ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УрО РАН
Contents

Том 6     N  1     2000

THE METALLIC MEANS FAMILY AND RENORMALIZATION GROUP TECHNIQUES
Vera W. de Spinadel

    We present the new family of Metallic Means (MMF), being the most paramount of its members, the Golden Mean $\phi$ and, in the second place, the Silver Mean $\sigma_{Ag}$. Why do we call them a family? Because, besides carrying the name of a metal--the Golden Mean, the Silver Mean, the Copper Mean, the Bronze Mean, the Nickel Mean--they enjoy common mathematical properties that attach a fundamental importance to them in modern investigations about the search of universal roads to chaos. Among these applications, we have chosen the analysis of the main renormalization group techniques, which have the purpose of getting the quantitative microcharacterization of the transition from order to quantum chaos.

1. Fibonacci sequences

2. The family of metallic means

3. Additive properties

4. Pisot and Salem numbers

5. Renormalization group method of Kohmoto-Kadanoff-Tang (KKT)

6. Density states for a two-dimensional quasiperiodic Penrose lattice

7. Wave functions on a Fibonacci lattice.

8. Behavior of a particle in an almost periodic potential: a renormalization group technique applied to Schrodinger's and Harper's equations

9. Renormalization group of El Naschie

10. Golden and Silver Mean dimensions

11. The Subtle Mean

12. Concluding remarks

Bibliography