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Sidorov, N. V.

doi:10.1023/a:1013104016392

Cited by 16 publications

(1 citation statement)

References 19 publications

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“…(3) Other kinds of Pisot toral automorphisms (=algebraic automorphisms of T m given by matrices whose characteristic polynomial is irreducible and has a Pisot root) have been studied in [19] and [16] (including higher dimensions). Various results concerning βtransformations with holes-with β > 2 [1] and β < 2 [4,5]-could be possibly used to build a consistent theory.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 99%

The baker’s map with a convex hole

2018

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We consider the baker's map B on the unit square X and an open convex set H ⊂ X which we regard as a hole. The survivor set J (H) is defined as the set of all points in X whose B-trajectories are disjoint from H. The main purpose of this paper is to study holes H for which dim H J (H) = 0 (dimension traps) as well as those for which any periodic trajectory of B intersects H (cycle traps).We show that any H which lies in the interior of X is not a dimension trap. This means that, unlike the doubling map and other one-dimensional examples, we can have dim H J (H) > 0 for H whose Lebesgue measure is arbitrarily close to one. Also, we describe holes which are dimension or cycle traps, critical in the sense that if we consider a strictly convex subset, then the corresponding property in question no longer holds.We also determine δ > 0 such that dim H J (H) > 0 for all convex H whose Lebesgue measure is less than δ.This paper may be seen as a first extension of our work begun in [5,6,10,11,20] to higher dimensions.

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Section: Final Remarks and Open Problemsmentioning

confidence: 99%

The baker’s map with a convex hole

2018

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On the Pisot Substitution Conjecture

Akiyama

Barge

Berthé

et al. 2015

Mathematics of Aperiodic Order

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Symbolic representations of nonexpansive group automorphisms

Lindenstrauss

Schmidt

2005

Isr. J. Math.

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Abstract. If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in X. In spite of this we are able to construct a symbolic space V and a class of shift-invariant probability measures on V each of which corresponds to an α-invariant probability measure on X. Moreover, every α-invariant probability measure on X arises essentially in this way.The last part of the paper deals with the connection between the twosided beta-shift V β arising from a Salem number β and the nonhyperbolic ergodic toral automorphism α arising from the companion matrix of the minimal polynomial of β, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures on V β and certain α-invariant probability measures on X. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.

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Cited by 16 publications

References 19 publications

The baker’s map with a convex hole

The baker’s map with a convex hole

On the Pisot Substitution Conjecture

Symbolic representations of nonexpansive group automorphisms

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