Periodic points and transitivity on dendrites

Abstract

Abstract. We study relations between transitivity, mixing and periodic points on dendrites. We prove that when there is a point with dense orbit which is not an endpoint, then periodic points are dense and there is a terminal periodic decomposition (we provide an example of a dynamical system on a dendrite with dense endpoints satisfying this assumption). We also show that it may happen that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It may also happen that dynamical system is transitive but there is a unique periodic point, which in fact is the unique fixed point. We also prove that on almost meshed-continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all nonempty compact subsets.

Transitive dendrite map with zero entropy

“…This also provides a new example of a transitive map of a dendrite which does not have dense periodic points (because proximality excludes existence of more than one minimal point and every periodic point is minimal). This property of the Hoehn-Mouron example is also noted in [1]. Observe that dendrites have the fixed point property, hence every continuous dendrite map has at least one fixed point.…”

“…Assume that we have defined N K and M K > N K for some K ∈ N. We set N K+1 = (2 M K + 1) · M K and we take M K+1 large enough so that Claims 17.2 and 17.5 of [7] are satisfied. 1 Take…”

“…Then f Z as defined in [7] is weakly mixing but not mixing. 1 In the notation of [7], it means that if Z ⊂ N is a thick set such that [N K+1 , M K+1 ] ∩ N ⊂ Z, then for each 0 ≤ i ≤ K + 1 there are some m, n ∈ N such that α(s) is defined for each s ∈ T si, K+1, m and α n (T si, K+1, m ) = T ∅, K+1, 0 .…”

“…4.2]). It is also known that maps on dendrites share some dynamical properties with graph maps (see, for example, [1,9,10,12,14,15]). In particular, if a dendrite contains a free arc then a transitive map necessarily has positive topological entropy (see [6]).…”

Dendritations of surfaces

“…Let us show that for a generic point p ∈ A we have that Q(p) is a nonseparating arc. The proof follows the ideas in [1,Lemma 3.1]. Let D n be the closed disc of radius 1 − 1/n centered at the origin.…”