Closure measures for coarse-graining of the tent map

Area of Science:

• Complex Systems
• Information Theory
• Dynamical Systems Theory

Background:

• Dynamical systems exhibit complex behaviors that can be simplified using symbolic representations.
• Understanding the relationship between continuous dynamics and discrete symbolic representations is crucial for analyzing emergent properties.

Purpose of the Study:

• To quantify the relationship between the dynamics of a time-discrete system (tent map) and its symbolic representation.
• To evaluate and compare closure measures for identifying emergent macro-levels.
• To derive all Markov partitions for the tent map system.

Main Methods:

• Utilizing information theory to analyze the dynamics of the tent map and its iterations.
• Applying a threshold crossing technique to generate symbolic dynamics (binary strings).
• Quantitatively evaluating closure measures based on partition points.

Main Results:

• Developed a novel information-theoretical approach to link system dynamics and symbolic representations.
• Successfully derived all Markov partitions with full support for the tent map system.
• Demonstrated examples of Markovian symbol dynamics with non-Markovian partitions and hierarchies.

Conclusions:

• The information-theoretical framework provides a robust method for analyzing discrete dynamical systems and their symbolic counterparts.
• This approach enables a deeper understanding of emergent macro-levels and the identification of Markov partitions.
• The findings offer new insights into the construction of symbolic dynamics and their properties.