Ledrappier-Young formula and exact dimensionality of self-affine measures

TL;DR

This paper proves that all planar self-affine measures are exact dimensional and satisfy the Ledrappier-Young formula, resolving a long-standing open problem and extending results to higher dimensions under specific conditions.

Contribution

It establishes the exact dimensionality of self-affine measures on the plane and higher dimensions under certain assumptions, confirming the Ledrappier-Young formula.

Findings

All planar self-affine measures are exact dimensional.

Self-affine measures in higher dimensions are exact dimensional under certain conditions.

Measures satisfy the Ledrappier-Young formula in these cases.

Abstract

In this paper, we solve the long standing open problem on exact dimensionality of self-affine measures on the plane. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system. In higher dimensions, under certain assumptions, we prove that self-affine and quasi self-affine measures are exact dimensional. In both cases, the measures satisfy the Ledrappier-Young formula.