Pointwise convergence of Birkhoff averages for global observables

Area of Science:

• Dynamical Systems
• Ergodic Theory
• Mathematical Physics

Background:

• The Birkhoff Ergodic Theorem is trivial for infinite measure-preserving systems, stating averages are zero.
• Existing rescalings of Birkhoff sums do not yield non-degenerate pointwise limits in these systems.

Purpose of the Study:

• To develop a non-trivial version of Birkhoff's theorem for infinite measure-preserving dynamical systems.
• To introduce and analyze the concept of "global observables" as a new class of functions for this theorem.

Main Methods:

• Developed a generalized Birkhoff theorem using "global observables" in conservative, ergodic, infinite-measure-preserving systems.
• Introduced a hypothesis of "approximate partial averaging" for these observables.
• Illustrated the applicability of the core idea with a general example.

Main Results:

• Presented a main theorem applicable to general systems under the "approximate partial averaging" hypothesis.
• Demonstrated that the concept of global observables allows for non-degenerate limits where integrable functions fail.
• Explored the conditions under which a Birkhoff theorem can hold for infinite measure-preserving systems.

Conclusions:

• The study successfully extends Birkhoff's theorem to infinite measure-preserving systems using global observables.
• Further research is needed to identify the optimal class of observables for these systems.
• Counterexamples and simulations guide the ongoing investigation into optimal observable classes.