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A generalization of mixed invariant sets
Author(s) -
Feiste Uwe
Publication year - 1988
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015199
Subject(s) - mathematics , invariant (physics) , bounded function , compact space , pure mathematics , metric space , discrete mathematics , combinatorics , mathematical analysis , mathematical physics
Summary The concept of mixed invariant set is due to Bandt [1], Bedford [2], Dekking [3, 4], Marion [4] and Schulz [10]. An m ‐tuple B = ( B 1 , …, B m ) of closed and bounded subsets B i of a complete finitely compact (bounded and closed subsets are compact) metric space X is called a mixed invariant set with respect to contractions f 1 , …, f m and a transition matrix M = ( m ij ), if, and only if,B i = ∪ { j | m i j = 1 }f i ( B j )for every i ∈ {1, …, m }. In the papers quoted an essential condition is that all mappings f 1 , …, f m be contractions. We will show that, under certain conditions, the construction of mixed invariant sets also works in cases where some of the mappings are isometries or even expanding mappings.