Search

A continuum X is a dendrite if it is locally connected and contains no simple closed curve, a self mapping f of X is called monotone if the preimage of any connected subset of X is connected. If X is a dendrite and f : X → X is a monotone continuous map then we prove that any ω -limit set is approximated by periodic orbits and the family of all ω -limit sets is closed with respect to the Hausdorff metric. Second, we prove that the equality between the closure of the set of periodic points, the set of regularly recurrent points and the union of all ω -limit sets holds for the induced maps F n ( f ) : F n ( X ) → F n ( X ) and T n ( f ) : T n ( X ) → T n ( X ) where F n ( X ) denotes the family of finite subsets of X with at most n points, T n ( X ) denotes the family of subtrees of X with at most n endpoints and F n ( f ) = 2 | F n ( X ) f , T n ( f ) = 2 | T n ( X ) f , in particular there is no Li–Yorke pair for these maps. However, we will show that this rigidity in general is not exhibited by the induced map C ( f ) : C ( X ) → C ( X ) where C ( X ) denotes the family of sub-continua of X and C ( f ) = 2 | C ( X ) f , we will discuss an example of a homeomorphism g on a dendrite S which is dynamically simple whereas its induced map C ( g ) is ω -chaotic and has infinite topological entropy. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by , 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. [Copyright &y& Elsevier]