Properties of Topologically Transitive Maps on the Real Line

Abstract

We prove that every topologically transitive map f on the real line must satisfy the following properties:(1) The set C of critical points is unbounded.(2) The set f (C) of critical values is also unbounded.(3) Apart from the empty set and the whole set, there can be at most one open invariant set.(4) With a single possible exception, for every element x the backward orbit {y ∈ R : f n (y) = x for some n in N} is dense in R.

Variations on the concept of topological transitivity

“…An example of a transitive map on R such that f −1 (0) = {0} and all other backward orbits are dense has been constructed in [15]. It has been also proved that there can be at most one such point, whose backward orbit is not dense, for any transitive map on R in [14].…”

Topological entropy of compact subsystems of transitive real line maps

“…They may be proved in much the same way as in original references, or else can be deduced from previously known results as noted in [6, Section 7]. For other properties of transitive map of the real line see also [24,25]. Proposition 4.…”

Families of transitive maps on $\mathbb{R}$ with horizontal asymptotes

“…For example, in [9] it is proved that any Anosov diffeomorphism defined in the plane R 2 onto R 2 (R 2 −→ R 2 ) cannot be transitive; on the other hand, there are transitive diffeomorphisms of the plane R 2 minus a line of discontinuities (see [4,10]). In the case of non-bounded intervals the situation is similar to the case of non-compact manifolds; in this direction, in [12] it is shown that continuous transitive maps from R to R (R −→ R) must have infinite critical points; also in [13] there is a large class of examples of transitive maps on R. In any case, a characterization for transitivity is not shown and, in the context of such articles, small perturbations of those maps lose the transitive property. Recently, [11] shows the existence of a class of maps, leading to a geometric model of the well known Boole transformation T (x) = x − 1…”