---
abstract: 'Let $\{S_i\}_{i\in \Lambda}$ be a finite contracting affine iterated function system (IFS) on $\R^d$. Let $(\Sigma,\sigma)$ denote the two-sided full shift over the alphabet $\Lambda$, and $\pi:\Sigma\to \R^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic $\sigma$-invariant measure on $\Sigma$ under $\pi$ is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier-Young type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.'
address: |
     Department of Mathematics\
    The Chinese University of Hong Kong\
    Shatin, Hong Kong\
author:
- 'De-Jun Feng'
title: Dimension of invariant measures for affine iterated function systems
---

13.7pt

[^1]

Introduction {#S-1}
============

Motivation and the main result
------------------------------

Let $\R^{d\times d}$ denote the set of real $d\times d$ matrices. By an [*affine iterated function system*]{} (affine IFS) on $\R^d$ we mean a finite family ${\mathcal S}=\{S_j\}_{j\in \Lambda}$ of affine mappings from $\R^d$ to $\R^d$, taking the form $$\label{e-form}
S_j(x)=M_jx+a_j,\qquad j\in \Lambda,$$ where $M_j\in \R^{d\times d}$ and $a_j\in \R^d$. Here, in contrast to the usual definition of 