$\overline{R} = \overline{P}$ FOR MAPS OF DENDRITES X WITH Card(End(X)) < c

Abstract

Let X be a dendrite and f : X → X be a continuous map. Denote by R(f ) and P (f ) the sets of recurrent points and periodic points of f respectively. In this paper we show that, if the cardinal number Card(End(X)) of the set of endpoints of X is less than the cardinal number c of the continuum, then R = P . From this we derive that, if Card(End(X)) < c, then f is chaotic in the sense of Devaney if and only if f is transitive.

Some Chaos Notions on Dendrites

“…Surprisingly, these results are true on dendrites whose set of branch points is not dense [16,21]. For dendrites whose set of endpoints is countable, transitivity = Devaney chaos [20].…”

“…The interest in the study of chaotic behaviour on dendrites has been increasing for a few years. For examples, researches focus on shadowing [17], distributional chaos [18], induced map [19], and relations between some chaotic notions [20][21][22][23]. It turns out that most of the results on the interval do not necessarily hold true on dendrites.…”

Example of the Smooth Skew Product in the Plane with the One-dimensional Ramified Continuum as the Global Attractor

“…During last years dynamic systems on one-dimensional ramified continua, in particular, on dendrites (i. e. on locally connected continua containing no simple closed curves), are intensively investigating (see e.g. papers [21] - [24]). Give the main definitions (see [25]).…”

Equicontinuity of maps on a dendrite with finite branch points

“…In [12], Kato constructed a dendrite T and a map f ∈ C 0 (T ) such that R(f ) = P (f ). However, Mai and Shi [15] proved that if T is a dendrite with Card(End(T )) < c, then R(f ) = P (f ) for any f ∈ C 0 (T ). In [23] we proved that if T is a dendrite with finite branch points and f ∈ C 0 (T ), then R(f ) = P (f ) and the depth of f is at most 3.…”