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

Abstract

We will prove the existence of a class of transitive maps on the real line R, with a discontinuity and horizontal asymptotes, whose set of periodic orbits is dense in R; that is, a class of chaotic families. In addition, we will show a rare phenomenon: the existence of periodic orbits of period three prevents the existence of transitivity.

Remark on Transitivity for piecewise increassing maps on R

“…1. Podemos observar que las diferentes hipótesis impuestas a los sistemas alternantes crecientes en los artículos [10,11] correspondiente al Teorema 1.1 y al artículo [7] correspondiente al Teorema 1.2 implican la condición…”

Global attractors in the parametrized Hénon–Devaney map

“…(1) There are two simple curves L + ⊂ R + γ ∩ R + δ and L − ⊂ R − γ ∩ R − δ satisfying the following conditions: (a) F (L + ) and F (L − ) are long curves contained in R + δ and R − δ , respectively, which are graphs of strictly decreasing functions intersecting γ transversely at single points. (b) (4) Consider the region (a, c) : 0 < c < a < 1 and a 2 < c < √ a 3 and let P be the union of all successive pre-images from γ, that is (a) If s is a long curve which is the graph of a strictly decreasing function contained in R 2 \ δ, then P ∩ s is dense in s. (b) Γ is a set formed by long curves which are graphs of strictly decreasing functions; in particular, P ∩ Γ is dense in Γ.…”

“…Following [6], there is an important relationship between F a,c and the (onedimensional) Boole-like expanding maps, via the projection to {x = 0} along a C 1 foliation. Similarly, Hénon-Devaney-like maps, as considered in [4], are (twodimensional) topological versions of expansive (one-dimensional) alternating systems [8,3] projecting along a continuous foliation.…”