Algorithmic information for interval maps with an indifferent fixed point and infinite invariant measure

Area of Science:

• Dynamical Systems Theory
• Information Theory
• Statistical Mechanics

Background:

• The Kolmogorov-Sinai entropy quantifies the average information rate for chaotic systems.
• Systems with null entropy exhibit complex behaviors where information does not increase linearly with time.
• Infinite natural invariant measures are crucial for understanding the long-term dynamics of certain systems.

Purpose of the Study:

• To generalize the concept of Kolmogorov-Sinai entropy for systems with null entropy.
• To investigate the information growth rate in dynamical systems with an infinite natural invariant measure.
• To characterize the sub-linear information increase in systems with an indifferent fixed point.

Main Methods:

• Analysis of a class of maps of the interval with an indifferent fixed point at the origin.
• Calculation of the average information required to describe the system's orbital behavior.
• Asymptotic analysis of the information growth with respect to time (n).

Main Results:

• A generalization of Kolmogorov-Sinai entropy is proposed for systems with null entropy.
• The average information necessary to describe orbital behavior increases sub-linearly with time, approximately as n^alpha.
• The exponent alpha is determined by the map's asymptotic behavior near the origin.

Conclusions:

• The study provides a new framework for analyzing information dynamics in low-entropy systems.
• The findings reveal a specific power-law relationship (n^alpha) governing information growth.
• The asymptotic behavior near an indifferent fixed point is key to understanding this information scaling.