Transitivity and the centre for maps of the circle

Coven, Ethan M.; Mulvey, Irene

doi:10.1017/s0143385700003254

Cited by 36 publications

(17 citation statements)

References 7 publications

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“…Coven and Mulvey in [23] extended Theorem 1.1 to the circle proving that for transitive circle maps with periodic points the set of all periodic points is dense in S 1 . Moreover, Blokh in [20] showed that transitive circle maps with periodic points have positive topological entropy.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Entropy and periodic points for transitive maps

Alsedà¹,

Kolyada

Llibre³

et al. 1999

Trans. Amer. Math. Soc.

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Abstract. The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the n-star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Entropy and periodic points for transitive maps

Alsedà¹,

Kolyada

Llibre³

et al. 1999

Trans. Amer. Math. Soc.

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“…We now refer the reader to the third paragraph of the proof of Theorem A of [2]. That paragraph, with some cosmetic changes, completes the proof of the present lemma.…”

Section: Preliminary Lemmasmentioning

confidence: 81%

“…This result fails for maps of the circle, as we show in an example. The notion of a one-way interval for a map of the circle is introduced in [2] where dynamic properties of circle maps are explored. In the present paper, we continue the study of one-way intervals for maps of the circle.…”

Section: Introductionmentioning

confidence: 99%

One-way intervals of circle maps

Ancel

Hero²

1998

Proc. Amer. Math. Soc.

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Abstract. An interval in the circle S 1 is one-way with respect to a map f : S 1 → S 1 if under repeated applications of f all points of the interval move in the same direction. The main result is that every locally one-way interval is either one-way or is the union of two overlapping one-way subintervals. An example is given which illustrates that the latter case can occur; however, it is proved that the latter case cannot occur if the interval is covered by the image of the map. As a corollary, it is shown that if f has periodic points, then every interval which contains no periodic points is either one-way or is the union of two overlapping one-way subintervals.

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“…Define x, arbitrarily for 2"(j2)< r < P"(Jl). For i >Jl the term p"(i) is the highest subscript occurring in (2), while for i< J2 the term 2"(0 is the lowest subscript occurring in (2). Therefore the system can be solved consistently, showing that I -0" is surjective.…”

Section: Jek(ni)mentioning

confidence: 94%

“…Note that if 2 is a character then U"(2) = 20" = 0n (2). Let f~ = ~ cij2~, where j=O 4o = la, the cijsC, and the 2f~G.…”

Section: The Endomorphism Theoremmentioning

confidence: 99%

Ergodic endomorphisms of compact abelian groups

Shirvani

Rogers

1988

Commun.Math. Phys.

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Abstract.We show that for a surjective endomorphism of a compact abelian group ergodicity is equivalent to a condition which implies r-mixing for all r > 1, and we characterize such maps algebraically. This is then used in proving the ergodicity of an extensive class of endomorphisms of the binary sequence space. As a simple corollary it is found that one-dimensional linear cellular automata and the accumulator automata are r-mixing for all r > 1.

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Transitivity and the centre for maps of the circle

Cited by 36 publications

References 7 publications

Entropy and periodic points for transitive maps

Entropy and periodic points for transitive maps

One-way intervals of circle maps

Ergodic endomorphisms of compact abelian groups

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