Lecture22 Tue 2024 11 21  Fractals (Transversal structure of Rossler/Lorenz)
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     Additional info in:
     "Lecture21_2021_scan.pdf" and "Lecture21_whiteboard353.pdf"
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Question: How do we characterize fractals?
Want some measure that allows a characterization of fractals; similar to
the way "genus" characterizes surfaces [surfaces with the same genus are
equivalent topologically] ...
Dimension was born from this purpose, though it failed to achieve this objective.
   In this stuff I will be following
   "Lecture21_2021_scan.pdf" and "Lecture21_whiteboard353.pdf"
Self-similar dimension [definition, motivation, examples].
Topological Cantor set.
Box dimension.  Problem: Box(rationals in [0 1]) = 1.
Hausdorf dimension.
Pointwise and correlation dimensions.
Mention multifractals [see the end of chapter 11 in Strogatz].
Liapunov
Takens' embedding theorem.
"Fun" stuff: dimension of music composers.
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x_{n+1} = f(x_n);
Linear stability.
COBWEBS
Illustrate it with f(x) = r x (1-x); r < 1 [x = 0 is a global attractor].
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