Notes for Lecture 22 (Tue 2024 11 21)  DIMENSION.
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                                                            REVIEW OF DIMENSION
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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).
Examples: Cantor; Two fifths Cantor; Koch curve; Sierpinski gasket; etc.
Note:     self-similar for rationals in [0 1] is 1.

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BOX-dimension: Cover the fractal with balls [or intervals/squares/cubes or ...
              boxes] of size r. Let N=N(r) = minimum number of boxes needed.
Then, again, N ~ r^{-d}, so that d = - lim_{r \to 0} log(N)/log(r).
Examples: a non-self-similar Cantor (or Sierpinski gasket) where deleted parts
          are "random".
Again:    Box dimension for self-similar rationals in [0 1] is 1.

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HAUSDORFF dimension: See BELOW for (messy) definition. Attempt ~1918, to have a
                     notion of dimension that would pin down fractals precisely.
It does not, and it is "practically" uncomputable. 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 dim. of rationals in [0 1] is 0 (better than the others on this).
      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 the covering vanishes as
epsilon \to 0, since it is epsilon/1-epsilon (geometrical series).

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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}.
Cons and pros:
1) Relatively easy to compute when the attractor is given via the output of a
   numerical solution of a dynamical system [or a sequence of measurements].
2) Does not work for infinite sequences. For example: order the rationals in
   [0, 1]. Then N(epsilon) = infinity, for any epsilon > 0!
3) To measure d accurately need a lot of data.
   Do "typical" graph of log(N) versus log(epsilon).
      For large epsilon (attractor size) N saturates.
      For small epsilon (discretization scale) N becomes zero.
   To measure d "accurately" one needs a significant section of the graph to be
   close to a straight line with slope d; but because of the logs, this means
   that one needs at small epsilon over several orders of magnitude ... lots of
   data needed.
4) Embedding issue. When the data comes from a numerical calculation, one has
   access to the full attractor (see note##). But with data one may have access
to only some variables. Takens' embedding theorem can then be used to "recover"
the full attractor. But this is highly non-trivial. The theorem involves two
parameters (delay time and number of delays to apply) not known a-priori, and
finding them can be highly non-trivial. Further, the theorem does not guarantee
that the recovered attractor would not be "flattened" in some direction, which
will make the corresponding contribution to the dimension hard to capture, and
will introduce errors.
   Note## Even when, in principle, one has access to all the variables of the
   dynamical system, problems arise. Imagine that you are solving a pde ...
   Then the numerical phase space will be huge [and the actual one infinite in
   dimension]. Calculating N(epsilon) in such a situation would be extremely
   expensive ... so you try to select only a small number of variables [this is
   the opposite of Takens' embedding, but equally problematic].

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CORRELATION DIMENSION: The point-wise dimension gives a dimension near a point
   only. To define a dimension for the whole abstract, one takes an average of
   point-wise dimensions over the attractor.
   But, what if these point-wise dimensions vary significantly from point to
   point? More than would be expected from the errors in the calculation [not
   enough data, etc.]?  This leads into the "multi-fractal" issue.

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LIAPUNOV dimension. Yet another notion of dimension. This one is based on the
   idea that the rates of contraction/expansion along the attractor determine a
dimension.   Like the correlation dimension, it is specially suited for dynamical
systems.

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FINAL COMMENT:  There are many notions of dimension, and they do not all give
                the same number! Overlooking issues due to poor writing [e.g.:
papers where it is not entirely to what parameter (in the equation used) values
the reported dimension corresponds], the actually unsolved problem is 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?" Further, the dimension is just one number attached to a very complex
object; are there other quantities that one can measure from a fractal? Things
that may provide an answer to  "what does it mean and what do we do with it?"
As far as I know, currently there is no such thing (unfortunately, because
fractals seem to be quite common in nature).
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                                          DEFINITION of the HAUSDORFF dimension.
Let S be the set whose dimension we want.
For any del > 0, let {U_n} be a countable covering of S by open sets such that
    diam(U_n) \leq del  [covering means that the union of the U_n includes S].
Define now
     H_1(d, del) = inf \sum (diam U_n)^d; where the inf is over all possible
                                          coverings.
     H_2(d) = lim_{del \to 0} H_1(d, del).

     Then dim_H(S) = inf{ d \geq 0, H_2(d) = 0}.

INTUITION
     Consider the example where S is a surface. Then, for d = 2, (diam U_n)^d is
     (roughly) the area of U_n, and \sum (diam U_n)^d is an upper bound on the
     area, while H_1(d, del) will be the area. But, if d > 2, then for del small
     H_1(d, del) will become very small [divide a square in small equal squares
     and sum the lengths of the squares cubed, this behaves like area*length],
     and very large if d < 2.
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                                                             EXAMPLE DIMENSIONS
Lorenz (sigma=10, r=28, b=8/3)  D_correlation = 2.049 \pm 0.096
                                D_Lyapunov    = 2.062
Rossler (a = b = 0.2, c = 5.7)  D_correlation = 1.986 \pm 0.078
                                D_Lyapunov    = 2.013
Logistic map at r_inf           D_correlation = 0.500 \pm 0.005
                                D_box         ~ 0.538
The Lyapunov dimension is based on the idea that the rates of contraction and
expansion determine a dimension. Like the correlation dimension, it is
specially suited for dynamical systems [give a bit of the intuition here].
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