Renormalisation of curlicues

@article{Berry1988RenormalisationOC,
  title={Renormalisation of curlicues},
  author={Michael V Berry and J Goldberg},
  journal={Nonlinearity},
  year={1988},
  volume={1},
  pages={1-26}
}
The recursively spiralling patterns drawn in the complex plane by the values of SL( tau )= Sigma Ln=1 exp(i pi tau n2) as L to infinity with tau fixed in the range 0<or= tau <or=1, depend on the arithmetic of tau . A compendious understanding of the patterns is obtained by iterating an explicit asymptotic renormalisation transformation relating SL( tau ) to a similar sum, magnified by 1/ square root tau and rotated or reflected with a smaller number L tau of terms and a new parameter tau 1( tau… Expand
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References

SHOWING 1-10 OF 18 REFERENCES
Disorder, renormalizability, theta functions and Cornu spirals
Abstract The partial sums of the trigonometrical series S = Σ∞-∞ eiπαnp for α small mod 2 lead to distributions of points in the complex plane C composed of Cornu-like spirals. For p ≠ 2 and p > 1Expand
Quantization of linear maps on a torus-fresnel diffraction by a periodic grating
Abstract Quantization on a phase space q, p in the form of a torus (or periodized plane) with dimensions Δ q , Δ p requires the Planck's constant take one of the values h = ΔqΔp / N , where N is anExpand
Level clustering in the regular spectrum
  • M. Berry, M. Tabor
  • Mathematics
  • Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1977
In the regular spectrum of an f-dimensional system each energy level can be labelled with f quantum numbers originating in f constants of the classical motion. Levels with very different quantumExpand
Closed orbits and the regular bound spectrum
  • M. Berry, M. Tabor
  • Mathematics
  • Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1976
The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K.Expand
Semiclassical approximations in wave mechanics
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with weExpand
Uniform distribution modulo one: a geometrical viewpoint.
At an early stage of our work we tacitly assumed that the curve Γ (u) generated by an equidistributed sequence u would be unbounded. This is not the case, and it seems that there are no simpleExpand
Diffraction in crystals at high energies
By means of several distinct stages of approximation, the way in which wave propagation in a lattice becomes classical at high energies is analysed. First, the principle that deflection anglesExpand
Fractal Geometry of Nature
Phase transitions in the thermodynamic formalism of multifractals.
Les non-analyticites dans les dimensions generalisees d'ensembles multifractals d'interet physique sont interpretees comme des transitions de phase
Number theory
TLDR
The author briefly reviews earlier uses of number theory and then examines recent applications to music, cryptography, and error-correction codes. Expand
...
1
2
...