Fractals/IFS, dimension, coding/symbolic dynamics

• what happened before the 1st photo: TBA;
  general idea: chaotic systems (e.g., the Lorenz attractor), how to understand/describe them
  for example, the middle-third Cantor set, denoted for now \(C\), its "remarkable" properties:
  • non-empty, compact (i.e., closed and bounded)
  • perfect (each point in \(C\) is limit of other points in \(C\)); Thm: a perfect set is uncountable.
  • it is nowhere dense: its closure, here \(C\) itself, contains no intervals
  • has measure (here "length") zero
  
  \(C\) can be obtained by an IFS, see next photo.
• photo 1 Iterated Function Systems (IFS)
• photo 2 Box-counting dimension
• photo 3 Box-counting dimension continued
• photo 4 Coding, the doubling map
• photo 5 Coding continued
• photo 6 The \(\beta\)-transformation
• photo 7 Extension of the \(\beta\)-transformation, the logistic family
• next time: bifurcations

  https://en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension
	a.k.a. Box-dimension

  https://en.wikipedia.org/wiki/Hausdorff_dimension

  http://math.arizona.edu/~shankar/efa/efa2.pdf

  https://www.amazon.com/Fractals-Everywhere-Michael-F-Barnsley/dp/0120790610#reader_0120790610

  http://wwwf.imperial.ac.uk/~jswlamb/M345PA46/%5BB%5D%20chap%20IX.pdf
	  Barnsley CH 10: measures on fractals

  http://www.davidmartipete.cat/outreach/140122-FractalsPresentation.pdf

  https://en.wikipedia.org/wiki/Iterated_function_system

  https://en.wikipedia.org/wiki/Hausdorff_distance

  https://en.wikipedia.org/wiki/Collage_theorem

  the logistic family

  period doubling in the logistic family, how it leads to chaos

  the universality of this phenomenon, discovered and explained by M. Feigenbaum

  https://en.wikipedia.org/wiki/Mitchell_Feigenbaum#Work