Let ℳ be the set of pairs (T,g) such that T ⊂ ℝ is compact, g: T → T is continuous, g is minimal on T and has a piecewise monotone extension to convT. Two pairs (T,g),(S,f) from ℳ are equivalent if the map h: orb(minT,g) → orb(minS,f) defined for each m ∈ ℕ₀ by is increasing on orb(minT,g). An equivalence class of this relation-a minimal (oriented) pattern A-is exhibited by a continuous interval map f:I → I if there is a set T ⊂ I such that (T,f|T) = (T,f) ∈ A. We define the forcing relation on minimal patterns: A forces B if all continuous interval maps exhibiting A also exhibit B. In Theorem 3.1 we show that for each minimal pattern A there are maps exhibiting only patterns forced by A. Using this result we prove that the forcing relation on minimal patterns is a partial ordering. Our Theorem 3.2 extends the result of [B], where pairs (T,g) with T finite are considered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm169-2-5,
author = {Jozef Bobok},
title = {Forcing relation on minimal interval patterns},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {161-173},
zbl = {1058.37005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm169-2-5}
}
Jozef Bobok. Forcing relation on minimal interval patterns. Fundamenta Mathematicae, Tome 167 (2001) pp. 161-173. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm169-2-5/