The construction of P-expansive maps of regular continua: A geometric approach

The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval

“…The constructions of the dendrite Z(f, P ). In [1], a regular curve Z has been constructed from a continuous map f from a finite graph to itself and an finvariant subset of the finite graph. In this section, under some natural restriction, the dendrite Z(f, P ) is constructed from a continuous map f : I → I and a periodic orbit P of f .…”

“…it is surjective and satisfies π•f = g•π. See [1] for details. We notice that g(p i0,i1,...,im ) = p i1,i2,...,im , thus, g m (p i0,i1,...,im ) ∈ π(P ).…”

“…By the result of [2], we can extend Theorem 4.4 for tree maps. For a continuous map f from a tree T to itself and a periodic orbit P of f , we can construct a dendrite Z(f, P ) and a continuous map g : Z(f, P ) −→ Z(f, P ) which is semi-conjugate to f by the same way as in Section 3 (see [1] for detail). When speaking about tree maps, the role of odd periodic orbit played for interval maps is replaced by non-divisible periodic orbits (see [2] for the notion of a division).…”

“…Introduction. In [1], a new space Z and a continuous map from Z to itself have been constructed from a given system by the geometrical method. The structure of Z changes corresponding to the behavior of a continuous map f from a finite graph to itself and the way of choosing an invariant subset of f .…”