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Renormalizable two-parameter piecewise isometries.

J H Lowenstein1, F Vivaldi1

  • 1Department of Physics, New York University, 2 Washington Place, New York, New York 10003, USA.

Chaos (Woodbury, N.Y.)
|July 3, 2016
PubMed
Summary
This summary is machine-generated.

We found two renormalization scenarios for piecewise isometries. One scenario shows flexible self-similarity, while the other reveals a weak rigidity dependent on specific parameter fields.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Number Theory
  • Ergodic Theory

Background:

  • Piecewise isometries exhibit complex dynamics.
  • Renormalization is a key concept for understanding self-similarity in dynamical systems.
  • Previous work established renormalizability for one-parameter systems linked to algebraic number fields.

Purpose of the Study:

  • To investigate renormalization scenarios for two-parameter piecewise isometries.
  • To explore the emergence of new dynamical features with increased parameters.
  • To analyze the interplay between parameter rigidity and self-similarity.

Main Methods:

  • Analysis of two distinct renormalization scenarios for two-parameter piecewise isometries.
  • Utilizing the established renormalizability of a one-parameter triangle map.
  • Investigating parameter-dependent translations and rhombus rotations.
  • Drawing analogies with Rauzy-Veech renormalization of interval exchange transformations.

Main Results:

  • The first scenario demonstrates non-rigid renormalizability, allowing one parameter to vary continuously while the other remains in a specific algebraic field K=Q(5).
  • This non-rigidity is linked to the recombination of neighboring elements after distinct return paths.
  • The second scenario reveals a weak form of rigidity, where phase space splits into invariant components, but simultaneous self-similarity requires both parameters to belong to K.
  • The phenomenon of recombination is also observed in interval exchange transformations.

Conclusions:

  • Two-parameter piecewise isometries exhibit richer and more varied renormalization behaviors than their one-parameter counterparts.
  • The concept of rigidity in renormalization can be relaxed, allowing for continuous parameter variations under certain conditions.
  • The algebraic number field K=Q(5) plays a crucial role in maintaining self-similarity in these systems.
  • Analogies with interval exchange transformations provide further insight into these dynamical phenomena.