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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Statistical Mechanics
  • Information Theory

Background:

  • Wootters' distinguishability provides a framework for quantifying differences between quantum states.
  • Statistical distances are crucial for analyzing the behavior of dynamical systems, particularly chaotic maps.
  • Understanding the properties of invariant densities and wandering sets is key to characterizing map dynamics.

Purpose of the Study:

  • To extend Wootters' distinguishability notion to probability density functions for discrete maps.
  • To define a novel statistical distance metric (d¯) for arbitrary discrete maps.
  • To characterize the wandering set and identify dissipative regions in phase space using the d¯ metric.

Main Methods:

  • Adaptation of Wootters' distinguishability for probability density functions.
  • Development of a statistical distance metric (d¯) for discrete maps.
  • Association of metric spaces with invariant densities and analysis of their properties as iterations approach infinity.
  • Analytical and numerical investigation using logistic and circle maps.

Main Results:

  • A new metric d¯ is defined for arbitrary discrete maps.
  • Metric spaces associated with invariant densities are constructed, revealing distinguished points at infinite iterations.
  • The wandering set of maps is characterized using d¯, enabling identification of dissipative regions.
  • The metric is extended to arbitrary probability distributions, with applications to histogram analysis and the uncertainty principle.

Conclusions:

  • The d¯ metric offers a robust tool for analyzing statistical properties of chaotic maps.
  • The framework provides a novel way to understand dissipative behavior and phase space structure.
  • The study connects concepts from information theory and dynamical systems, with implications for statistical mechanics and quantum mechanics.