indices i R 2 (f, 0) = i R 2 (f 2 , 0) = 1 then there exists an orientation preserving dissipative homeomorphism ϕ : R 2 → R 2 such that f 2 = ϕ in a small neighbourhood of 0 and {0} is a global attractor for ϕ. As a corollary we have that for orientation reversing planar homeomorphisms a fixed point, which is an isolated fixed point for f 2 , is asymptotically stable if and only if it is stable. We also present an application to periodic differential equations with symmetries where orientation reversing homeomorphisms appear naturally.

Displacement sequence of an orientation preserving circle homeomorphism

We give a complete description of the behaviour of the sequence of displacements etan(z)=Phin(x)−Phin−1(x)rmod1\eta_n(z)=\Phi^n(x) - \Phi^{n-1}(x) \ \rmod \ 1etan(z)=Phin(x)−Phin−1(x)rmod1, z=exp(2pirmix)z=\exp(2\pi \rmi x)z=exp(2pirmix), along a trajectory varphin(z)\{\varphi^{n}(z)\}varphin(z), where varphi\varphivarphi is an orientation preserving circle homeomorphism and Phi:mathbbRtomathbbR\Phi:\mathbb{R} \to \mathbb{R}Phi:mathbbRtomathbbR its lift. If the rotation number varrho(varphi)=fracpq\varrho(\varphi)=\frac{p}{q}varrho(varphi)=fracpq is rational then etan(z)\eta_n(z)etan(z) is asymptotically periodic with semi-period qqq. This convergence to a periodic sequence is uniform in zzz if we admit that some points are iterated backward instead of taking only forward iterations for all zzz. If varrho(varphi)notinmathbbQ\varrho(\varphi) \notin \mathbb{Q}varrho(varphi)notinmathbbQ then the values of etan(z)\eta_n(z)etan(z) are dense in a set which depends on the map gamma\gammagamma (semi-)conjugating varphi\varphivarphi with the rotation by varrho(varphi)\varrho(\varphi)varrho(varphi) and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if varphi\varphivarphi is C1C^1C1-diffeomorphism and show approximation of the displa...

On the Regularity of the Displacement Sequence of an Orientation Preserving Circle Homeomorphism

2015

We investigate the regularity properties of the displacement sequence () () () (), 2 exp , 1 mod 1 ix z x x z n n n π = Φ − Φ = η − where R R → Φ : is a lift of an orientation preserving circle homeomorphism. If the rotation number () q p = ϕ is rational, then () z n η is asymptotically periodic with semi-period q. This WACŁAW MARZANTOWICZ and JUSTYNA SIGNERSKA 12 convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. This leads to the notion of an basins'-ε edge, which we illustrate by the numerical example. If () , Q ∈ / ϕ then some classical results in topological dynamics yield that the displacement sequence also exhibits some regularity properties, which we define and prove in the second part of the paper.

A Recurrent Nonrotational Homeomorphism on the Annulus

Transactions of the American Mathematical Society, 1992

We construct an area-and orientation-preserving recurrent diffeomorphism on the annulus without periodic points, which is not conjugate to a rotation. The mapping is, however, semiconjugate to an irrational rotation of a circle. Our example is a counterexample to the Birkhoff Conjecture.

On the existence of attractors

Transactions of the American Mathematical Society, 2012

On every compact 3-manifold, we build a non-empty open set U of Diff 1 (M ) such that, for every r ≥ 1, every C r -generic diffeomorphism f ∈ U ∩ Diff r (M ) has no topological attractors. On higher dimensional manifolds, one may require that f has neither topological attractors nor topological repellers. Our examples have finitely many quasi attractors. For flows, we may require that these quasi attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics. * This work has been done during the stays of Li Ming and Yang Dawei at the IMB, Université de Bourgogne and we thank the IMB for its warm hospitality. M. Li is supported by a post doctoral grant of the Région Bourgogne, and D. Yang is supported by CSC of Chinese Education Ministry.

Rotation numbers of discontinuous orientation-preserving circle maps revisited

The theory of circle homeomorphisms has a great number of deep results. However, sometimes continuity or single-valuedness of a circle map may be restrictive in theoretical constructions or applications. In this paper it is shown that some principal properties of circle homeomorphisms are inherited by the class of orientation-preserving circle maps. The latter class is rather broad and contains not only circle homeomorphisms but also a variety of non continuous maps arising in applications. Of course, even in cases when a property remains to be valid for orientation-preserving circle maps, absence of continuity sometimes results in noticeable changes of related proofs.

Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Nonlinearity, 2013

We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.

On conjugacy of homeomorphisms of the circle possessing periodic points

Journal of Mathematical Analysis and Applications, 2007

We give a necessary and sufficient condition for topological conjugacy of homeomorphisms of the circle having periodic points. As an application we get the following theorem on the representation of homeomorphisms. The homeomorphism F : S 1 → S 1 has a periodic point of period n iff there exist a positive integer q < n relatively prime to n and a homeomorphism Φ : S 1 → S 1 such that the lift of Φ −1 • F • Φ restricted to [0, 1] has the form

The geometric index and attractors of homeomorphisms of

Ergodic Theory and Dynamical Systems, 2021

In this paper we focus on compacta$K \subseteq \mathbb {R}^3$which possess a neighbourhood basis that consists of nested solid tori$T_i$. We call these sets toroidal. Making use of the classical notion of the geometric index of a curve inside a torus, we introduce the self-geometric index of a toroidal setK, which roughly captures how each torus$T_{i+1}$winds inside the previous$T_i$as$i \rightarrow +\infty .Wethenusethisindextoobtainsomeresultsabouttherealizabilityoftoroidalsetsasattractorsforhomeomorphismsof. We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of.Wethenusethisindextoobtainsomeresultsabouttherealizabilityoftoroidalsetsasattractorsforhomeomorphismsof\mathbb {R}^3$.