Premium
Families, filters and chaos
Author(s) -
Oprocha Piotr
Publication year - 2010
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdq034
Subject(s) - mathematics , topological entropy , mixing (physics) , chaos (operating system) , compact space , metric space , set (abstract data type) , sequence (biology) , topological conjugacy , entropy (arrow of time) , pure mathematics , metric (unit) , topological space , topology (electrical circuits) , discrete mathematics , combinatorics , computer science , physics , computer security , quantum mechanics , biology , genetics , programming language , operations management , economics
In this paper, we use the definition of ( ℱ 1 , ℱ 2 )‐chaos introduced recently by Tan and Xiong together with the properties of residual relations as a tool in construction of various kinds of scrambled sets. In particular, we show that a continuous map acting on a compact metric space has an ε ‐scrambled set if and only if it has a distributionally ε ‐scrambled set with respect to a sequence. We also provide an example of a topologically mixing map with positive topological entropy but without DC1 pairs.