Dynamics of measurable functions on the interval

Let A⊂I=[0,1] such that λA=0 and A is a dense Gδ subset of [0,1], and take M to be the set of measurable self-maps of I. There exists a residual set R⊂M such that for each f in R, the following hold:(1)The range of f is contained in A, and f is a one-to-one function.(2)For any x∈[0,1], the ω-limit s...

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Bibliographic Details
Published in:Topology and its applications 2021-05, Vol.295, p.107664, Article 107664
Main Authors: Pierce, Pamela, Steele, T.H.
Format: Article
Language:English
Subjects:
Generic
Measurable function
Trajectory
ω-limit set
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Summary:Let A⊂I=[0,1] such that λA=0 and A is a dense Gδ subset of [0,1], and take M to be the set of measurable self-maps of I. There exists a residual set R⊂M such that for each f in R, the following hold:(1)The range of f is contained in A, and f is a one-to-one function.(2)For any x∈[0,1], the ω-limit set ω(x,f) is nowhere dense.(3)The Hausdorff dimension of the ω-limit points Λ(f)‾=∪x∈Iω(x,f) is zero.(4)The function f is nowhere continuous. Any closed set E contained in [0,1] is an ω-limit set for some measurable function f:I→I. Moreover, there exists a measurable function f:[0,1]→[0,1] such that for any ε>0, x∈[0,1] and closed set E⊂[0,1], there is a function g:I→I such that ∥g−f∥
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2021.107664