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Subexponential instability in one-dimensional maps implies infinite invariant measure.

Takuma Akimoto1, Yoji Aizawa

  • 1Department of Applied Physics, Advanced School of Science and Engineering, Waseda University, Okubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan. akimoto@z8.keio.jp

Chaos (Woodbury, N.Y.)
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Summary

Dynamical instability in weak chaos is defined as subexponential instability. This study shows that such systems possess infinite invariant measures and introduces a generalized Lyapunov exponent for characterization.

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Area of Science:

  • Dynamical systems theory
  • Chaos theory
  • Ergodic theory

Background:

  • Characterizing the dynamics of chaotic systems is crucial for understanding their behavior.
  • Weak chaos, a less turbulent form of chaos, presents unique challenges in dynamical analysis.
  • Subexponential instability offers a new perspective on the divergence of trajectories in these systems.

Purpose of the Study:

  • To define and characterize dynamical instability in weak chaos.
  • To establish a link between subexponential instability and the properties of invariant measures.
  • To introduce a novel mathematical tool for quantifying subexponential instability.

Main Methods:

  • Analysis of one-dimensional, conservative, ergodic measure-preserving maps.
  • Investigation of the properties of invariant measures in systems exhibiting subexponential instability.
  • Development and application of a generalized Lyapunov exponent.

Main Results:

  • Subexponential instability is formally characterized in weak chaos.
  • Systems with subexponential instability are shown to possess an infinite invariant measure.
  • A generalized Lyapunov exponent is presented as a tool to quantify subexponential instability.

Conclusions:

  • Subexponential instability provides a precise measure for weak chaos dynamics.
  • The existence of an infinite invariant measure is a key consequence of subexponential instability.
  • The generalized Lyapunov exponent offers a powerful method for analyzing these complex systems.