Abstract
Many processes in life arise from the iteration of some simple rule or function. As an instance, the amount of money in your savings account is determined by the fact that the bank pays r percent interest per year.
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Notes
- 1.
To be fair to history, it should be noted that Brooks and Matelski [BM 81] described and produced the Mandelbrot set a couple of years before Mandelbrot studied it. But Mandelbrot and his followers did a lot more with it.
- 2.
An interval is a contiguous set of numbers in the real line, for example, the unit interval [0, 1] consisting of the numbers 0, 1, and all numbers between them.
- 3.
If you begin with a negative number, then the first time you hit the square root button you will get an error message. If you begin with 0, then the string of your answers will all be 0s. Thus the phenomenon we are describing only applies to positive numbers.
- 4.
If we wished to be more pedantic we would say the “mass of the cart.”
- 5.
Three decimal places of output seems crude nowadays, but in 1961 multiplication was often done with the aid of a slide rule and two decimal places was one’s typical precision with a slide rule.
- 6.
We use the formula \(\sum _{j=0}^{\infty }{r}^{j} = 1/(1 - r)\), valid for − 1 < r < 1.
- 7.
This is not to say that Mandelbrot was right and Newton was wrong. Rather, Mandelbrot offered a new way to look at things.
References and Further Reading
Brooks, R., Matelski, J.P.: The dynamics of 2-generator subgroups of P S L(2, ℂ). In: Kra, I., Maskit, B. (eds.) Riemann Surfaces and Related Topics. Annals of Mathematics Studies, vol. 97, pp. 65–71. Princeton University Press, Princeton (1981)
Devaney, R.: Classical Mechanics and Dynamical Systems. CRC Press, Boca Raton (1981)
Devaney, R.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley, Redwood City (1989)
Diacu, F., Holmes, P.: Celestial Encounters: The Origins of Chaos and Stability. Princeton University Press, Princeton (1996)
Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes. Prépublications mathématiques d’Orsay 2/4 (1984/1985)
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (1990)
Fleick, J.: Chaos: Making a New Science. Viking, New York (1987)
Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman, San Francisco (1982)
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Krantz, S.G., Parks, H.R. (2014). Dynamical Systems. In: A Mathematical Odyssey. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8939-9_4
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