Benjamini-Schramm convergence of periodic orbits

Area of Science:

• Mathematics
• Dynamical Systems
• Geometric Group Theory

Background:

• Periodic orbits are fundamental in understanding the structure of dynamical systems.
• Benjamini-Schramm convergence provides a powerful framework for studying the statistical properties of geometric objects.
• Lie groups and their homogeneous spaces are central objects in modern mathematics.

Purpose of the Study:

• To develop a general criterion for Benjamini-Schramm convergence of periodic orbits.
• To apply this criterion to specific mathematical structures like homogeneous spaces and translation surfaces.
• To advance the understanding of convergence properties in geometric and dynamical settings.

Main Methods:

• Development of a novel criterion for Benjamini-Schramm convergence.
• Application of the criterion to the theory of Lie groups.
• Analysis of convergence for periodic orbits on homogeneous spaces.
• Investigation of convergence properties on the space of translation surfaces.

Main Results:

• A precise criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups is established.
• The criterion is successfully applied to yield new insights into homogeneous spaces.
• The convergence properties of periodic orbits on translation surfaces are analyzed using the new criterion.

Conclusions:

• The developed criterion offers a versatile tool for studying Benjamini-Schramm convergence in various mathematical contexts.
• This work deepens the understanding of the interplay between group theory, geometry, and dynamical systems.
• The results open avenues for further research into the statistical behavior of orbits in complex mathematical structures.