# Entropy and dimension of disintegrations of stationary measures

## Abstract

We extend a result of Ledrappier, Hochman, and Solomyak on exact dimensionality of stationary measures for to disintegrations of stationary measures for onto the one dimensional foliations of the space of flags obtained by forgetting a single subspace.

The dimensions of these conditional measures are expressed in terms of the gap between consecutive Lyapunov exponents, and a certain entropy associated to the group action on the one dimensional foliation they are defined on. It is shown that the entropies thus defined are also related to simplicity of the Lyapunov spectrum for the given measure on .

### 1. Introduction

It was shown by Ledrappier Reference Led84, Hochman and Solomyak Reference HS17, that if is a probability on the projective space of which is stationary with respect to a probability on with finite Lyapunov exponents, then is exact dimensional and its dimension is where is the Furstenberg entropy and is the largest Lyapunov exponent (hence is the gap between the two Lyapunov exponents).

Suppose now that is a probability on and is a probability on the space of flags in -stationary (i.e. pairs where , is a one dimensional subspace, and is a two dimensional subspace), which is a three-dimensional manifold.

We consider here the two foliations of the space of flags obtained by partitioning into sets of flags sharing the same one dimensional subspace on the one hand, and flags sharing the same two dimensional subspace on the other. These are foliations by circles, and furthermore the action of any invertible linear self mapping of preserves both foliations.

In this context we show that the conditional measures obtained by disintegrating with respect to these two foliations, are exact dimensional. Furthermore we express the dimension of these disintegrations in terms of the gap between consecutive Lyapunov exponents as well as two entropies Before establishing the dimension formula we show that the entropies . bound the gaps between exponents from below and therefore, in principle, yield a criteria for simplicity of the Lyapunov spectrum.

We prove our results in a slightly more general context, that of actions of on the space of complete flags in In this context there are . associated one dimensional foliations which correspond to “forgetting” the subspace of all flags for some -dimensional .

#### 1.1. Preliminaries

Let denote the singular values of an element with respect to the standard inner product.

We denote by the space of complete flags in an element , is of the form where is an subspace of -dimensional for each and for .

Let denote the space of flags missing their subspace. For a given complete flag -dimensional we denote by its projection to (i.e. the sequence obtained by removing from ).

We use the notation for equality in distribution between random elements and And . to mean that the probability is absolutely continuous with respect to .

If and are random elements taking values in complete separable metric spaces (a version of) the conditional distribution of given is a random probability -measurable on the range of such that

for all continuous bounded real functions (here the right-hand side is the conditional expectation of with respect to the generated by -algebra Such a conditional distribution is well defined up to sets of zero measure but we will abuse notation slightly referring to ‘the conditional distribution’. ).

It is always the case that there exists a Borel mapping from the range of to the space of probabilities on the range of such that is a version of the conditional distribution of given Fixing such a mapping one may speak of . for non-random in the range of .

The lower local dimension of a probability measure on a metric space at a point is defined by

while the upper local dimension is defined by

where

If the lower and upper dimensions of

#### 1.2. Statement of main results

Suppose that

and let

The existence of such a pair

The Lyapunov exponents

where

The Lyapunov exponents given by the multiplicative ergodic theorem of Reference Ose68 for a product of i.i.d. random matrices of distribution

Fix

In the case

Theorem 1 implies that the Lyapunov spectrum is simple (i.e. all exponents are different) if there does not exist a family of conditional probabilities

## Part 1. Entropy, mutual information, and Lyapunov exponent gaps

### 2. Entropy and mutual information

We will define below

The purpose of this section is to prove that:

This result reduces the problem of showing that

A general reference covering mutual information including Dobrushin’s theorem and the Gelfand-Yaglom-Perez theorem is Reference Pin64.

#### 2.1. Conditional mutual information

##### 2.1.1. Mutual information

Let

The mutual information between

where the supremum is over all finite partitions

Directly from the definition one sees that

By Jensen’s inequality

It was shown in Reference Dob59 that

It was shown in Reference GfY59 and Reference Per59 that if

Conversely, if

whether the right hand side is finite or not.

These results are usually called the Gelfand-Yaglom-Perez Theorem.

In our context, when

##### 2.1.2. Conditional mutual information

Let

The mutual information between

One still has

In general there is no relation between

To see this suppose for example that

On the other hand for any Markov chain

The following semi-continuity property holds:

The following monotonicity property follows immediately from the definition of mutual information

A more precise version of monotonicity is the following:

#### 2.2. Proof of Lemma 1

We will calculate the marginal distributions and the joint distribution of

To begin we simply let

By stationarity of

For the joint distribution notice that the distribution of

Hence the joint conditional distribution of