On the natural extensions of dynamics with a Siegel or Cremer point

Cabrera, Carlos; Kawahira, Tomoki

doi:10.1080/10236198.2012.681780

Cited by 2 publications

(4 citation statements)

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“…Indeed, for any Cremer fixed point x 0 of f we have some open neighborhood Ω 0 of x 0 in which f is univalent and in which there exists a univalent branch g of f −1 with g(x 0 ) = x 0 . Hence, by Theorem 4.1 of [2], for any open neighborhood U 0 of x 0 , with Ū0 ⊂ Ω 0 , there exists a compact, connected set H ⊂ Ū0 , containing the Cremer point x 0 and one or more points from the boundary of U 0 , which is a full continuum in a Julia set of f and satisfies f (H) = H and g(H) = H. Such H is called a hedgehog of x 0 . Theorem 4.2 of [2] asserts that the invariant lift of H to S ∞ consists only of irregular points.…”

Section: If For Two Integersmentioning

confidence: 79%

“…Hence, by Theorem 4.1 of [2], for any open neighborhood U 0 of x 0 , with Ū0 ⊂ Ω 0 , there exists a compact, connected set H ⊂ Ū0 , containing the Cremer point x 0 and one or more points from the boundary of U 0 , which is a full continuum in a Julia set of f and satisfies f (H) = H and g(H) = H. Such H is called a hedgehog of x 0 . Theorem 4.2 of [2] asserts that the invariant lift of H to S ∞ consists only of irregular points. Thus, any open neighborhood U of x, with Ū0 ⊂ Ω 0 , will contain an uncountable number of irregular points, different from x.…”

Section: If For Two Integersmentioning

confidence: 79%

“…It is know from Theorems 4.1 and 4.2 of [2] that the invariant lift x of a Cremer cycle is not an isolated irregular point. Indeed, for any Cremer fixed point x 0 of f we have some open neighborhood Ω 0 of x 0 in which f is univalent and in which there exists a univalent branch g of f −1 with g(x 0 ) = x 0 .…”

Section: If For Two Integersmentioning

confidence: 99%

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Plaque Inverse Limit of a Dynamical System - Dynamics, Signatures and Local Topology

Goldstein¹

2015

Preprint

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The Plaque Inverse Limit of a branched covering self-map of a Riemann surface was introduced and studied in [1]. A point x of P.I.L. was called regular if P.I.L. has the natural Riemann Surface structure at x and was called irregular otherwise. The notion of the signature sign(x, c) of x with respect to a critical point c, which was shown to be a local invariant of P.I.L. was introduced and developed. It was shown that sign(x, c) is nontrivial for some critical points c if and only if x is an irregular point. It was shown that the local topology of P.I.L. at an irregular point x has a property, that removing x from any its neighborhood breaks some pathconnected component of that neighborhood into an uncountable number of path-connected components. Finally, various signatures, including signatures of the invariant lifts of super-attracting and attracting cycles and certain signatures of the invariant lift of a parabolic cycle, were computed. All these signatures had a maximal element.In this work we show that the local topology of P.I.L. at irregular points with different types of signatures is different. Namely, we prove that the local topology at an irregular point x has a property, that for any neighborhood V of x and for some point y = x in V , the open set V − {y} consists of uncountable number of path-connected components, if and only if the signature sign(x, c), for some critical point c, has no maximal element. Next, for a polynomial functions, we compute the signature of the invariant lift of a parabolic cycle with respect to a certain recurrent critical point. This signature, unlike the cases studied in [1], has no maximal element. We show that all other irregular points, except the invariant lifts of super-attracting, attracting, and parabolic cycles, have no maximal element with respect to some recurrent critical point.

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Section: If For Two Integersmentioning

confidence: 79%

Section: If For Two Integersmentioning

confidence: 79%

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Plaque Inverse Limit of a Dynamical System - Dynamics, Signatures and Local Topology

Goldstein¹

2015

Preprint

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“…The following theorem relates dynamical properties of a cycle and the signature of its lift to the P.I.L. The part of this theorem, dealing with the invariant lifts of Cremer and Siegel cycles, was also stated, in a different manner, and proven in [4]. Here we present another proof.…”

Section: From Proposition 23 Of [12]mentioning

confidence: 90%

On the topology of the inverse limit of a branched covering over a Riemann surface

Cabrera

Cherif

Goldstein³

2014

Topology and its Applications

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We introduce the Plaque Topology on the inverse limit of a branched covering self-map of a Riemann surface of a finite degree greater than one. We present the notions of regular and irregular points in the setting of this Plaque Inverse Limit and study its local topological properties at the irregular points. We construct a certain Boolean algebra and a certain sigma-lattice, derived from it, and use them to compute local topological invariants of the Plaque Inverse Limit. Finally, we obtain several results interrelating the dynamics of the forward iterations of the self-map and the topology of the Plaque Inverse Limit.

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On the natural extensions of dynamics with a Siegel or Cremer point

Cited by 2 publications

References 9 publications

Plaque Inverse Limit of a Dynamical System - Dynamics, Signatures and Local Topology

Plaque Inverse Limit of a Dynamical System - Dynamics, Signatures and Local Topology

On the topology of the inverse limit of a branched covering over a Riemann surface

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