On the ω-limit sets of tent maps

• Autorzy: Andrew D. Barwell , Gareth Davies , Chris Good
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54H20: Topological dynamics
• 54C05: Continuous maps
• 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
• 37B10: Symbolic dynamics
• 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2012
• Tom: 217
• Numer: 1
• Strony: 35-54

Daty

wydano

2012

Twórcy

autor

Andrew D. Barwell

• School of Mathematics, University of Bristol, Howard House, Queens Avenue, Bristol, BS8 1SN, UK
• School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

autor

Gareth Davies

• Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 3LB, UK

autor

Chris Good

• School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

Identyfikatory

DOI

10.4064/fm217-1-4