Canonical metrics of commuting maps

TL;DR

This paper proves that for two commuting maps on an algebraic variety, their canonical metrics, heights, and measures are identical, establishing a fundamental equivalence in their dynamical properties.

Contribution

It establishes the equality of canonical metrics, heights, and measures for commuting maps on algebraic varieties, a novel result in dynamical systems.

Findings

Canonical metrics of commuting maps are equal.

Canonical heights and measures associated with commuting maps are identical.

Provides a fundamental link between the dynamics of commuting maps.

Abstract

In the present work we establish the equality of the canonical metric of two commuting maps on an algebraic variety X. As a consequence the canonical height and measure associated to both maps are identical.