Generalized uniform laws for tied-down occupation times of infinite ergodic transformations

TL;DR

This paper proves a new limit theorem for occupation times in infinite ergodic systems, extending uniform laws to tied-down conditions and applying operator renewal theory, with implications for intermittent maps.

Contribution

It introduces a generalized limit law for occupation times under tied-down conditions in infinite ergodic transformations, expanding previous uniform distribution results.

Findings

Established a conditional limit theorem for occupation times.

Generalized the uniform distribution law for tied-down conditions.

Applicable to intermittent maps with multiple indifferent fixed points.

Abstract

We establish a conditional limit theorem for occupation times of infinite ergodic transformations under a tied-down condition, that is, the condition that the orbit returns to a reference set with finite measure at the final observation time. The class of limit distributions is the generalization of the uniform distribution which was discovered by M. Barlow, J. Pitman and M. Yor in [S\'eminaire de Probabilit\'es XXIII. Lecture Notes in Mathematics, volume 1372 (1989), 294--314]. For the proof we utilize operator renewal theory. Our result can be applied to intermittent maps with two or more indifferent fixed points.