Construction of the Jordan basis for the Baker map

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Dynamical systems

Background:

• The Baker map and Bernoulli map are fundamental chaotic systems.
• Understanding their spectral properties is crucial for analyzing chaotic dynamics.
• Previous methods for spectral decomposition were complex.

Purpose of the Study:

• To construct Jordan canonical form basis states for chaotic maps.
• To develop a direct and simple method for spectral decomposition of the Frobenius-Perron operator.
• To explore the physical significance of Jordan states via entropy evolution.

Main Methods:

• Construction of Jordan canonical form basis states.
• Derivation of a recursion formula for Jordan states.
• Application of spectral decomposition to the Frobenius-Perron operator.
• Analysis of entropy evolution equations.

Main Results:

• A straightforward recursion formula for Jordan states and spectral decomposition was obtained.
• The new method is significantly more direct and simpler than subdynamics techniques.
• The physical significance of Jordan states was approached through entropy evolution.
• The method was successfully applied to the Bernoulli map, simplifying eigenstate derivation.

Conclusions:

• The developed method provides a more efficient approach to analyzing chaotic maps.
• Jordan states offer insights into the physical behavior of chaotic systems.
• This technique simplifies the study of spectral properties in dynamical systems.