Rotational chaos and strange attractors on the 2-torus

Boroński, Jan P.; Oprocha, Piotr

doi:10.1007/s00209-014-1388-1

Cited by 15 publications

(15 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Birkhoff attractor [LC88] provides an example of an invariant essential cofrontier K such that ρ − (F, K) = ρ + (F, K). An example with similar properties where K is a pseudo-circle is given in [BO15]. The next result, which is a corollary of the main theorem from [KLCN15], shows that there is no area-preserving analogue of the Birkhoff attractor 1 Theorem 2.8.…”

Section: Prime Ends Rotation Numbers Suppose That K ⊂ Intmentioning

confidence: 87%

Realizing rotation numbers on annular continua

Koropecki

2016

Math. Z.

View full text Add to dashboard Cite

An annular continuum is a compact connected set K which separates a closed annulus A into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where K = A, showing that if K is an invariant annular continuum of a homeomorphism of A isotopic to the identity, then the rotation set in K is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in K (and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum K is minimal with the property of being annular (what we call a circloid), then every rational number between the extrema of the rotation set in K is realized by a periodic orbit in K. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in K. This improves a previous result of Barge and Gillette.

show abstract

Section: Prime Ends Rotation Numbers Suppose That K ⊂ Intmentioning

confidence: 87%

Realizing rotation numbers on annular continua

Koropecki

2016

Math. Z.

View full text Add to dashboard Cite

show abstract

“…For such an endomorphisms it was proved in [8] that every rotation number ρ ∈ ρ( f ) can always be realised by some point. More precisely, we can replace the original definition (7) with the more naturally defined pointwise rotation number.…”

Section: Auxiliary Lemma On Circle Endomorphisms With Two Turnsmentioning

confidence: 99%

“…Proof. For endomorphisms with two turns, since the definitions (7) and (8) are equivalent, there always exists some point y 0 with a lift y 0 , such that,…”

Section: Auxiliary Lemma On Circle Endomorphisms With Two Turnsmentioning

confidence: 99%

Prime ends dynamics in parametrised families of rotational attractors

Boroński

Činč

Liu

2020

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We provide several new examples in dynamics on the 2-sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrized families. First, motivated by a topological version of the Poincaré-Bendixson Theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of S 2 with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux. Second, from reduced Arnold's family we construct a parametrised family of Birkhoff-like cofrontier attractors, where at least for uncountably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This example complements the resolution of Walker's Conjecture by Koropecki, Le Calvez and Nassiri from 2015. Third, answering a question of Boyland, we show that there exists a non-transitive Birkhoff-like attracting cofrontier which is obtained from a BBM embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor so that the two induced prime ends rotation numbers are exactly the two endpoints of the rotation interval.

show abstract

“…In [17] the authors showed that there exists a 2-torus homeomorphism h homotopic to the identity with an attracting R.H. Bing's pseudocircle C such that the rotation set of h|C is not a unique vector. It follows from Theorem 4.3 that the topological entropy of this homeomorphism is infinite.…”

Section: Theorem 43 Ifmentioning

confidence: 99%

“…1 Strange attractors, supporting annulus homeomorphisms with non-unique rotation numbers, obtained by a construction on an inverse limit of circles, as described in [17] is topologically conjugate to the induced map on an inverse limit space based on a branched one-dimensional manifold, but Barge [5] proved that certain dynamical systems with Hénon-type attractors cannot me modeled on inverse limits. In addition, authors' recent example of torus homeomorphism with an attracting pseudo-circle as Birkhoff-type attractor in [17] is non-differentiable and has infinite entropy (see Fig. 1).…”

Section: Introductionmentioning

confidence: 99%

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

Boroński

Oprocha

2015

J Dyn Diff Equat

Self Cite

View full text Add to dashboard Cite

We prove that if f : G → G is a map on a topological graph G such that the inverse limit lim ← − (G, f ) is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Chéritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if G = [0, 1], then h( f ) ∈ {0, ∞}, and combined with a result of Ito shows that certain dynamical systems on compact finitedimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.

show abstract

Rotational chaos and strange attractors on the 2-torus

Cited by 15 publications

References 34 publications

Realizing rotation numbers on annular continua

Realizing rotation numbers on annular continua

Prime ends dynamics in parametrised families of rotational attractors

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

Contact Info

Product

Resources

About