# Forcing relation on minimal interval patterns

Fundamenta Mathematicae (2001)

- Volume: 169, Issue: 2, page 161-173
- ISSN: 0016-2736

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topJozef Bobok. "Forcing relation on minimal interval patterns." Fundamenta Mathematicae 169.2 (2001): 161-173. <http://eudml.org/doc/281968>.

@article{JozefBobok2001,

abstract = {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 $h(g^m(minT)) = f^m(minS)$ 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.},

author = {Jozef Bobok},

journal = {Fundamenta Mathematicae},

keywords = {interval map; minimal pattern; forcing relation},

language = {eng},

number = {2},

pages = {161-173},

title = {Forcing relation on minimal interval patterns},

url = {http://eudml.org/doc/281968},

volume = {169},

year = {2001},

}

TY - JOUR

AU - Jozef Bobok

TI - Forcing relation on minimal interval patterns

JO - Fundamenta Mathematicae

PY - 2001

VL - 169

IS - 2

SP - 161

EP - 173

AB - 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 $h(g^m(minT)) = f^m(minS)$ 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.

LA - eng

KW - interval map; minimal pattern; forcing relation

UR - http://eudml.org/doc/281968

ER -

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