The number and linking of periodic solutions of periodic systems

Matsuoka, Toshifumi

doi:10.1007/bf01391795

Cited by 25 publications

(18 citation statements)

References 6 publications

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“…We claim that m = p. To prove this claim, recall that x 1 ∈ R 1 and that f (x i ) = x i+1 for 1 ≤ i < p and f (x p ) ∈ L. Then, from (21) it follows that p = tm for some t ≥ 1 and x i ∈ R i mod m for 1 ≤ i ≤ p. Since p, m ≥ 2, the claim is trivially true when p = 2 or p = 3. So, assume that p ≥ 4.…”

Section: A Lower Bound For the Entropy Of No Division Patterns With Tmentioning

confidence: 96%

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On the minimum positive entropy for cycles on trees

Alsedà

Juher

Mañosas

2016

Trans. Amer. Math. Soc.

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Abstract. Consider, for any n ∈ N, the set Posn of all n-periodic tree patterns with positive topological entropy and the set Irrn Posn of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families Posn and Irrn. Let λn be the unique real root of the polynomial x n − 2x − 1 in (1, +∞). We explicitly construct an irreducible n-periodic tree pattern Qn whose entropy is log(λn). For n = m k , where m is a prime, we prove that this entropy is minimum in the set Posn. Since the pattern Qn is irreducible, Qn also minimizes the entropy in the family Irrn.

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Section: A Lower Bound For the Entropy Of No Division Patterns With Tmentioning

confidence: 96%

“…As another important example, when F X is the family of surface homeomorphisms, the pattern (or braid type) of a cycle P of a map f : M −→ M in F X is characterized by the isotopy class, up to conjugacy, of f M\P [16,21].…”

Section: Introductionmentioning

confidence: 99%

On the minimum positive entropy for cycles on trees

Alsedà

Juher

Mañosas

2016

Trans. Amer. Math. Soc.

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“…Therefore, in the special case where/ is a diffeomorphism of an annulus homotopic to the identity, Theorem 4 becomes a result of Kawakami [12], which are applied to the study of periodic systems of differential equations. The maps treated in Example 2 appear naturally in the theory of periodic systems (see [16,Lemma 2]). Hence Theorem 4 and Proposition 3 can be applied to the study of such systems.…”

Section: R (N)=-£ (*(D)ilt(rn/d)mentioning

confidence: 99%

The number of periodic points of smooth maps

Matsuoka

1989

Ergod. Th. Dynam. Sys.

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Abstract. Let/: M -* M be a C 1 map on a compact manifold. We give a topological condition under which / has an even number of periodic points with a given period.We also obtain a sufficient condition, in terms of homology, for / to have infinitely many periodic points. In this paper, we give two applications of the Dold's equalities to the study of periodic points of smooth maps. The first application is concerned with the problem of whether the number v(n) of periodic points of a given period n is even or odd. We give a condition on n and on the homotopy class of the map under which v{n) is even. This generalizes results in [14,15,18].In the second application, we generalize some results of Franks [8,9] on the existence of infinitely many periodic points.In the last section, we generalize the Dold's equalities to isolated sets of periodic points.

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“…Joan S. Birman and R. F. Williams have been studying the knotted periodic orbits in the Lorenz systems in [2] and [3]. See also [7] for an application to periodic systems.…”

Section: Y Togawamentioning

confidence: 99%