Lecture 24 - Tue 2023 12 05  Fractals, transversal structure of Rossler.
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                                                                       FRACTALS.
Finish material in Lecture 23; review notion of dimension and introduce
the pointwise and correlation dimension.
In pointwise: N(epsilon) = number of points in ball of radius epsilon
                           centered at some point. Then N ~ epsilon^d
Make drawing of the typical log(N) versus log(epsilon) plot one gets
"in practice". The difficulty in measuring d accurately.
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                                                            REVIEW OF DIMENSION
Self-similar: split fractal in pieces of size r, reproducing the whole fractal.
              Let N = N(r) the number of pieces.    The self-similar dimension
              follows from the scaling N ~ r^{-d}
              i.e.:  d = - lim_{r \to 0} log(N)/log(r).
              Note: self-similar rationals in [0 1] is 1.
              Do a Sierpinski gasket example.

Box-dimension: Cover the fractal with balls [or intervals/squares/cubes or ...
              boxes] of size r. Let N=N(r) be the minimum number of boxes
              needed. Then, again, N ~ r^{-d}, so that
              d = - lim_{r \to 0} log(N)/log(r).
              Note: self-similar rationals in [0 1] is 1.
              Do a non-self-similar Sierpinski gasket example.

Hausdorff dimension: Will not define it, a bit messy. But also based on coverings
              by boxes, except that the boxes need not be all the same size; but
              their max. size goes to zero.
              Note: Hausdorff rationals in [0 1] is 0!
              Reason: can enumerate them, say {x_n}_0^infty. Then cover each x_n
              with an interval of length epsilon^n. Then total length of covering
              epsilon/1-epsilon can be made to vanish.

Point-wise dimension: Start with attractor in the form of a sequence [this notion
              is special for dynamical systems]. Then take a point in the attractor
              and count how many points in the sequence fit within a ball of radius
              epsilon:  N = N(epsilon). Then, again, N ~ epsilon^{d}.
              Can measure by graphing log(N).

              Explain problems that arise because {x_n} is finite. Need enough
              points to have a "clear" straight line piece in the plot.
              Do drawing obtained "in practice". The difficulty in measuring d
              accurately.

Correlation dimension: take average of point-wise dimension over attractor.

Lyapunov dimension: based on the idea that the rates of contraction/expansion
              determine a dimension.
and many more!

There are many notions of dimension, and they do not all give the same number!
In addition, often papers are not entirely clear as to what the parameters in
the equation used where, exactly [e.g.: information displayed elsewhere in the
paper, not where the dimension is reported].
The worse part is, of course, that the physical/biological/whatever meaning
(and usefulness) of any dimension extracted from a natural process is often
unclear: "what does it mean and what do we do with it?"
In addition: are there other quantities that one can measure from a fractal?
Things that may have an answer to  "what does it mean and what do we do with
it?"
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