Distinguishability notion based on Wootters statistical distance: Application to discrete maps

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.