10. Prove that T, is topologically conjugate to the quadratic map F(z) = 4z(1 - z).
10. Prove that T, is topologically conjugate to the quadratic map F(z) = 4z(1 - z).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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
Transcribed Image Text:The following exercises apply to the tent map
2z
T2(z) =
2(1-z) 1/2 S=S1
. To describe the
Note that the maximum of T is 1 and occurs at z =
dynamics of T via symbolic dy namics, we thus need to modify E2 somewhat
since there is an ambiguity in the sequence associated to any rational number
of the form p/2 where p is an integer. For example, 1/2 may be described by
either (11000...) or (01000...). To remedy this, we identify any two sequences
of the form (so ...3-1 1000...), where 0 or 1. For example, the sequences
(1101000...) and (1111000...) are to be thought of as representing the same
point. Let E denote E2 with these identifications.
Prove that S:I→ E, is one-to-one, where S(z) is defined as in $1.7.
7. Prove that o o S = So T2.
8. Prove that Tz has exactly 2" periodic points of period n.
6.
%3D
9. Prove that T2 is chaotic on 1.
10. Prove that Tz is topologically conjugate to the quadratic map F(z) =
4z(1 – z).
%3D
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