# Fractals and self-similarity

November 14, 2011 — September 22, 2021

Objects with a fractional Hausdorff dimension, AFAICT. For some equivocation on this theme, see wikipedia.

## 1 Iterated function systems

Connection to design grammars.

Fun fact: there is a fractal renderer built into GIMP.

## 2 Mandelbrot sets I guess

TBD

## 3 Non-local derivatives

## 4 Long memory time series

See, for example, fractional brownian motion, long memory time series.

## 5 In nature

Pattern formation in nature often looks fractal to some approximation, or at some range of scales.

## 6 relationship to dimensions

David S. Richeson, A Mathematician’s Guided Tour Through High Dimensions does a nice job of relating fractals to measure-theoretic notions of dimension.

## 7 Estimating fractal dimension

Question: How closely related is this to estimating a Hurst exponent? How close to grammatical induction? Various classic methods based on naïve plug-in versions of mathematical definitions are given in Theiler (1990). A new one, which I am curious about, is Chamorro-Posada (2016) based on some kind of compression argument, basically, gzipping copies of the image that have been downsampled by various ratios and watching how the file size changes as a kind of entropy estimate.

I am reminded of Cosma Shalizi’s cautionary note on estimating Entropies and Information using Lempel-Ziv, and the caveat:

Jose M. Amigo, Janusz Szczepanski, Elek Wajnryb and Maria V. Sanchez-Vives, “Estimating the Entropy Rate of Spike Trains via Lempel-Ziv Complexity”,

Neural Computation16(2004): 717–736: Normally, I have strong views on using Lempel-Ziv to measure entropy rates, but here they are using the 1976 Lempel-Ziv definitions, not the 1978 ones. The difference is subtle, but important; 1978 leads to gzip and practical compression algorithms, but very bad entropy estimates; 1976 leads, as they show numerically, to reasonable entropy rate estimates, at least for some processes. Thanks to Dr. Szczepanski for correspondence about this paper.]

## 8 References

*Dynamics–the geometry of behavior*.

*The Annals of Probability*.

*Architectural Design*.

*Constructive Approximation*.

*Fractals Everywhere*.

*SuperFractals*.

*Constructive Approximation*.

*Fractals*.

*Advances in Mathematics*.

*Chaos, Solitons & Fractals*.

*Chaos, Solitons & Fractals*.

*IEEE Transactions on Image Processing*.

*Nature*.

*Physical Review Letters*.

*Fractal Geometry: Mathematical Foundations and Applications*.

*Fractal Image Compression: Theory and Application*.

*Image and Text Compression*. The Kluwer International Series in Engineering and Computer Science.

*Physics Letters A*.

*arXiv:1108.1325 [Physics]*.

*Phys. Rev. A*.

*Indiana University Mathematics Journal*.

*IEEE Transactions on Image Processing*.

*Proceedings of the IEEE*.

*Mathematical Proceedings of the Cambridge Philosophical Society*.

*Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976)*.

*Fractals and Scaling In Finance: Discontinuity, Concentration, Risk*.

*Nature*.

*The Misbehavior of Markets: A Fractal View of Financial Turbulence*.

*PLoS ONE*.

*Biometrika*.

*Physica A: Statistical and Theoretical Physics*.

*SIGGRAPH Comput. Graph.*

*Physica A: Statistical and Theoretical Physics*.

*Nature*.

*Journal of the Optical Society of America A*.

*Journal of Physics A: Mathematical and General*.

*Fractal Growth Phenomena*.

*Physical Review Letters*.

*Science*.

*Applications of Random Matrices in Physics*.

*The European Physical Journal B - Condensed Matter and Complex Systems*.

*Journal of Theoretical Probability*.