Topological and symbolic dynamics for hyperbolic systems with holes

Bundfuss, S.; Krüger, Tyll; Troubetzkoy, Serge

doi:10.1017/s0143385710000556

Cited by 18 publications

(32 citation statements)

References 25 publications

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“…Note that X U is compact and f -invariant, therefore, (X U , f U ) where f U = f | X U is a dynamical system on its own right. Dynamical systems of this form are called open dynamical systems, exclusion maps, maps with holes or maps with gaps or interval exclusion systems if X is an interval [8].…”

Section: Introductionmentioning

confidence: 99%

Topological and ergodic properties of symmetric sub-shifts

Barrera¹

2014

DCDS-A

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The family of symmetric one sided sub-shifts in two symbols given by a sequence a is studied. We analyse some of their topological properties such as transitivity, the specification property and intrinsic ergodicity. It is shown that almost every member of this family admits only one measure of maximal entropy. It is shown that the same results hold for attractors of the family of open dynamical systems arising from the doubling map with a centred symmetric hole depending on one parameter, and for the set of points that have unique β-expansion for β ∈ (ϕ, 2) where ϕ is the Golden Ratio.

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Section: Introductionmentioning

confidence: 99%

Topological and ergodic properties of symmetric sub-shifts

Barrera¹

2014

DCDS-A

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“…They were concerned with iterating maps on a manifold M and the set of x ∈ M that were never mapped into some hole. Theorem 2.4 is essentially a consequence of Proposition 4.1 [2]. However, all of our results regarding calculating dim H (U) and the identification of the univoque set with a graph-directed self-similar set are completely new.…”

Section: Introductionmentioning

confidence: 78%

“…After completion of this paper the authors were made aware of the work of Bundfuss, Krüger and Troubetzkoy [2]. They were concerned with iterating maps on a manifold M and the set of x ∈ M that were never mapped into some hole.…”

Section: Introductionmentioning

confidence: 99%

On univoque points for self-similar sets

2015

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Let K ⊆ R be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence of this, we can show that the set of points with a unique coding is a graph-directed selfsimilar set in the sense of Mauldin and Williams [15]. The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Our algorithm can be applied generically, and our result generalises the work of [4], [10], [11], and [5].

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“…The study of dynamical systems with holes, also termed as open dynamical systems, first proposed by Pianigiani and Yorke [16], has recently attracted attention on account of both its dynamical interest and applications, we refer to [2,4,6,7,8,9,10] for some related work. The study of billiards with holes is the foundational example in this field [10].…”

Section: Introductionmentioning

confidence: 99%