A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

     

Group automorphisms with few and with many periodic points

For each a compact group automorphism is constructed with the property that

This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms with any given topological entropy exist.

     

Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Let and suppose that f : K ⁿ →K ⁿ is nonexpansive with respect to the l ₁-norm, , and satisfies f (0) = 0. Let P ₃(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P ₃(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : D _g→D _g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.     

Computing topological entropy for periodic sequences of unimodal maps

In this paper we introduce an algorithm which allows us to compute the topological entropy of a class of piecewise monotone continuous interval maps. The algorithm can be applied to a class of economic models called duopolies, and it can be useful to compute the topological entropy of periodic sequences of continuous maps which have been used in some population growth models.     

Common fixed points for weakly compatible maps

The purpose of this paper is to prove a common fixed point theorem, from the class of compatible continuous maps to a larger class of maps having weakly compatible maps without appeal to continuity, which generalizes the results of Jungck [3], Fisher [1], Kang and Kim [8], Jachymski [2], and Rhoades [9].     

On the global dynamics of periodic triangular maps

This paper is an extension of an earlier paper that dealt with global dynamics in autonomous triangular maps. In the current paper, we extend the results on global dynamics of autonomous triangular maps to periodic non-autonomous triangular maps. We show that, under certain conditions, the orbit of every point in a periodic non-autonomous triangular map converges to a fixed point (respectively, periodic orbit of period p) if and only if there is no periodic orbit of prime period two (respectively, periodic orbits of prime period greater than p).     

The triangular maps with closed sets of periodic points

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F(x,y)=(f(x),gx(y)), satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987-997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.     

Periodic points of holomorphic maps via Lefschetz numbers

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.     

Over-rotation numbers for unimodal maps

We introduce twist unimodal maps of the interval and describe their structure. Sufficient conditions for the growth of over-rotation interval in families of maps are given.     

Existence theorems for saddle points of vector-valued maps

In this paper, we prove some new existence theorems for loose saddle points and for saddle points of set-valued maps or vector-valued functions. These theorems generalize the corresponding results of Tanaka and those of Luc and Vargas via different proofs.The authors would like to thank two referees for their careful reading of the paper and helpful comments.     

真拟弱几乎周期点和拟正则点

T为紧致度量空间X上的连续映射,M（X）为X上所有Borel概率测度.设x∈X,记Mx（T）为概率测度序列{1n∑n 1i=0δTi（x）}在M（X）中的极限点的集合,其中δx表示支撑集是{x}的点测度.记W（T）和QW（T）分别为T的弱几乎周期点和拟弱几乎周期点集.本文证明,如果（X,T）非平凡且满足specifcation性质,则存在x,y∈QW（T）／W（T）（称为真拟弱几乎周期点）,分别满足μ∈Mx（T）,x∈Supp（μ）和ν∈My（T）,y∈/Supp（ν）,回答了周作领等提出的公开问题.Mx（T）在弱拓扑中是紧致连通集,所以,要么是单点集,要么是不可数集.如果x∈QW（T）／W（T）,则Mx（T）是不可数集.一个自然的问题是,怎么刻画M x（T）是单点集的点x（这时x称为拟正则点）.本文给出M x（T）是单点集的充要条件.     

Essential maps and coincidence theory

Using an elementary approach, we present new coincidence principles for general classes of maps.     

Entropy formula and continuity of entropy for piecewise expanding maps

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.     

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.     

On Weyl almost periodic selections of multivalued maps

We prove that Weyl almost periodic multivalued maps R∋t→F(t)∈clU have Weyl almost periodic selections, where clU is the collection of non-empty closed sets of a complete metric space U.     

图映射拓扑序列熵的可交换性

主要研究图上连续自映射拓扑序列熵的可交换性,证明了对任意无界的正整数递增序列A=(ai)∞i=1和任意的连续图映射f,g都有hA(fog):hA(g of).解决了Balibrea F等人在相关文献中提出的一个猜想.     

Dynamics of elementary maps of dendrites

The notion of elementary map of a dendrite into itself is introduced. Arithmetical relations between the periods of periodic points are given; the structure ofω-limit sets, sets of periodic and nonwandering points is described; the topological entropy of elementary maps is shown to be equal to 0. Examples are given illustrating the differences in the entropic properties of continuous maps of dendrites with a countable set of branch points and continuous maps of their retracts which are finite trees. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 183–195, February, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01755.     

Essential extension type maps for general classes of maps

This paper discusses admissible (and more general) maps between topological spaces. We show that if F is H-essential and F ≅ G then G has a fixed point.