Pesin's dimension for Poincare recurrences

Area of Science:

• Dynamical Systems and Chaos Theory
• Ergodic Theory

Background:

• Poincare recurrences are fundamental in understanding the long-term behavior of dynamical systems.
• Existing measures may not fully capture the complexity of return times in diverse systems.

Purpose of the Study:

• Introduce a novel characteristic for Poincare recurrences.
• Develop a general framework for dimension-like characteristics.
• Analyze the behavior of this new characteristic in specific dynamical systems.

Main Methods:

• Introduction of a new average return time characteristic.
• Application of a general construction for dimension-like characteristics.
• Case studies including rotations on the circle and the Denjoy example.

Main Results:

• A new characteristic quantifying average return times in dynamical systems is presented.
• The framework accommodates various systems, including well-known examples.
• The study provides insights into the properties of Poincare recurrences.

Conclusions:

• The new characteristic offers a valuable tool for analyzing Poincare recurrences.
• The general construction framework is applicable to a range of dynamical systems.
• Further research can explore this characteristic in more complex systems.