On the ω-limit sets of tent maps

@article{AndrewD2012,
abstract = {For a continuous map f on a compact metric space (X,d), a set D ⊂ X is internally chain transitive if for every x,y ∈ D and every δ > 0 there is a sequence of points ⟨x = x₀,x₁,...,xₙ = y⟩ such that $d(f(x_i),x_\{i+1\})< δ$ for 0 ≤ i< n. In this paper, we prove that for tent maps with periodic critical point, every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for tent maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.},
author = {Andrew D. Barwell, Gareth Davies, Chris Good},
journal = {Fundamenta Mathematicae},
keywords = {internal chain transitivity; pseudo orbits tracing property; -limit set; tent map; shadowing},
language = {eng},
number = {1},
pages = {35-54},
title = {On the ω-limit sets of tent maps},
url = {http://eudml.org/doc/282713},
volume = {217},
year = {2012},
}

TY - JOUR
AU - Andrew D. Barwell
AU - Gareth Davies
AU - Chris Good
TI - On the ω-limit sets of tent maps
JO - Fundamenta Mathematicae
PY - 2012
VL - 217
IS - 1
SP - 35
EP - 54
AB - For a continuous map f on a compact metric space (X,d), a set D ⊂ X is internally chain transitive if for every x,y ∈ D and every δ > 0 there is a sequence of points ⟨x = x₀,x₁,...,xₙ = y⟩ such that $d(f(x_i),x_{i+1})< δ$ for 0 ≤ i< n. In this paper, we prove that for tent maps with periodic critical point, every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for tent maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.
LA - eng
KW - internal chain transitivity; pseudo orbits tracing property; -limit set; tent map; shadowing
UR - http://eudml.org/doc/282713
ER -