Relatively compact Siegel disks with non-locally connected boundaries

Abstract

International audienc

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

“…We show that 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 [26], Herman [28] and Chéritat [21] cannot be modeled on inverse limits. This resembles a known fact for Hénon-type attractors: Williams [48] showed that every hyperbolic, one-dimensional, expanding attractor for a discrete dynamical system Fig.…”

“…For the pseudo-circle, there is also an important example of Handel [26] who constructed a homeomorphism on the pseudo-circle (extendable to the whole plane) with zero topological entropy. Related results in complex dynamics were obtained by Herman [28] and Chéritat [21]. The natural question is whether the example of [27] can be somehow generalized to obtain the pseudo-circle as an inverse limit with one bonding map and with zero entropy.…”

“…A related example was given by Herman [28] in the complex plane. Quite recently Herman's construction has been extended by Chéritat [21] to show that the pseudocircle can also emerge as the boundary of a Siegel disk. Other prominent examples include those given by Kennedy and Yorke, who exhibited that there exist C ∞ -smooth dynamical systems in dimensions greater than 2 with uncountably many minimal pseudo-circles, and any small C 1 perturbation of which manifests the same property (see [33,34]).…”

Positive area and inaccessible fixed points for hedgehogs

“…In [PM93], [PM95], [PM00], Perez-Marco developed techniques using "tube-log Riemann surfaces" for the construction of interesting examples of indifferent dynamics. These were used by the author to construct examples of hedgehogs containing smooth combs and hedgehogs of minimal Hausdorff dimension one ( [Bis05], [Bis08]), and further developed by Cheritat to construct Siegel disks with pseudo-circle boundaries ( [Che09]).…”

Rotational chaos and strange attractors on the 2-torus

“…The pseudocircle admits periodic homeomorphisms (rational rotations) that extend to orientation-preserving and orientation-reversing homeomorphisms of the annulus or 2-sphere [9,24]. It occurs also as an attracting minimal set for a C ∞ -smooth planar diffeomorphism [21,22], the minimal set of a volume-preserving planar diffeomorphism [21], or as the boundary of a Siegel disk for a holomorphic map in the complex plane [14]. It is worth mentioning that the diffeomorphism constructed in [21] can be easily modified to an annulus homeomorphism (equal to the identity on the boundary), which gives a homeomorphism of the 2-torus, with a pseudocircle as an attractor.…”