# Aspects of convergence of random walks on finite volume homogeneous spaces

Roland Prohaska  Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092
Zürich, Switzerland<EMAIL_ADDRESS>
(Date: )

###### Abstract.

We investigate three aspects of weak* convergence of the $n$-step
distributions of random walks on finite volume homogeneous spaces $G/\Gamma$
of semisimple real Lie groups. First, we look into the obvious obstruction to
the upgrade from Cesàro to non-averaged convergence: periodicity. We give
examples where it occurs and conditions under which it does not. In a second
part, we prove convergence towards Haar measure with exponential speed from
almost every starting point. Finally, we establish a strong uniformity
property for the Cesàro convergence towards Haar measure for uniquely ergodic
random walks.

###### Key words and phrases:

Random walk, homogeneous space, aperiodic, spectral gap, recurrence

###### 2010 Mathematics Subject Classification:

Primary 60B15; Secondary 22F30, 60G50, 22E30

##  1\. Introduction

Let $G$ be a real Lie group and $\Gamma$ a lattice in $G$, that is, a discrete
subgroup of $G$ such that the homogeneous space $X=G/\Gamma$ admits a
$G$-invariant Borel probability measure $m_{X}$. This measure $m_{X}$ is
unique and we refer to it as the (normalized) _Haar measure_ on $X$. A good
example to have in mind is $G=\operatorname{SL}_{d}(\mathbb{R})$ and
$\Gamma=\operatorname{SL}_{d}(\mathbb{Z})$.

The objects of study in this paper are _random walks_ on $X$, given by
probability measures $\mu$ on $G$: A step corresponds to randomly choosing a
group element $g\in G$ according to $\mu$ and then moving from the current
location $X\ni x$ to $gx$. Starting at $x_{0}\in X$, the distribution of the
location after $n$ steps is given by the convolution

$\displaystyle\mu^{*n}*\delta_{x_{0}},$

which is the push-forward of the product measure $\mu^{\otimes
n}\otimes\delta_{x_{0}}$ under the multiplication map $G^{n}\times
X\ni(g_{n},\dots,g_{1},x)\mapsto g_{n}\dotsm g_{1}x\in X$.

Understanding the limiting behavior of these $n$-step distributions is a
notoriously difficult problem, investigated e.g. by Eskin–Margulis in their
work on non-divergence [13], and Benoist–Quint in their series of articles [2,
4, 5, 6]. We reproduce one of the main results of [6] as motivating example.
For the statement, recall that a probability measure $\nu$ on $X$ is called
_homogeneous_ if there exists a closed subgroup $H$ of $G$ and a point $x\in
X$ such that $\operatorname{supp}(\nu)=Hx$ is a closed orbit and $\nu$ is
$H$-invariant.

######  Theorem 1.1 (Benoist–Quint [6]).

Let $\mu$ be a compactly supported probability measure on $G$. Denote by
$\mathcal{S}$ and $\mathcal{G}$ the closed subsemigroup and subgroup of $G$
generated by $\operatorname{supp}(\mu)$, respectively, and suppose that the
Zariski closure of $\operatorname{Ad}(\mathcal{G})$ in
$\operatorname{Aut}(\mathfrak{g})$ is Zariski connected, semisimple, and has
no compact factors. Then for every $x_{0}\in X$ there is a homogeneous
probability measure $\nu_{x_{0}}$ on $X$ with
$\operatorname{supp}(\nu_{x_{0}})=\overline{\mathcal{S}x_{0}}=\overline{\mathcal{G}x_{0}}$
and such that

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x_{0}}\longrightarrow\nu_{x_{0}}$
(1.1)

as $n\to\infty$ in the weak* topology.

Here the weak* convergence (1.1) more explicitly means that for every
compactly supported continuous function $f\in C_{c}(X)$ we have

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\int_{X}f\mathop{}\\!\mathrm{d}(\mu^{*k}*\delta_{x_{0}})=\frac{1}{n}\sum_{k=0}^{n-1}\int_{G^{k}}f(g_{k}\dotsm
g_{1}x_{0})\mathop{}\\!\mathrm{d}\mu^{\otimes
k}(g_{1},\dots,g_{k})\longrightarrow\int_{X}f\mathop{}\\!\mathrm{d}\nu_{x_{0}}$

as $n\to\infty$.

Some questions left open by the above result are listed by Benoist–Quint at
the end of their survey [3]. A major one is the following.

######  Question 1.2.

In the setting of Theorem 1.1, is it also true that

$\displaystyle\mu^{*n}*\delta_{x_{0}}\longrightarrow\nu_{x_{0}}$ (1.2)

as $n\to\infty$?

Answers are available only in special cases: Breuillard [9] established (1.2)
for certain measures $\mu$ supported on unipotent subgroups, Buenger [10]
proved it for some sparse solvable measures, and in previous work the author
dealt with the case of spread out measures [16].

The purpose of this article is to discuss three (largely independent) aspects
of random walk convergence related to Theorem 1.1 and Question 1.2, mainly
having in mind the case that $G$ is a semisimple real Lie group. We are going
to use the following terminology.

######  Definition 1.3.

Let $\nu$ be a probability measure on $X$ and $x_{0}\in X$. We say that the
random walk on $X$ given by $\mu$ _converges to $\nu$ on average_ (resp.
_converges to $\nu$_) from the starting point $x_{0}$ if
$\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x_{0}}\rightarrow\nu$ (resp.
$\mu^{*n}*\delta_{x_{0}}\rightarrow\nu$) as $n\to\infty$ in the weak*
topology.

Convergence on average is also commonly referred to as _Cesàro convergence_.
We use the two terms interchangeably.

The article is organized as follows.

In §2, we look into the obvious obstruction to the upgrade from Cesàro
convergence to (non-averaged) convergence: periodicity. We show in Example 2.1
how (1.2) can fail when $x_{0}$ has finite orbit under $\mathcal{S}$. Using a
product construction, we can also produce a counterexample in which the orbit
closure $\overline{\mathcal{S}x_{0}}$ has positive dimension (Example 2.2). In
both cases, the periodic behavior occurs at the level of the connected
components of the orbit closure. As it turns out, this is no coincidence: If,
in the setting of Theorem 1.1, the orbit closure $\overline{\mathcal{S}x_{0}}$
is connected, there can be no periodicity (Theorem 2.5) and we can show that
the Cesàro convergence (1.1) also holds along arithmetic progressions
(Corollary 2.7).

In §3, we establish effective convergence of random walks to the normalized
Haar measure $m_{X}$ for typical starting points $x_{0}$: When
$\operatorname{supp}(\mu)$ generates a Zariski dense subgroup of a semisimple
real Lie group $G$ without compact factors, for any fixed $L^{2}$-function $f$
on $X$ the convergence

$\displaystyle\int_{X}f\mathop{}\\!\mathrm{d}(\mu^{*n}*\delta_{x_{0}})\overset{n\to\infty}{\longrightarrow}\int_{X}f\mathop{}\\!\mathrm{d}m_{X}$

not only holds but is in fact exponentially fast for $m_{X}$-almost every
$x_{0}\in X$ (Theorem 3.2, Proposition 3.4). The proof relies on an
$L^{2}$-spectral gap of the convolution operator

$\displaystyle\pi(\mu)\colon
f\mapsto\Bigl{(}x\mapsto\int_{G}f(gx)\mathop{}\\!\mathrm{d}\mu(g)\Bigr{)}$

acting on measurable functions on $X$. Taking into account regularity of the
function $f$, the above can be further strengthened to the statement that
almost every $x\in X$ is _exponentially generic_ (Definition 3.12): Up to a
constant factor depending on derivatives of $f$, the exponential speed of
convergence holds uniformly over all compactly supported smooth functions
(Theorem 3.13). Key to this upgrade are the definition of suitable Sobolev
norms and a functional analytic argument involving relative traces, first
exploited in a dynamical context by Einsiedler–Margulis–Venkatesh [11].

Finally, in §4 we prove that convergence on average to $m_{X}$ happens locally
uniformly in $x_{0}$ in a strong way when the random walk is uniquely ergodic
and admits a Lyapunov function (Theorem 4.13). For example, this is the case
when $G$ is a connected semisimple real algebraic group and
$\operatorname{supp}(\mu)$ generates a non-discrete Zariski dense subgroup,
and also in the setup of Simmons–Weiss [20], which has connections to
Diophantine approximation problems on fractals. To this end, we introduce the
new concept of _$(K_{n})_{n}$ -uniform recurrence_ (Definition 4.10), which
refines recurrence properties of random walks previously studied in [4, 13].

### Standing Assumptions & Notation

As many of our arguments work in greater generality, in the remainder of the
article we will relax the assumptions stated at the beginning of this
introduction. The following setup shall be in place whenever nothing else is
specified: $G$ is a locally compact $\sigma$-compact metrizable group acting
ergodically on a locally compact $\sigma$-compact metrizable space $X$ endowed
with a $G$-invariant probability measure $m_{X}$; and $\mu$ is a Borel
probability measure on $G$.

### Acknowledgments

The author is grateful to Andreas Wieser for valuable comments on preliminary
versions of the article, and to Manfred Einsiedler for explaining how relative
traces can be used to make separability effective. Thanks also go to HIM Bonn
and the organizers of the trimester program “Dynamics: Topology and Numbers”,
in the course of which parts of this manuscript were completed, for
hospitality and providing an excellent working environment.

##  2\. Periodicity

In this section, we start with two simple counterexamples to (1.2), which
illustrate ways in which a random walk may exhibit periodic behavior (§2.1).
Analyzing these examples for their common feature, we are led to a simple
condition ensuring aperiodicity, stated and proved in §2.2.

###  2.1. Examples

The first example with periodicity is on finite periodic orbits. In the
following, for $d\geq 2$ we denote by $\mathbf{1}_{d}$ the $d\times
d$-identity matrix.

######  Example 2.1.

Consider the _principal congruence lattice_

$\displaystyle\Gamma=\Gamma(2)=\\{g\in\operatorname{SL}_{2}(\mathbb{Z})\nonscript\>|\allowbreak\nonscript\>\mathopen{}g\equiv\mathbf{1}_{2}\bmod
2\\}$

in $G=\operatorname{SL}_{2}(\mathbb{R})$. Being the kernel of the reduction
homomorphism from $\operatorname{SL}_{2}(\mathbb{Z})$ to
$\operatorname{SL}_{2}(\mathbb{Z}/2\mathbb{Z})$, we recognize $\Gamma(2)$ as a
finite-index normal subgroup of $\operatorname{SL}_{2}(\mathbb{Z})$. In
particular, $\Gamma(2)$ is a lattice in $G$. Let
$\mu=\tfrac{1}{2}(\delta_{h_{1}}+\delta_{h_{2}})$ with

$\displaystyle h_{1}=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},\,h_{2}=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}.$

Then the closed subgroup $\mathcal{G}$ generated by
$\operatorname{supp}(\mu)=\\{h_{1},h_{2}\\}$ is
$\mathcal{G}=\operatorname{SL}_{2}(\mathbb{Z})$, which is Zariski dense in
$G$. The $\mathcal{G}$-orbit of $x_{0}=\mathbf{1}_{2}\Gamma\in G/\Gamma$ is

$\displaystyle\mathcal{O}=\Bigl{\\{}$ $\displaystyle
x_{0},h_{1}x_{0},h_{2}x_{0},h_{2}h_{1}x_{0}=\bigl{(}\begin{smallmatrix}1&1\\\
1&2\end{smallmatrix}\bigr{)}x_{0},h_{1}h_{2}x_{0}=\bigl{(}\begin{smallmatrix}2&1\\\
1&1\end{smallmatrix}\bigr{)}x_{0},$ $\displaystyle
h_{1}h_{2}h_{1}x_{0}=h_{2}h_{1}h_{2}x_{0}=\bigl{(}\begin{smallmatrix}2&-1\\\
1&0\end{smallmatrix}\bigr{)}x_{0}\Bigr{\\}},$

with transitions as shown in the following diagram:

${h_{2}h_{1}x_{0}}$${h_{1}x_{0}}$${h_{1}h_{2}h_{1}x_{0}}$${x_{0}}$${h_{1}h_{2}x_{0}}$${h_{2}x_{0}}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{1}}$$\scriptstyle{h_{1}}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{1}}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{1}}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{1}}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{1}}$

Consequently, we see that the random walk with starting point $x_{0}$
alternates between the two sets

$\displaystyle\mathcal{O}_{1}=\\{x_{0},h_{1}h_{2}x_{0},h_{2}h_{1}x_{0}\\}\text{
and }\mathcal{O}_{2}=\\{h_{1}x_{0},h_{2}x_{0},h_{1}h_{2}h_{1}x_{0}\\}.$

The $2$-step random walks on these sets constitute irreducible, aperiodic,
finite state Markov chains, so that

$\displaystyle\mu^{*2n}*\delta_{x_{0}}$
$\displaystyle\longrightarrow\frac{1}{3}\sum_{p\in\mathcal{O}_{1}}\delta_{p},$
$\displaystyle\mu^{*(2n+1)}*\delta_{x_{0}}$
$\displaystyle\longrightarrow\frac{1}{3}\sum_{p\in\mathcal{O}_{2}}\delta_{p},$

as $n\to\infty$ in the weak* topology. ∎

In the example above, the support of $\mu$ generates a Zariski dense subgroup
of $G$ and the lattice $\Gamma$ in $G$ is irreducible. (Recall that, loosely
speaking, “irreducibility” of $\Gamma$ means that it does not arise from a
product construction, cf. [17, Definition 5.20].) By the work of Benoist–Quint
([6, Corollary 1.8]), these properties force any orbit closure
$\overline{\mathcal{S}x_{0}}$ to be either finite or all of $X$. As soon as
intermediate orbit closures are possible, however, one can also construct
examples with periodic behavior on non-discrete orbit closures.

######  Example 2.2.

Let $G$, $\Gamma$, $X=G/\Gamma$, $h_{1},h_{2}$, $x_{0}$ and $\mathcal{G}$ be
as in Example 2.1 and choose a diagonal matrix
$a\in\operatorname{SL}_{2}(\mathbb{R})$ such that the diagonal entries of
$a^{2}$ are irrational. We are going to consider the random walk on the
product space

$\displaystyle X\times X=(G\times G)/(\Gamma\times\Gamma)$

given by the probability measure
$\mu=\tfrac{1}{4}\sum_{i=1}^{4}\delta_{g_{i}}$ on $G\times G$ with

$\displaystyle g_{1}$
$\displaystyle=(h_{1},ah_{1}a^{-1}),\,g_{2}=(h_{1},\mathbf{1}_{2}),$
$\displaystyle g_{3}$
$\displaystyle=(h_{2},ah_{2}a^{-1}),\,g_{4}=(h_{2},\mathbf{1}_{2}).$

The (closed) subgroup generated by the support of this measure $\mu$ is given
by $\mathcal{G}\times
a\mathcal{G}a^{-1}=\operatorname{SL}_{2}(\mathbb{Z})\times
a\operatorname{SL}_{2}(\mathbb{Z})a^{-1}$. Indeed, the correct entry in the
second copy of $G$ can be arranged using a finite product of $g_{1}^{\pm
1},g_{3}^{\pm 1}$, and then the entry in the first copy can be corrected using
$g_{2}^{\pm 1},g_{4}^{\pm 1}$. By Theorem 1.1 we thus know that for the
starting point $(x_{0},x_{0})\in X\times X$ we have the weak* convergence

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{(x_{0},x_{0})}\longrightarrow\nu_{(x_{0},x_{0})}$

as $n\to\infty$, where $\nu_{(x_{0},x_{0})}$ is the homogeneous probability
measure on the closure of the $\mathcal{G}\times a\mathcal{G}a^{-1}$-orbit of
$(x_{0},x_{0})$. (Recall that it makes no difference for the closure whether
one considers the orbit under the generated subgroup or subsemigroup.)

Let us identify this orbit closure. In the first copy of $X$, we recognize the
finite orbit $\mathcal{O}$ from Example 2.1. In the second copy, we see the
action of irrational conjugates of $h_{1},h_{2}$. As the acting group has
product structure, the orbit closure in question is the product of these two
orbit closures in the components:

$\displaystyle\overline{(\mathcal{G}\times
a\mathcal{G}a^{-1})(x_{0},x_{0})}=\mathcal{O}\times\overline{a\mathcal{G}a^{-1}x_{0}}.$

Since the orbit $a\mathcal{G}a^{-1}x_{0}$ is infinite by our choice of the
matrix $a$, it follows from [6, Corollary 1.8] that
$\overline{a\mathcal{G}a^{-1}x_{0}}=X$, so that

$\displaystyle\overline{(\mathcal{G}\times
a\mathcal{G}a^{-1})(x_{0},x_{0})}=\mathcal{O}\times X\text{ and
}\nu_{(x_{0},x_{0})}=m_{\mathcal{O}}\otimes m_{X}$

for the normalized counting measure $m_{\mathcal{O}}$ on $\mathcal{O}$ and the
normalized Haar measure $m_{X}$ on $X$. However, in analogy to Example 2.1,
the random walk is found to alternate between the sets

$\displaystyle\mathcal{O}_{1}\times X\text{ and }\mathcal{O}_{2}\times X,$

in the sense that
$\operatorname{supp}(\mu^{*2n}*\delta_{(x_{0},x_{0})})\subset\mathcal{O}_{1}\times
X$ and
$\operatorname{supp}(\mu^{*(2n+1)}*\delta_{(x_{0},x_{0})})\subset\mathcal{O}_{2}\times
X$ for all $n\in\mathbb{N}$. Hence, we conclude that the random walk starting
from $(x_{0},x_{0})$ does not converge to $\nu_{(x_{0},x_{0})}$. ∎

######  Remark 2.3.

The same behavior as in the previous example can be arranged inside a
homogeneous space $X^{\prime}=G^{\prime}/\Gamma^{\prime}$ that is the quotient
of a semisimple real Lie group $G^{\prime}$ by an irreducible lattice
$\Gamma^{\prime}$. Indeed, this is only a matter of choosing suitable
embeddings $G\times G\hookrightarrow G^{\prime}$ and $X\times X\hookrightarrow
X^{\prime}$, where $G$ and $X$ are as in Example 2.2. Concretely, one can e.g.
consider the $4\times 4$-congruence lattice

$\displaystyle\Gamma^{\prime}=\Gamma(2)=\\{g\in\operatorname{SL}_{4}(\mathbb{Z})\nonscript\>|\allowbreak\nonscript\>\mathopen{}g\equiv\mathbf{1}_{4}\bmod
2\\}$

in $G^{\prime}=\operatorname{SL}_{4}(\mathbb{R})$ and the diagonal embeddings

$\displaystyle\begin{aligned} G\times G&\hookrightarrow G^{\prime},\\\
(g,h)&\mapsto\begin{pmatrix}g&\\\
&h\end{pmatrix},\end{aligned}\qquad\begin{aligned} X\times X&\hookrightarrow
X^{\prime},\\\ (g\Gamma,h\Gamma)&\mapsto\begin{pmatrix}g&\\\
&h\end{pmatrix}\Gamma^{\prime}.\end{aligned}$

We therefore see that Example 2.2, i.e. periodic behavior on a non-discrete
orbit closure, can be realized inside $X^{\prime}=G^{\prime}/\Gamma^{\prime}$.
Of course, after applying this embedding, the subgroup generated by the
support of $\mu$ will no longer be Zariski dense in $G^{\prime}$. ∎

###  2.2. An Aperiodicity Criterion

Inspecting the examples above, one may notice that their common salient
feature is that the orbit closure $\overline{\mathcal{S}x_{0}}$ is
disconnected. This naturally raises the question whether periodic behavior can
also occur when this orbit closure is connected. In what follows, we answer
this question in the negative. We shall use the following formalization of
periodicity.

######  Definition 2.4.

Assume that the random walk on $X$ given by $\mu$ converges on average to a
probability measure $\nu$ on $X$ from the starting point $x_{0}\in X$. We say
that this convergence is _periodic_ if there exists an integer $d\geq 2$ and
pairwise disjoint measurable subsets $D_{0},\dots,D_{d-1}\subset X$ with
$\nu(\partial D_{i})=0$ for $0\leq i<d$ and such that
$(\mu^{*n}*\delta_{x_{0}})(D_{n\bmod d})=1$ for every $n\in\mathbb{N}$.
Otherwise, we call the convergence _aperiodic_.

The requirement on the boundaries of the sets $D_{i}$ is needed to ensure that
the cyclic behavior is witnessed by the limit measure $\nu$. Without a
condition of this sort, one could try to artificially define $D_{i}$ as the
set of all points in $X$ that can be reached from $x_{0}$ precisely in
$n\equiv i\bmod d$ steps. Indeed, this construction is possible for example
when $\mu$ is finitely supported with the property that its support freely
generates a discrete subsemigroup $\mathcal{S}$ of $G$ and the starting point
$x_{0}\in X$ has a free $\mathcal{S}$-orbit. The latter is the case e.g. for
$X=\operatorname{SL}_{2}(\mathbb{R})/\operatorname{SL}_{2}(\mathbb{Z})$,
$\mu=\tfrac{1}{2}(\delta_{h_{1}}+\delta_{h_{2}})$ with
$h_{1}=(\begin{smallmatrix}1&2\\\ 0&1\end{smallmatrix})$ and
$h_{2}=(\begin{smallmatrix}1&0\\\ 2&1\end{smallmatrix})$, and
$x_{0}=a\operatorname{SL}_{2}(\mathbb{Z})$ for a diagonal matrix
$a\in\operatorname{SL}_{2}(\mathbb{R})$ such that the diagonal entries of
$a^{2}$ are irrational.

We are now ready to state the announced aperiodicity theorem.

######  Theorem 2.5.

Retain the notation and assumptions from Theorem 1.1 and let $x_{0}\in X$ be
such that the orbit closure $\overline{\mathcal{S}x_{0}}$ is connected. Then
the Cesàro convergence to $\nu_{x_{0}}$ of the random walk on $X$ given by
$\mu$ starting from $x_{0}$ is aperiodic.

For the proof we need the following simple lemma.

######  Lemma 2.6.

Let $H$ be a Zariski connected real algebraic group and $S$ a subset of $H$
generating a Zariski dense subsemigroup. Then for every $d\in\mathbb{N}$, also
the $d$-fold product set $S^{d}=\\{g_{d}\dotsm
g_{1}\nonscript\>|\allowbreak\nonscript\>\mathopen{}g_{1},\dots,g_{d}\in S\\}$
generates a Zariski dense subsemigroup of $H$. In particular, if
$\operatorname{supp}(\mu)$ generates a Zariski dense subsemigroup for some
probability measure $\mu$ on $H$, the same is true for
$\operatorname{supp}(\mu^{*d})$.

###### Proof.

Let $U\subset H$ be a non-empty Zariski open subset and consider the map
$\phi\colon H\to H,\,g\mapsto g^{d}$. Since $\phi$ is Zariski continuous,
$\phi^{-1}(U)$ is Zariski open. Moreover, this preimage is non-empty because
$U$ is dense in the Lie group topology and $\phi$ is a diffeomorphism near the
identity. By the assumption that $S$ generates a Zariski dense subsemigroup,
we can thus find an element $g\in\phi^{-1}(U)$ that is the product of finitely
many elements of $S$. It follows that $\phi(g)=g^{d}$ lies in the intersection
of $U$ with the subsemigroup generated by $S^{d}$.

The second claim involving $\mu$ immediately follows from the above together
with the inclusion
$\operatorname{supp}(\mu^{*d})\supset\operatorname{supp}(\mu)^{d}$. ∎

###### Proof of Theorem 2.5.

Suppose $d\in\mathbb{N}$ is an integer such that there are pairwise disjoint
$D_{0},\dots,D_{d-1}\subset X$ with $\nu_{x_{0}}(\partial D_{i})=0$ for all
$0\leq i<d$ and such that $(\mu^{*n}*\delta_{x_{0}})(D_{n\bmod d})=1$ for all
$n\in\mathbb{N}$ as in the definition of periodicity. We have to show that
$d=1$.

First note that from Theorem 1.1 and the properties of the sets $D_{i}$ it
follows that

$\displaystyle\nu_{x_{0}}(D_{0})=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(\mu^{*k}*\delta_{x_{0}})(D_{0})=\frac{1}{d},$
(2.1)

where the application of weak* convergence to the set $D_{0}$ is justified
since it has negligible boundary with respect to the limit measure
$\nu_{x_{0}}$. In view of Lemma 2.6, Theorem 1.1 also applies to the $d$-step
random walk given by $\mu^{*d}$. Assuming for the moment that the limit
measure for this $d$-step random walk starting from $x_{0}$ coincides with
$\nu_{x_{0}}$, we deduce that

$\displaystyle\nu_{x_{0}}(D_{0})=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(\mu^{*dk}*\delta_{x_{0}})(D_{0})=1.$
(2.2)

Together, (2.1) and (2.2) imply $d=1$, the desired conclusion.

It thus remains to show that the $d$-step random walk starting from $x_{0}$
does indeed have the same limit measure as the $1$-step random walk. Denoting
by $\mathcal{S}$ and $\mathcal{S}_{d}$ the closed subsemigroups of $G$
generated by $\operatorname{supp}(\mu)$ and $\operatorname{supp}(\mu^{*d})$,
respectively, this statement is equivalent to the equality
$\overline{\mathcal{S}x_{0}}=\overline{\mathcal{S}_{d}x_{0}}$ of orbit
closures. To prove this, let $g\in\operatorname{supp}(\mu)$ be arbitrary. We
claim that

$\displaystyle\overline{\mathcal{S}x_{0}}=\bigcup_{k=0}^{d-1}g^{-k}\overline{\mathcal{S}_{d}x_{0}}.$

Indeed, since $\overline{\mathcal{S}x_{0}}$ is homogeneous, it is invariant
under the group generated by $\mathcal{S}$. As $\overline{\mathcal{S}x_{0}}$
clearly contains $\overline{\mathcal{S}_{d}x_{0}}$, the inclusion “$\supset$”
follows. For the reverse inclusion let
$g_{n},\dots,g_{1}\in\operatorname{supp}(\mu)$ for some $n\in\mathbb{N}$.
Choose $0\leq k<d$ such that $n+k\equiv 0\bmod d$. Then $g^{k}g_{n}\dotsm
g_{1}x_{0}\in\overline{\mathcal{S}_{d}x_{0}}$ and hence $g_{n}\dotsm
g_{1}x_{0}\in g^{-k}\overline{\mathcal{S}_{d}x_{0}}$, giving the claim.

We already noted that Theorem 1.1 applies to $\mu^{*d}$. In particular, the
orbit closure $\overline{\mathcal{S}_{d}x_{0}}$ and its translates by
$g^{-k}$, $0\leq k<d$, are submanifolds of $\overline{\mathcal{S}x_{0}}$.
Necessarily, all these translates have the same dimension, and since together
they make up $\overline{\mathcal{S}x_{0}}$ by the claim above, their shared
dimension coincides with that of $\overline{\mathcal{S}x_{0}}$. This implies
that $\overline{\mathcal{S}_{d}x_{0}}$ is open in
$\overline{\mathcal{S}x_{0}}$. However, it is also closed, so that the assumed
connectedness of $\overline{\mathcal{S}x_{0}}$ forces
$\overline{\mathcal{S}x_{0}}=\overline{\mathcal{S}_{d}x_{0}}$. This completes
the proof. ∎

We end this section by recording a corollary of the proof above.

######  Corollary 2.7.

Retain the notation and assumptions from Theorem 1.1 and suppose that
$\overline{\mathcal{S}x_{0}}$ is connected. Let $d\in\mathbb{N}$ and denote by
$\mathcal{S}_{d}$ the closed subsemigroup of $G$ generated by
$\operatorname{supp}(\mu^{*d})$. Then
$\overline{\mathcal{S}x_{0}}=\overline{\mathcal{S}_{d}x_{0}}$, and for the
homogeneous probability measure $\nu_{x_{0}}$ on this orbit closure we have
for arbitrary $r\in\mathbb{N}_{0}$ that

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*(dk+r)}*\delta_{x_{0}}\longrightarrow\nu_{x_{0}}$
(2.3)

as $n\to\infty$ in the weak* topology.

###### Proof.

The statement about orbit closures was established as part of the proof of
Theorem 2.5. From Theorem 1.1 we thus get the weak* convergence

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*dk}*\delta_{x_{0}}\overset{n\to\infty}{\longrightarrow}\nu_{x_{0}},$
(2.4)

which is (2.3) for $r=0$. Given $f\in C_{c}(X)$, the general case follows by
applying (2.4) to the compactly supported continuous function $f_{r}$ defined
by

$\displaystyle
f_{r}(x)\coloneqq\int_{G}f(gx)\mathop{}\\!\mathrm{d}\mu^{*r}(g)=\int_{G^{r}}f(g_{r}\dotsm
g_{1}x)\mathop{}\\!\mathrm{d}\mu^{\otimes r}(g_{1},\dots,g_{r})$

for $x\in X$. ∎

This corollary sharpens the convergence statement in Theorem 1.1 in the case
of a connected orbit closure: The Cesàro convergence to $\nu_{x_{0}}$ holds
along arbitrary arithmetic progressions. Although this does not provide an
answer to Question 1.2, it at least allows the following conclusion to be
drawn: If $(n_{i})_{i}$ is a sequence of indices such that
$\mu^{*n_{i}}*\delta_{x_{0}}$ converges to a weak* limit different from
$\nu_{x_{0}}$ as $i\to\infty$, then $(n_{i})_{i}$ cannot contain a density $1$
subset of an infinite arithmetic progression.

##  3\. Spectral Gap

In this section, we will explain how a spectral gap of the convolution
operator $\pi(\mu)$ associated to a random walk entails the convergence of
$\mu^{*n}*\delta_{x}$ towards $m_{X}$ for $m_{X}$-a.e. $x\in X$. In its
simplest form, the involved argument works in great generality and also
produces an exponential rate of convergence from almost every starting point
when the test function $f$ is fixed. This is done in §3.1. The following
subsections §3.2–§3.4 are dedicated to a substantial refinement of this
spectral gap argument for random walks on homogeneous spaces of real Lie
groups, making the exponentially fast convergence uniform over smooth test
functions.

###  3.1. Generic Points

Recall that $\pi(\mu)\colon L^{\infty}(X,m_{X})\to L^{\infty}(X,m_{X})$ is
defined by

$\displaystyle\pi(\mu)f(x)\coloneqq\int_{X}f\mathop{}\\!\mathrm{d}(\mu*\delta_{x})=\int_{G}f(gx)\mathop{}\\!\mathrm{d}\mu(g)$

for $f\in L^{\infty}(X,m_{X})$ and $x\in X$, and that it extends to a
continuous contraction on each $L^{p}$-space (see [7, Corollary 2.2]). We
shall study its behavior on $L^{2}(X,m_{X})$. By ergodicity, the $G$-fixed
functions are the constant functions, so we restrict our attention to their
orthogonal complement $L_{0}^{2}(X,m_{X})$ of $L^{2}$-functions with mean $0$.

######  Definition 3.1.

We say that $\mu$ has a _spectral gap_ on $X$ if the associated convolution
operator $\pi(\mu)$ restricted to $L_{0}^{2}(X,m_{X})$ has spectral radius
strictly less than $1$.

We note that by the spectral radius formula, $\mu$ having a spectral gap on
$X$ can be reformulated as the requirement that

$\displaystyle\lim_{n\to\infty}\sqrt[n]{\lVert\pi(\mu)|^{n}_{L_{0}^{2}}\rVert_{\mathrm{op}}}<1.$

Given the existence of a spectral gap, we obtain an almost everywhere
convergence result in a quite general setup.

######  Theorem 3.2.

Suppose that $\mu$ has a spectral gap on $X$. Then $m_{X}$-a.e. $x\in X$ is
_generic_ for the random walk on $X$ given by $\mu$, meaning that

$\displaystyle\mu^{*n}*\delta_{x}\longrightarrow m_{X}$

as $n\to\infty$ in the weak* topology. This convergence is exponentially fast
in the sense that for every fixed $f\in L^{2}(X,m_{X})$ we have

$\displaystyle\limsup_{n\to\infty}\left\lvert\int_{X}f\mathop{}\\!\mathrm{d}(\mu^{*n}*\delta_{x})-\int
f\mathop{}\\!\mathrm{d}m_{X}\right\rvert^{1/n}\leq\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}^{1/2}$

for $m_{X}$-a.e. $x\in X$, where $\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}$
denotes the spectral radius of $\pi(\mu)$ restricted to $L_{0}^{2}(X,m_{X})$.

###### Proof.

By separability of $C_{c}(X)$, for the statement about weak* convergence it
suffices to prove $m_{X}$-a.s. convergence for one fixed function $f\in
C_{c}(X)$. Consequently, it is enough to prove the second assertion of the
theorem. To this end, fix a function $f\in L^{2}(X,m_{X})$ and a number
$\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}<\alpha<1$, and consider the
$L_{0}^{2}$-function $f_{0}=f-\int f\mathop{}\\!\mathrm{d}m_{X}$. Then in view
of the spectral radius formula we have

$\displaystyle\Big{\lVert}\pi(\mu)^{n}f-\int
f\mathop{}\\!\mathrm{d}m_{X}\Big{\rVert}_{L^{2}}=\lVert\pi(\mu)^{n}f_{0}\rVert_{L^{2}}\leq\lVert\pi(\mu)|^{n}_{L_{0}^{2}}\rVert_{\mathrm{op}}\lVert
f_{0}\rVert_{L^{2}}\leq\alpha^{n}\lVert f_{0}\rVert_{L^{2}}$

for sufficiently large $n\in\mathbb{N}$. A standard Borel–Cantelli argument
now implies the statement. As we shall need similar estimates later on, we
include the details: By Chebyshev’s inequality, the above implies that for
large $n$ we have

$\displaystyle m_{X}\left(\left\\{x\in
X\nonscript\>\middle|\allowbreak\nonscript\>\mathopen{}\Big{\lvert}\pi(\mu)^{n}f(x)-\int
f\mathop{}\\!\mathrm{d}m_{X}\Big{\rvert}\geq\alpha^{n/2}\lVert
f_{0}\rVert_{L^{2}}\right\\}\right)$
$\displaystyle\leq\frac{\left\lVert\pi(\mu)^{n}f-\int
f\mathop{}\\!\mathrm{d}m_{X}\right\rVert_{L^{2}}^{2}}{\alpha^{n}\lVert
f_{0}\rVert_{L^{2}}^{2}}$ $\displaystyle\leq\alpha^{n}.$

By Borel–Cantelli it follows that for $m_{X}$-a.e. $x\in X$, the inequality

$\displaystyle\Big{\lvert}\pi(\mu)^{n}f(x)-\int
f\mathop{}\\!\mathrm{d}m_{X}\Big{\rvert}\geq\alpha^{n/2}\lVert
f_{0}\rVert_{L^{2}}$

holds only for finitely many $n\in\mathbb{N}$. Since $\pi(\mu)^{n}f(x)=\int
f\mathop{}\\!\mathrm{d}(\mu^{*n}*\delta_{x})$, letting $\alpha$ approach
$\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}$ gives the result. ∎

######  Remark 3.3.

In the second conclusion of Theorem 3.2, how long it takes for the exponential
rate of convergence to kick in depends on the point $x$. However, the measure
of sets on which one has to wait for a long time can be controlled as follows:
Given $\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}<\alpha<1$, choose
$N\in\mathbb{N}$ such that
$\lVert\pi(\mu)|^{n}_{L_{0}^{2}}\rVert_{\mathrm{op}}\leq\alpha^{n}$ for all
$n\geq N$. Then if we denote

$\displaystyle B_{\alpha,n,f}=\left\\{x\in
X\nonscript\>\middle|\allowbreak\nonscript\>\mathopen{}\Big{\lvert}\pi(\mu)^{n^{\prime}}f(x)-\int
f\mathop{}\\!\mathrm{d}m_{X}\Big{\rvert}\geq\alpha^{n^{\prime}/2}\lVert
f_{0}\rVert_{L^{2}}\text{ for some }n^{\prime}\geq n\right\\},$

the proof above gives the bound

$\displaystyle
m_{X}\left(B_{\alpha,n,f}\right)\leq\frac{\alpha^{n}}{1-\alpha}$

for every $n\geq N$. In other words, the measure of the set on which the
exponential convergence does not start during the first $n$ steps decays
exponentially in $n$. ∎

We now demonstrate that the previous result covers the case announced in §1.

######  Proposition 3.4.

Let $G$ be a connected semisimple real Lie group without compact factors and
with finite center, $\Gamma\subset G$ a lattice, and $X$ the homogeneous space
$G/\Gamma$ endowed with the Haar measure $m_{X}$. Suppose that the closed
subsemigroup $\mathcal{S}$ generated by $\operatorname{supp}(\mu)$ has the
property that $\operatorname{Ad}(\mathcal{S})$ is Zariski dense in
$\operatorname{Ad}(G)$. Then $\mu$ has a spectral gap on $X$.

###### Proof.

Consider the regular representation of $G$ on $L_{0}^{2}(X,m_{X})$. By [1,
Lemma 3] it doesn’t weakly contain the trivial representation. From this, in
view of [18, Theorem C], the result follows if we can argue that the
projection of $\mu$ to any simple factor of $G$ is not supported on a closed
amenable subgroup. However, since amenability passes to the Zariski closure
(see e.g. [21, Theorem 4.1.15]) the latter would imply that one of the simple
factors of $\operatorname{Ad}(G)$ is amenable, hence compact by a classical
result of Furstenberg (see e.g. [21, Proposition 4.1.8]). ∎

###  3.2. Good Height Functions

Inspecting the proof of Theorem 3.2, one observes that every step is
effective, with explicit bounds and good control over the measure of
exceptional sets, except for the very first one: separability of the space
$C_{c}(X)$ of compactly supported continuous functions. In the remainder of
this section, we aim to also make effective this step, the goal being
exponentially fast convergence $\mu^{*n}*\delta_{x}\rightarrow m_{X}$ from
almost every starting point, uniformly over functions $f$ on $X$. As merely
continuous functions can behave arbitrarily badly (with respect to the
convergence problem at hand), there is no hope of achieving this feat for all
$f\in C_{c}(X)$. We shall therefore restrict our attention to smooth functions
of compact support, and take into account their regularity by considering not
just their $L^{2}$, but also certain Sobolev norms. Built into the definition
of these norms will be what we call a _good height function_ , the concept of
which is introduced in this subsection.

Our setup is as follows: Let $G$ be a real Lie group with Lie algebra
$\mathfrak{g}$. We endow $\mathfrak{g}$ with a scalar product, which we use to
define a right-invariant metric $\operatorname{d}^{G}$ on $G$. Given a lattice
$\Gamma\subset G$, this metric descends to a metric $\operatorname{d}^{X}$ on
$X=G/\Gamma$ such that the projection $G\to X$ is locally an isometry.
Moreover, we fix an orthonormal basis of $\mathfrak{g}$, using which we will
identify $\mathfrak{g}$ with $\mathbb{R}^{\dim\mathfrak{g}}$. Here is the
crucial definition.

######  Definition 3.5.

We call a measurable function $\operatorname{ht}\colon X\to(0,\infty)$ a _good
height function_ if there exists $0<R\leq 1$ and a function $r\colon
X\to(0,R]$ with the following properties:

  1. (i)

The restriction of the exponential map $\exp\colon(-R,R)^{\dim\mathfrak{g}}\to
G$ is a diffeomorphism onto its image and we have
$\exp((-r/2,r/2)^{\dim\mathfrak{g}})\subset B_{r}^{G}(e)$ for all $r\leq R$,
where $B_{r}^{G}(e)$ denotes the open ball of radius $r$ around the identity
$e\in G$ with respect to the metric $\operatorname{d}^{G}$ on $G$.

  2. (ii)

For all $x\in X$, the projection $G\supset B_{r(x)}^{G}(e)\to X,\,g\mapsto gx$
is injective.

  3. (iii)

There exist constants $c,\kappa>0$ such that $r(x)\geq
c\operatorname{ht}(x)^{-\kappa}$ for all $x\in X$.

  4. (iv)

There exists a constant $\sigma>1$ such that
$\operatorname{ht}(x)\leq\sigma\operatorname{ht}(gx)$ for all $x\in X$ and all
$g\in B_{r(x)}^{G}(e)$.

The definition suggests to think of a good height function as reciprocal of
the injectivity radius. And indeed, this viewpoint allows their construction
on any homogeneous space $X=G/\Gamma$.

######  Proposition 3.6.

Let $G$ be a real Lie group and $\Gamma$ a lattice in $G$. Then $X=G/\Gamma$
admits a good height function.

###### Proof.

Choose $R>0$ such that condition (i) of the definition is satisfied and set
$r(x)=\min\\{R,r_{\mathrm{inj}}(x)\\}$, where $r_{\mathrm{inj}}(x)$ is the
injectivity radius at $x\in X$, i.e. the maximal radius such that (ii) holds
at $x$. Define

$\displaystyle\operatorname{ht}(x)=r(x)^{-1}.$

Then the only thing that needs to be verified is the validity of (iv). We
claim that it holds with $\sigma=2$. This will follow if we can show that

$\displaystyle r_{\mathrm{inj}}(gx)\leq 2r_{\mathrm{inj}}(x)$ (3.1)

whenever $g\in B_{r(x)}^{G}(e)$. To this end, let $r>r_{\mathrm{inj}}(x)$.
Then by definition, there are distinct $g_{1},g_{2}\in B_{r}^{G}(e)$ such that
$g_{1}x=g_{2}x$. As $g\in B_{r(x)}^{G}(e)$, right-invariance of the metric
implies

$\displaystyle\operatorname{d}^{G}(g_{i}g^{-1},e)=\operatorname{d}^{G}(g_{i},g)\leq\operatorname{d}^{G}(g_{i},e)+\operatorname{d}^{G}(g,e)<r+r(x)<2r$

for $i=1,2$, and we also have $(g_{1}g^{-1})gx=(g_{2}g^{-1})gx$. This shows
that $r_{\mathrm{inj}}(gx)\leq 2r$, and as $r>r_{\mathrm{inj}}(x)$ was
arbitrary, we see that (3.1) holds. ∎

Often, however, one might want to work with different, naturally occurring
height functions. The flexibility in our definition of a good height function
accommodates this possibility.

In the examples below, we denote by $\lambda_{1}(\Lambda)$ the length of a
shortest non-zero vector in a lattice $\Lambda\subset\mathbb{R}^{d}$.

######  Example 3.7.

Let $G=\operatorname{SL}_{d}(\mathbb{R})$ and
$\Gamma=\operatorname{SL}_{d}(\mathbb{Z})$. Then $X=G/\Gamma$ can be
identified with the space of lattices in $\mathbb{R}^{d}$ with covolume 1 via

$\displaystyle X\ni g\operatorname{SL}_{d}(\mathbb{Z})\longleftrightarrow
g\mathbb{Z}^{d}\subset\mathbb{R}^{d}.$

Then the function $\operatorname{ht}=\lambda_{1}^{-1}$, defined on $X$ via the
above identification, is a good height function. Indeed, one can first choose
$R>0$ such that (i) is satisfied, and then set
$r(x)=\min\\{R,r_{\mathrm{inj}}(x)\\}$ as in the proof of Proposition 3.6.
Then (ii) is automatically satisfied, and (iv) is valid for a suitable choice
of $\sigma$ due to the inequality $\lambda_{1}(gx)\leq\lVert
g\rVert\lambda_{1}(x)$ for $g\in G$ and $x\in X$, where $\lVert\cdot\rVert$
denotes any matrix norm. To see that also (iii) holds, let $x=g\Gamma$ and
suppose that $hx=x$ for some $h\in G$ with $h\neq e$. Then for all
$\gamma\in\operatorname{SL}_{d}(\mathbb{Z})$, the matrix
$(g\gamma)^{-1}h(g\gamma)$ fixes the lattice $\mathbb{Z}^{d}$ but is not the
identity, so that

$\displaystyle\lVert g\gamma\rVert^{\kappa_{1}}\lVert
h-e\rVert\geq\lVert(g\gamma)^{-1}(h-e)(g\gamma)\rVert=\lVert(g\gamma)^{-1}h(g\gamma)-e\rVert\geq
c_{1}$

for some constants $c_{1},\kappa_{1}>0$. For a basis change
$\gamma\in\operatorname{SL}_{d}(\mathbb{Z})$ such that $g\gamma$ consists of a
reduced basis of the lattice $x$ we have $\lVert g\gamma\rVert\leq
c_{2}\lambda_{1}(x)^{-\kappa_{2}}$ for some $c_{2},\kappa_{2}>0$ (cf. e.g.
[19, Chapter III]). With this choice, the above inequality implies

$\displaystyle\lVert h-e\rVert\geq c\lambda_{1}(x)^{\kappa}$

for $c=c_{1}/c_{2}$ and $\kappa=\kappa_{1}\kappa_{2}$. Since near the
identity, the metric $\operatorname{d}^{G}$ on $G$ is Lipschitz-equivalent to
the distance induced by $\lVert\cdot\rVert$, this establishes (iii). ∎

A similar construction is possible in a more general context.

######  Example 3.8 ([11]).

Let $G=\mathbb{G}(\mathbb{R})$ be the group of real points of a semisimple
$\mathbb{Q}$-group $\mathbb{G}$ and $\Gamma$ an arithmetic lattice in $G$.
Choose a rational $\operatorname{Ad}(\Gamma)$-stable lattice
$\mathfrak{g}_{\mathbb{Z}}\subset\mathfrak{g}$. Then, using similar reasoning
as in the previous example, the function $\operatorname{ht}$ on $X=G/\Gamma$
defined by

$\displaystyle\operatorname{ht}(x)=\lambda_{1}(\operatorname{Ad}(g)\mathfrak{g}_{\mathbb{Z}})^{-1}$

for $x=g\Gamma\in X$ is seen to be a good height function (cf. [11, §3.6]). ∎

###  3.3. Sobolev Norms

Given a good height function $\operatorname{ht}$ on $X$, the associated
_Sobolev norm_ of degree $\ell\geq 0$ of a compactly supported smooth function
$f\in C_{c}^{\infty}(X)$ is defined by

$\displaystyle\mathcal{S}_{\ell}(f)^{2}=\sum_{\deg\mathcal{D}\leq\ell}\lVert\operatorname{ht}(\cdot)^{\ell}\mathcal{D}f\rVert^{2}_{L^{2}},$

where the sum runs over differential operators $\mathcal{D}$ given by
monomials of degree at most $\ell$ in elements of the fixed orthonormal basis
of $\mathfrak{g}$ in the universal enveloping algebra.

In other words, the differential operators $\mathcal{D}$ appearing above are
$\partial_{v_{1}}\dotsm\partial_{v_{k}}$ for any $k$-tuple
$(v_{1},\dots,v_{k})$ of elements of the fixed basis of $\mathfrak{g}$, $0\leq
k\leq\ell$, where $\partial_{v}$ for $v\in\mathfrak{g}$ is defined by

$\displaystyle\partial_{v}f(x)=\lim_{t\to 0}\frac{f(\exp(tv)x)-f(x)}{t}$

for $f\in C_{c}^{\infty}(X)$ and $x\in X$.

Here are two immediate observations.

######  Lemma 3.9.

Let $\operatorname{ht}$ be a good height function on $X$ and
$\mathcal{S}_{\ell}$ the associated Sobolev norms.

  1. (i)

The norms $\mathcal{S}_{\ell}$ are induced by inner products
$\langle\cdot,\cdot\rangle_{\ell}$ on $C_{c}^{\infty}(X)$.

  2. (ii)

Given $0\leq\ell_{0}\leq\ell_{1}$, there exists a constant $\tilde{c}>0$ such
that $\mathcal{S}_{\ell_{0}}\leq\tilde{c}\mathcal{S}_{\ell_{1}}$.

###### Proof.

Part (i) is clear. Part (ii) is also immediate from the definition of the
Sobolev norms, once we know that a good height function must be bounded away
from $0$. The latter, however, follows directly from property (iii) in the
definition of a good height function, as the function $r$ appearing there is
assumed to be bounded. ∎

The proof of our convergence result in §3.4 will depend on the following
proposition.

######  Proposition 3.10 ([11]).

For the Sobolev norms associated to a good height function on $X$, there
exists a non-negative integer $\ell_{0}\geq 0$ and a constant $C>0$ with the
following properties:

  1. (i)

(Sobolev embedding estimate, [11, (3.9)]) For every $f\in C_{c}^{\infty}(X)$
it holds that $\lVert f\rVert_{\infty}\leq C\mathcal{S}_{\ell_{0}}(f)$.

  2. (ii)

(Finite relative traces, [11, (3.10)]) For all integers $\ell\geq 0$ the
relative trace
$\operatorname{Tr}(\mathcal{S}_{\ell}^{2}|\mathcal{S}_{\ell+\ell_{0}}^{2})$ is
finite, meaning that for any orthogonal basis $(e^{(k)})_{k}$ in the
completion of $C_{c}^{\infty}(X)$ with respect to
$\mathcal{S}_{\ell+\ell_{0}}$

$\displaystyle\operatorname{Tr}(\mathcal{S}_{\ell}^{2}|\mathcal{S}_{\ell+\ell_{0}}^{2})\coloneqq\sum_{k}\frac{\mathcal{S}_{\ell}(e^{(k)})^{2}}{\mathcal{S}_{\ell+\ell_{0}}(e^{(k)})^{2}}<\infty.$

We refer to Bernstein–Reznikov [8] for a systematic treatment of relative
traces. In particular, it is proved in this reference that the above
expression is independent of the choice of orthogonal basis.

The proofs in [11] of the statements in the above proposition are given for
the height function from Example 3.8. However, the only properties used are
those in our definition of a good height function. In fact, the arguments only
depend on validity of the second statement in [11, Lemma 5.1], which holds in
our context, as we demonstrate below.

######  Lemma 3.11.

Let $\operatorname{ht}$ be a good height function on $X$. Then there exists a
non-negative integer $\ell_{0}\geq 0$ and a constant $C>0$ such that for every
non-negative integer $\ell\geq 0$ and every differential operator
$\mathcal{D}$ given by a monomial of degree at most $\ell$ in elements of the
fixed basis of $\mathfrak{g}$ we have

$\displaystyle\lvert\operatorname{ht}(x)^{\ell}\mathcal{D}f(x)\rvert\leq
C\mathcal{S}_{\ell+\ell_{0}}(f)$

for every $f\in C_{c}^{\infty}(X)$ and $x\in X$.

###### Proof.

We inspect the function $F=\mathcal{D}f$ in a chart around $x$ given by the
exponential map: We set $\varepsilon=r(x)/2$, where $r\colon X\to(0,R]$ is the
function from the definition of a good height function, $d=\dim\mathfrak{g}$,
and consider

$\displaystyle\tilde{F}\colon(-\varepsilon,\varepsilon)^{d}\to\mathbb{R},\,v\mapsto
F(\exp(v)x).$

Then by the first statement of [11, Lemma 5.1], which is simply a Sobolev
embedding estimate on $\mathbb{R}^{d}$, we know

$\displaystyle\lvert F(x)\rvert=\lvert\tilde{F}(0)\rvert\leq
C_{1}2^{d}r(x)^{-d}\mathcal{S}_{d,\varepsilon}(\tilde{F}),$ (3.2)

where $C_{1}>0$ is a constant depending only on the dimension $d$ of
$\mathfrak{g}$ and $\mathcal{S}_{d,\varepsilon}$ is the standard degree $d$
Sobolev norm on the open subset $(-\varepsilon,\varepsilon)^{d}$ of
$\mathbb{R}^{d}$, i.e.

$\displaystyle\mathcal{S}_{d,\varepsilon}(\tilde{F})^{2}=\sum_{\lvert\boldsymbol{\alpha}\rvert\leq
d}\lVert\partial_{\boldsymbol{\alpha}}\tilde{F}\rVert^{2}_{L^{2}((-\varepsilon,\varepsilon)^{d})},$

where the sum is over all multi-indices $\boldsymbol{\alpha}$ of degree at
most $d$ and $\partial_{\boldsymbol{\alpha}}\tilde{F}$ is the corresponding
standard partial derivative of $\tilde{F}$. Using property (iii) in the
definition of a good height function, (3.2) implies that

$\displaystyle\lvert\operatorname{ht}(x)^{\ell}F(x)\rvert\leq
C_{2}\operatorname{ht}(x)^{\ell+\ell_{0}}\mathcal{S}_{d,\varepsilon}(\tilde{F}),$
(3.3)

where $C_{2}>0$ is another constant and we used that $\operatorname{ht}$ is
bounded away from $0$ to replace $\kappa d$ appearing in the exponent by
$\ell_{0}=\max\\{\lceil\kappa d\rceil,d\\}$. Using properties (i) and (ii) in
the definition of a good height function, we find $C_{3}>0$ such that

$\displaystyle\mathcal{S}_{d,\varepsilon}(\tilde{F})\leq
C_{3}\sqrt{\sum_{\deg\mathcal{D}^{\prime}\leq
d}\lVert\mathcal{D}^{\prime}F|_{B_{r(x)}^{X}(x)}\rVert^{2}_{L^{2}}}.$ (3.4)

To see this, one needs to note two things: firstly, that by the chain rule the
partial derivatives of $\tilde{F}$ at a point
$v\in(-\varepsilon,\varepsilon)^{d}$ in the chart can be expressed as linear
combinations of derivatives $\mathcal{D}^{\prime}F$ appearing on the right-
hand side in (3.4) evaluated at the corresponding point $x^{\prime}=\exp(v)x$,
with fixed coefficient functions depending only on finitely many derivatives
of the exponential map on $(-\varepsilon,\varepsilon)^{d}$; and secondly, that
the Haar measure $m_{X}$ is a smooth measure, meaning that it has a smooth and
nowhere vanishing density w.r.t. Lebesgue measure in the chart.

Combining (3.3), (3.4), condition (iv) in the definition of a good height
function, and plugging back in the definition of $F$, we finally arrive at

$\displaystyle\lvert\operatorname{ht}(x)^{\ell}\mathcal{D}f(x)\rvert\leq
C_{4}\sqrt{\sum_{\deg\mathcal{D}^{\prime}\leq
d}\lVert\operatorname{ht}(\cdot)^{\ell+\ell_{0}}\mathcal{D}^{\prime}\mathcal{D}f|_{B_{r(x)}^{X}(x)}\rVert^{2}_{L^{2}}}\leq
C_{4}\mathcal{S}_{\ell+\ell_{0}}(f),$

for yet another constant $C_{4}>0$, which is the one appearing in the lemma. ∎

###  3.4. Exponentially Generic Points

Now we are ready to define the notion of effective genericity we wish to
establish, and to prove the main convergence result of this section.

Until the end of this section, we fix a good height function
$\operatorname{ht}$ on $X$. Moreover, given a bounded measurable function $f$
on $X$ and $n\in\mathbb{N}$ we will use the notation

$\displaystyle D_{n}(f)(x)=\pi(\mu)^{n}f(x)-\int f\mathop{}\\!\mathrm{d}m_{X}$

for $x\in X$. We refer to $D_{n}(f)$ as the _time $n$ discrepancy_ for the
function $f$.

######  Definition 3.12.

We say that a point $x\in X$ is _$(\ell,\beta)$ -exponentially generic_ if
$\ell\geq 0$ is a non-negative integer and $\beta$ a real number in $(0,1)$
satisfying

$\displaystyle\limsup_{n\to\infty}\sup_{f\in
C_{c}^{\infty}(X)\setminus\\{0\\}}\biggl{(}\frac{\lvert
D_{n}(f)(x)\rvert}{\mathcal{S}_{\ell}(f)}\biggr{)}^{1/n}\leq\beta,$

where $\mathcal{S}_{\ell}$ is the degree $\ell$ Sobolev norm associated to
$\operatorname{ht}$.

With this terminology, we have the following result, which quantifies the
dependence on the function $f$ in the effective part of Theorem 3.2.

######  Theorem 3.13.

Let $G$ be a real Lie group, $\Gamma\subset G$ a lattice and $X=G/\Gamma$
endowed with the Haar measure $m_{X}$. Suppose that $\mu$ has a spectral gap
on $X$. Then there exists a non-negative integer $\ell_{1}\geq 0$ such that
$m_{X}$-almost every point $x\in X$ is
$\bigl{(}\ell_{1},\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}^{1/2}\bigr{)}$-exponentially
generic.

Our argument uses ideas from the proof of [11, Proposition 9.2]. Recall that
$\langle\cdot,\cdot\rangle_{\ell}$ denotes the inner product associated to the
Sobolev norm $\mathcal{S}_{\ell}$.

###### Proof.

Set $\ell_{1}=2\ell_{0}$ with $\ell_{0}$ from Proposition 3.10. We denote by
$\mathcal{H}$ the completion of $C_{c}^{\infty}(X)$ with respect to
$\mathcal{S}_{\ell_{1}}$.

The first step of the proof is to argue that $\mathcal{H}$ admits an
orthonormal basis $(e^{(k)})_{k}$ with respect to $\mathcal{S}_{\ell_{1}}$
that is also orthogonal with respect to $\mathcal{S}_{\ell_{0}}$. To this end,
let us endow $\mathcal{H}$ with the scalar product
$\langle\cdot,\cdot\rangle_{\ell_{1}}$ associated to $\mathcal{S}_{\ell_{1}}$.
This makes $\mathcal{H}$ into a Hilbert space. As a consequence of Lemma
3.9(ii), $\langle\cdot,\cdot\rangle_{\ell_{0}}$ defines a bounded positive
definite Hermitian form on
$(\mathcal{H},\langle\cdot,\cdot\rangle_{\ell_{1}})$. Using Riesz
representation it follows that there is a bounded positive self-adjoint
operator $T$ on $(\mathcal{H},\langle\cdot,\cdot\rangle_{\ell_{1}})$ such that

$\displaystyle\langle v,w\rangle_{\ell_{0}}=\langle Tv,w\rangle_{\ell_{1}}$

for all $v,w\in\mathcal{H}$. Finiteness of the relative trace
$\operatorname{Tr}(\mathcal{S}_{\ell_{0}}^{2}|\mathcal{S}_{\ell_{1}}^{2})$
from Proposition 3.10(ii) then translates into the statement that $T$ is a
trace-class operator on $(\mathcal{H},\langle\cdot,\cdot\rangle_{\ell_{1}})$
(cf. [12, Proposition 6.44]); in particular, the operator $T$ is compact (cf.
[12, Proposition 6.42]). By the spectral theorem, $T$ is thus diagonalizable.
Hence, an orthonormal basis $(e^{(k)})_{k}$ of
$(\mathcal{H},\langle\cdot,\cdot\rangle_{\ell_{1}})$ consisting of
eigenvectors of $T$ is a basis with the desired properties.

Next, fix a number $\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}<\alpha<1$. As in
the proof of Theorem 3.2, using Chebyshev’s inequality we find that for every
$k\geq 0$ and large enough $n$ we have

$\displaystyle m_{X}\left(\left\\{x\in
X\nonscript\>\middle|\allowbreak\nonscript\>\mathopen{}\big{\lvert}D_{n}(e^{(k)})(x)\big{\rvert}\geq\alpha^{n/2}\mathcal{S}_{\ell_{0}}(e^{(k)})\right\\}\right)\leq\frac{\lVert
e_{0}^{(k)}\rVert_{L^{2}}^{2}}{\mathcal{S}_{\ell_{0}}(e^{(k)})^{2}}\alpha^{n}\leq\frac{\lVert
e^{(k)}\rVert_{L^{2}}^{2}}{\mathcal{S}_{\ell_{0}}(e^{(k)})^{2}}\alpha^{n},$

where $e_{0}^{(k)}=e^{(k)}-\int e^{(k)}\mathop{}\\!\mathrm{d}m_{X}$. Since the
relative trace
$\operatorname{Tr}(\mathcal{S}_{0}^{2}|\mathcal{S}_{\ell_{0}}^{2})$ is finite
by Proposition 3.10, the terms on the right-hand side above are summable over
$k,n\geq 0$. Borel–Cantelli thus implies that

$\displaystyle A=\limsup_{k,n\geq 0}\left\\{x\in
X\nonscript\>\middle|\allowbreak\nonscript\>\mathopen{}\big{\lvert}D_{n}(e^{(k)})(x)\big{\rvert}\geq\alpha^{n/2}\mathcal{S}_{\ell_{0}}(e^{(k)})\right\\}$

is a null set. We claim that any $x\notin A$ is
$(\ell_{1},\alpha^{1/2})$-exponentially generic. Fix such a point $x$ and let
$f\in C_{c}^{\infty}(X)\setminus\\{0\\}$. Write $f=\sum_{k}f_{k}e^{(k)}$ for
the expansion of $f$ in terms of the orthonormal basis $(e^{(k)})_{k}$. Then,
using the triangle inequality, we can estimate the time $n$ discrepancy for
$f$ as follows:

$\displaystyle\lvert D_{n}(f)(x)\rvert\leq\sum_{k}\lvert f_{k}\rvert\lvert
D_{n}(e^{(k)})(x)\rvert.$ (3.5)

The exchange of integral and summation involved in the above estimate is
justified by part (i) of Proposition 3.10: It ensures that the functions
$e^{(k)}$ are defined pointwise and the series expansion of $f$ converges
uniformly. Since $x\notin A$ we know that for large $n$ the inequality $\lvert
D_{n}(e^{(k)})(x)\rvert<\alpha^{n/2}\mathcal{S}_{\ell_{0}}(e^{(k)})$ holds for
all $k$. For such $n$, an application of the Cauchy–Schwarz inequality implies
that (3.5) is strictly less than

$\displaystyle\alpha^{n/2}\biggl{(}\sum_{k}\lvert
f_{k}\rvert^{2}\biggr{)}^{1/2}\biggl{(}\sum_{k}\mathcal{S}_{\ell_{0}}(e^{(k)})^{2}\biggr{)}^{1/2}=\alpha^{n/2}\mathcal{S}_{\ell_{1}}(f)\operatorname{Tr}(\mathcal{S}_{\ell_{0}}^{2}|\mathcal{S}_{\ell_{1}}^{2})^{1/2}.$
(3.6)

Again by Proposition 3.10, the relative trace
$\operatorname{Tr}(\mathcal{S}_{\ell_{0}}^{2}|\mathcal{S}_{\ell_{1}}^{2})$ is
finite. Hence, in view of our definition of exponential genericity, combining
(3.5) and (3.6) establishes the claim. Letting
$\alpha\searrow\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}$ gives the theorem. ∎

######  Remark 3.14.

In analogy to Remark 3.3, we can control the measure of the set of points
where exponentially generic behavior is not observed for a given number of
steps: If we define

$\displaystyle B_{\alpha,n}=\Bigl{\\{}x\in
X\nonscript\>\Big{|}\allowbreak\nonscript\>\mathopen{}\left\lvert
D_{n^{\prime}}(f)(x)\right\rvert\geq\alpha^{n^{\prime}/2}\mathcal{S}_{\ell_{1}}(f)$
$\displaystyle\operatorname{Tr}(\mathcal{S}_{\ell_{0}}^{2}|\mathcal{S}_{\ell_{1}}^{2})^{1/2}$
$\displaystyle\text{for some }n^{\prime}\geq n,f\in
C_{c}^{\infty}(X)\Bigr{\\}}$

for $\rho\bigl{(}\pi(\mu)|_{L_{0}^{2}}\bigr{)}<\alpha<1$ and $n\in\mathbb{N}$,
and $N\in\mathbb{N}$ is chosen such that
$\lVert\pi(\mu)|^{n}_{L_{0}^{2}}\rVert_{\mathrm{op}}\leq\alpha^{n}$ for all
$n\geq N$, then for every $n\geq N$ it holds that

$\displaystyle
m_{X}\left(B_{\alpha,n}\right)\leq\operatorname{Tr}(\mathcal{S}_{0}^{2}|\mathcal{S}_{\ell_{0}}^{2})\frac{\alpha^{n}}{1-\alpha}.$

Indeed, we have $B_{\alpha,n}\subset\bigcup_{n^{\prime}\geq n,k\geq
0}\big{\\{}x\in
X\nonscript\>\big{|}\allowbreak\nonscript\>\mathopen{}\big{\lvert}D_{n^{\prime}}(e^{(k)})(x)\big{\rvert}\geq\alpha^{n^{\prime}/2}\mathcal{S}_{\ell_{0}}(e^{(k)})\big{\\}}$,
as the proof of Theorem 3.13 demonstrates. Thus, again, the measure of the set
of “bad points”, on which exponential genericity takes more than $n$ steps to
manifest, is itself exponentially small in $n$. ∎

##  4\. Uniform Cesàro Convergence

In this last section, we explore the situation where the only possible limit
in Theorem 1.1 is the normalized Haar measure $m_{X}$. In this setting, by
analogy with the case of unique ergodicity in classical ergodic theory, it is
reasonable to expect the Cesàro convergence (1.1) to hold (locally) uniformly
in the starting point $x_{0}$. We shall prove in §4.1 below that this indeed
holds true. In §4.2, we conclude the article by showing that in many naturally
occurring situations something even stronger than locally uniform can be
achieved.

Before continuing with the pertinent definitions, let us recall that even
though the setup of Theorem 1.1 is our motivation and useful to have in mind,
formally we are working with the assumptions stated at the end of §1:
$(X,m_{X})$ is merely required to be a space with a $G$-action for which
$m_{X}$ is invariant and ergodic.

######  Definition 4.1.

A probability measure $\nu$ on $X$ is called _$\mu$ -stationary_ if
$\mu*\nu=\nu$. The random walk on $X$ induced by $\mu$ is called _uniquely
ergodic_ if $m_{X}$ is the unique $\mu$-stationary probability measure on $X$.

In particular, for a random walk to be uniquely ergodic, there must be no
finite $\mathcal{G}$-orbits in $X$, where $\mathcal{G}$ denotes the closed
subgroup of $G$ generated by $\mu$. In the case that $X=G/\Gamma$ for a
lattice $\Gamma$ in $G$, this happens if and only if $\mathcal{G}$ is not
virtually contained in a conjugate of $\Gamma$. (Recall that a subgroup $H$ of
$G$ is said to be _virtually contained_ in a subgroup $L$ of $G$ if $H\cap L$
has finite index in $H$.) In fact, in many cases of interest, finite orbits
are the only obstruction to unique ergodicity: For example, this is true when
$G$ is a connected semisimple Lie group without compact factors, $\Gamma$ is
an irreducible lattice, $X=G/\Gamma$, and $\operatorname{Ad}(\mathcal{S})$ is
Zariski dense in $\operatorname{Ad}(G)$ (see [6, Corollary 1.8]); and also in
the setting of [20], a special case of which is reproduced below as Example
4.8.

###  4.1. Locally Uniform Convergence

The notion of unique ergodicity introduced above coincides with the classical
property of unique ergodicity of the Markov operator $\pi(\mu)$. When the
space $X$ is compact, this is enough to guarantee that the Cesàro convergence
$\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x}\to m_{X}$ as $n\to\infty$ is
uniform in $x$ (see e.g. [14, §5.1]). Without compactness, we also need to
assume a form of recurrence.

######  Definition 4.2.

We say that the random walk on $X$ given by $\mu$ is _locally uniformly
recurrent_ if for every compact subset $K\subset X$ and $\varepsilon>0$ there
exists $n_{0}\in\mathbb{N}$ and a compact subset $M\subset X$ with

$\displaystyle\mu^{*n}*\delta_{x}(M)\geq 1-\varepsilon$

for all $n\geq n_{0}$ and $x\in K$. It is called _locally uniformly recurrent
on average_ if the above holds with the Cesàro averages
$\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x}$ in place of
$\mu^{*n}*\delta_{x}$.

It is a simple exercise to check that locally uniform recurrence implies
locally uniform recurrence on average. In concrete examples, recurrence
properties such as these are typically established by constructing a Lyapunov
function; see §4.2 below.

The following well-known fact explains why these properties are referred to as
“non-escape of mass”.

######  Lemma 4.3.

Let the sequence $\\{x_{n}\\}_{n}$ of points in $X$ be relatively compact and
suppose that the random walk on $X$ is locally uniformly recurrent (resp. on
average). Then every weak* limit of the sequence
$(\mu^{*n}*\delta_{x_{n}})_{n}$ (resp.
$\big{(}\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x_{n}}\big{)}_{n}$) is a
probability measure. ∎

The proof is immediate and left to the reader.

We are now ready to state and prove our first result on locally uniform Cesàro
convergence.

######  Theorem 4.4.

Suppose that the random walk on $X$ induced by $\mu$ is uniquely ergodic and
locally uniformly recurrent on average. Then for every $f\in C_{c}(X)$, every
compact $K\subset X$, and every $\varepsilon>0$, there exists
$n_{0}\in\mathbb{N}$ such that for every $n\geq n_{0}$ and $x\in K$ we have

$\displaystyle\bigg{\lvert}\frac{1}{n}\sum_{k=0}^{n-1}\int_{X}f\mathop{}\\!\mathrm{d}(\mu^{*k}*\delta_{x})-\int_{X}f\mathop{}\\!\mathrm{d}m_{X}\bigg{\rvert}<\varepsilon.$

Equivalently, considering the space of probability measures on $X$ as endowed
with the weak* topology, the sequence of functions

$\displaystyle X\ni x\mapsto\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x}$

converges to $m_{X}$ uniformly on compact subsets of $X$ as $n\to\infty$.

###### Proof.

The equivalence of the two formulations is due to the definition of
neighborhoods in the weak* topology by finitely many test functions in
$C_{c}(X)$.

To prove the statement for individual functions, we proceed by contradiction.
If the conclusion is false, then for some $f\in C_{c}(X)$, $K\subset X$
compact and $\varepsilon>0$ there exist indices $n(j)\to\infty$ and $x_{j}\in
K$ with

$\displaystyle\bigg{\lvert}\frac{1}{n(j)}\sum_{k=0}^{n(j)-1}\int_{X}f\mathop{}\\!\mathrm{d}(\mu^{*k}*\delta_{x_{j}})-\int_{X}f\mathop{}\\!\mathrm{d}m_{X}\bigg{\rvert}\geq\varepsilon$
(4.1)

for all $j\in\mathbb{N}$. Let $\nu$ be a weak* limit point of the sequence

$\displaystyle\bigg{(}\frac{1}{n(j)}\sum_{k=0}^{n(j)-1}\mu^{*k}*\delta_{x_{j}}\bigg{)}_{j}.$

Then $\nu$ is $\mu$-stationary, and a probability measure because of our
recurrence assumption and the fact that all $x_{j}$ lie in the fixed compact
set $K$ (Lemma 4.3). But by unique ergodicity this forces $\nu=m_{X}$,
contradicting (4.1). ∎

###  4.2. Lyapunov Functions & Stronger Uniformity

Loosely speaking, _( Foster–) Lyapunov functions_ are functions enjoying
certain contraction properties with respect to the random walk, to the effect
that (on average) its dynamics are directed towards the “center” of the space,
where the function takes values below some threshold. They were introduced
into the study of random walks on homogeneous spaces by Eskin–Margulis [13],
whose ideas were further developed by Benoist–Quint [4]. Although they can be
defined in greater generality, we shall work with the following definition.

######  Definition 4.5.

A proper continuous function $V\colon X\to[0,\infty)$ is called a _Lyapunov
function_ for the random walk on $X$ induced by $\mu$ if there exist constants
$\alpha<1$, $\beta\geq 0$ such that $\pi(\mu)V\leq\alpha V+\beta$, where
$\pi(\mu)$ is the convolution operator associated to $\mu$ introduced in §3.

######  Remark 4.6.

Let us collect some immediate observations about Lyapunov functions.

  1. (i)

If $V$ is a Lyapunov function, then so are $cV$ and $V+c$ for any constant
$c>0$. In particular, one may impose an arbitrary lower bound on $V$, so that
it is no restriction to assume that a Lyapunov function takes values $\geq 1$,
say.

  2. (ii)

Given a Lyapunov function $V^{\prime}\colon X\to[0,\infty)$ for the
$n_{0}$-step random walk (induced by the convolution power $\mu^{*n_{0}}$),
one can construct a Lyapunov function $V$ for the random walk given by $\mu$
itself by setting

$\displaystyle
V=\sum_{k=0}^{n_{0}-1}\alpha^{\frac{n_{0}-1-k}{n_{0}}}\pi(\mu)^{k}V^{\prime}.$

  3. (iii)

By enlarging $\alpha$ and using properness, the contraction inequality in the
definition of a Lyapunov function $V$ may be replaced by

$\displaystyle\pi(\mu)V\leq\alpha V+\beta\mathds{1}_{K}$

for some compact $K\subset X$, where $\mathds{1}_{K}$ denotes the indicator
function of $K$ (cf. [15, Lemma 15.2.8]). ∎

Two examples in which a Lyapunov function exists are the following.

######  Example 4.7 ([13]).

Identify
$X=\operatorname{SL}_{2}(\mathbb{R})/\operatorname{SL}_{2}(\mathbb{Z})$ with
the space of unimodular lattices in $\mathbb{R}^{2}$ as in Example 3.7 and
recall that we denote by $\lambda_{1}(x)$ the length of a shortest non-zero
vector in $x\in X$. Then for every compactly supported probability measure
$\mu$ on $G$ whose support generates a Zariski dense subgroup there exist
$\varepsilon,\delta>0$ such that
$V^{\prime}=1+\varepsilon\lambda_{1}^{-\delta}$ is a Lyapunov function for the
$n_{0}$-step random walk on $X$ induced by $\mu^{*n_{0}}$ for some
$n_{0}\in\mathbb{N}$. This construction can be generalized to higher
dimensions by taking into account the higher successive minima
$\lambda_{2},\dots,\lambda_{d}$ of lattices in $\mathbb{R}^{d}$. A more
advanced construction also ensures existence of Lyapunov functions for Zariski
dense probability measures with finite exponential moments when
$G=\mathbb{G}(\mathbb{R})$ is the group of real points of a Zariski connected
semisimple algebraic group $\mathbb{G}$ defined over $\mathbb{R}$ such that
$G$ has no compact factors. ∎

######  Example 4.8 ([20]).

Let $G=\operatorname{SL}_{d+1}(\mathbb{R})$,
$\Gamma=\operatorname{SL}_{d+1}(\mathbb{Z})$ and $X=G/\Gamma$. For $0\leq
i\leq m$ let $c_{i}>1$ be positive real numbers, $y_{i}\in\mathbb{R}^{d}$
vectors such that $y_{0}=0$ and $y_{1},\dots,y_{m}$ span $\mathbb{R}^{d}$,
$O_{i}\in\operatorname{SO}_{d}(\mathbb{R})$ and set

$\displaystyle g_{i}=\begin{pmatrix}c_{i}O_{i}&y_{i}\\\
0&c_{i}^{-d}\end{pmatrix}\in G.$

Then for any choice of $p_{0},\dots,p_{m}>0$ with $\sum_{i=0}^{m}p_{i}=1$, the
measure $\mu=\sum_{i=0}^{m}p_{i}\delta_{g_{i}}$ defines a uniquely ergodic
random walk on $X$ admitting a Lyapunov function. ∎

It is well known that existence of a Lyapunov function as above guarantees
locally uniform recurrence.

######  Lemma 4.9 ([13, Lemma 3.1]).

Suppose the random walk on $X$ given by $\mu$ admits a Lyapunov function $V$.
Then this random walk is locally uniformly recurrent.

The intuitive reason for this behavior is simple: The defining contraction
inequality means that after a step of the random walk, the value of the
Lyapunov function $V$ on average gets smaller by a constant factor, at least
when starting outside some compact set $K$ (cf. Remark 4.6(iii) above). It is
an exercise to show that $K$ can be taken to be an appropriate sublevel set
for $V$, and one thinks about it as the “center” of the space. By the
contraction property, the number of steps required to reach it is uniform over
starting points $x$ in compact subsets of $X$. This suggests that we might
even let the starting points diverge, as long as this divergence is
outcompeted by the geometric rate of contraction of $V$. We are led to the
following notion of recurrence.

######  Definition 4.10.

Let $(K_{n})_{n}$ be a sequence of subsets of $X$. We say that the random walk
on $X$ given by $\mu$ is _$(K_{n})_{n}$ -uniformly recurrent_ if for every
$\varepsilon>0$ there exists $n_{0}\in\mathbb{N}$ and a compact subset
$M\subset X$ with

$\displaystyle\mu^{*n}*\delta_{x}(M)\geq 1-\varepsilon$

for all $n\geq n_{0}$ and $x\in K_{n}$. It is called _$(K_{n})_{n}$ -uniformly
recurrent on average_ if the above holds with the Cesàro averages
$\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x}$ in place of
$\mu^{*n}*\delta_{x}$.

######  Remark 4.11.

We point out that contrary to the locally uniform situation, for the two
versions of this property (with/without average) it is generally not clear
whether one implies the other. ∎

We are now going to use Lyapunov functions to establish such recurrence
properties for certain slowly growing exhaustions of $X$ by compact sets
$K_{n}$. Recall that the _Lyapunov exponent_ of a function
$\varphi\colon\mathbb{N}\to[1,\infty)$ is the exponential growth rate

$\displaystyle\lambda(\varphi)=\limsup_{n\to\infty}\tfrac{1}{n}\log\varphi(n).$

If $\lambda(\varphi)=0$, we say that $\varphi$ has _sub-exponential growth_.

######  Proposition 4.12.

Let $\varphi\colon\mathbb{N}\to[1,\infty)$ be a function. Suppose that the
random walk on $X$ induced by $\mu$ admits a Lyapunov function $V$ with
contraction factor $\alpha<1$ and set $K_{n}=V^{-1}([0,\varphi(n)])$.

  1. (i)

If $\varphi$ has Lyapunov exponent $\lambda(\varphi)<\log(\alpha^{-1})$, then
the random walk on $X$ given by $\mu$ is $(K_{n})_{n}$-uniformly recurrent.
The number $n_{0}$ in the definition can be chosen independently of
$\varepsilon$.

  2. (ii)

If $\varphi$ has sub-exponential growth, then the random walk on $X$ given by
$\mu$ is $(K_{n})_{n}$-uniformly recurrent on average.

The proof is a refinement of the methods in [4, 13].

###### Proof.

Let $\alpha,\beta$ be the constants associated to $V$ as in the definition of
a Lyapunov function and set $B=\frac{\beta}{1-\alpha}$. We are going to use
the same set $M$ for both parts of the proposition, namely
$M=V^{-1}([0,2B/\varepsilon])$, which is compact since $V$ is proper. Then for
$n\in\mathbb{N}$ and $x\in K_{n}$ we find, by repeatedly using the contraction
property of $V$,

$\displaystyle\mu^{*n}*\delta_{x}(M^{c})$
$\displaystyle\leq\frac{\varepsilon}{2B}\pi(\mu)^{n}V(x)\leq\frac{\varepsilon}{2B}(\alpha^{n}V(x)+B)\leq\frac{\varepsilon}{2B}\alpha^{n}\varphi(n)+\frac{\varepsilon}{2}.$

When the exponential growth rate of $\varphi$ is less than
$\log(\alpha^{-1})$, for some $n_{0}\in\mathbb{N}$ we have
$\alpha^{n}\varphi(n)\leq B$ for all $n\geq n_{0}$. This proves (i).

In order to prove (ii) we use a similar estimate, but have to ensure that the
values $\mu^{*k}*\delta_{x}(M^{c})$ are small for a sufficiently large
proportion of $0\leq k<n$. For $x\in K_{n}$ we find, as above,

$\displaystyle\mu^{*k}*\delta_{x}(M^{c})\leq\frac{\varepsilon}{2B}\alpha^{k}\varphi(n)+\frac{\varepsilon}{2}.$
(4.2)

Using straightforward manipulations, we further see

$\displaystyle\alpha^{k}\varphi(n)\leq
B/2\iff\frac{k}{n}\geq\log(\alpha^{-1})^{-1}\left(\frac{1}{n}\log\varphi(n)-\frac{1}{n}\log(B/2)\right),$

the right-hand side of which tends to $0$ as $n\to\infty$ by sub-exponential
growth of $\varphi$. Hence, with $k(n)=\lfloor\varepsilon n/4\rfloor$, we may
choose $n_{0}$ large enough to ensure the above inequality holds for all
$k\geq k(n)$ for $n\geq n_{0}$. For such $n$ we conclude, using (4.2),

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x}(M^{c})$
$\displaystyle=\frac{1}{n}\sum_{k=0}^{k(n)-1}\mu^{*k}*\delta_{x}(M^{c})+\frac{1}{n}\sum_{k=k(n)}^{n-1}\mu^{*k}*\delta_{x}(M^{c})$
$\displaystyle\leq\frac{k(n)}{n}+\frac{3\varepsilon}{4}\leq\varepsilon,$

which ends the proof of (ii). ∎

Theorem 4.4 can now be strengthened in the following way.

######  Theorem 4.13.

In addition to the assumptions of Theorem 4.4, suppose that the random walk on
$X$ induced by $\mu$ admits a Lyapunov function $V$. Let
$\varphi\colon\mathbb{N}\to[1,\infty)$ have sub-exponential growth. Then for
every $f\in C_{c}(X)$ we have

$\displaystyle\lim_{n\to\infty}\sup_{V(x)\leq\varphi(n)}\bigg{\lvert}\frac{1}{n}\sum_{k=0}^{n-1}\int_{X}f\mathop{}\\!\mathrm{d}(\mu^{*k}*\delta_{x})-\int_{X}f\mathop{}\\!\mathrm{d}m_{X}\bigg{\rvert}=0.$

###### Proof.

Using $(K_{n})_{n}$-uniform recurrence on average for
$K_{n}=V^{-1}([0,\varphi(n)])$ from Proposition 4.12(ii), the proof of Theorem
4.4 goes through with the obvious modifications. ∎

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case of graded noncommutative non-Noetherian rings. That task lies beyond the
scope of this paper, though.

##  6\. Bounds on distinguished local-cohomological dimension.

The most fundamental and well-known vanishing theorem in classical local
cohomology is this: if $I$ is an ideal in a Noetherian commutative ring, then
the classical local cohomology groups $H^{n}_{I}(M)$ vanish for all
$R$-modules $M$ whenever $n$ is greater than the least number of generators
for $I$. One wants a generalization of this result which applies to
distinguished local cohomology. However, $h^{*}_{\operatorname{{\rm dist}}}$
is given by a colimit of $\operatorname{{\rm Ext}}$ groups associated to a set
$S$ of ideals which is not generally the sequence of powers of any single
ideal $I$, so it is not clear whether one ought to expect
$h^{n}_{\operatorname{{\rm dist}}}(M)$ to vanish for all $n$ larger than some
particular integer. It is even less clear what to expect from
$H^{n}_{\operatorname{{\rm dist}}}$ except when the relevant coalgebra
$\Gamma$ is either co-commutative, or finite-type and Mitchell, so that
$H^{n}_{\operatorname{{\rm dist}}}$ agrees with $h^{n}_{\operatorname{{\rm
dist}}}$.

The main result of this section is Theorem 6.1, which establishes that, for a
certain class of graded coalgebras $\Gamma$ (including those whose dual is a
$\mathcal{P}$-algebra, so in particular, including the dual Steenrod
algebras), there is no such vanishing theorem for distinguished local
cohomology. As a consequence we get that the graded comodule category over
such a coalgebra fails to be $AB4^{*}\mathchar 45\relax(n)$ for any $n$
whatsoever.

######  Theorem 6.1.

Suppose that $A$ is a field, and that $\Gamma^{*}$ is finite-type and
connected. Suppose furthermore that every bounded-below free graded
$\Gamma^{*}$-module is injective in $\operatorname{{\rm gr}}\operatorname{{\rm
Mod}}(\Gamma^{*})$. Then the following conditions are equivalent:

  1. (1)

The category of graded $\Gamma^{*}$-modules has distinguished local
cohomological dimension zero. That is, $H^{m}_{\operatorname{{\rm dist}}}(M)$
vanishes for all positive $m$ and all graded $\Gamma^{*}$-modules $M$.

  2. (2)

The category of graded $\Gamma^{*}$-modules has finite distinguished local
cohomological dimension. That is, there exists some integer $n$ such that
$H^{m}_{\operatorname{{\rm dist}}}(M)$ vanishes for all $m>n$ and all graded
$\Gamma^{*}$-modules $M$.

  3. (3)

The category of bounded-above graded $\Gamma$-comodules satisfies axiom
$AB4^{*}\mathchar 45\relax(n)$ for some nonnegative integer $n$. That is,
there exists some integer $n$ such that
$R^{m}\prod^{\Gamma}_{i}\left(\left\\{M_{i}\right\\}\right)$ vanishes for all
$m>n$, all countable sets $I$, and all sets $\\{M_{i}:i\in I\\}$ of bounded-
above graded $\Gamma$-comodules.

  4. (4)

The category of bounded-above graded $\Gamma$-comodules satisfies
Grothendieck’s property $AB4^{*}$. That is, countable products are exact on
bounded-above graded $\Gamma$-comodules.

###### Proof.

(1) implies (2):

Immediate.

(2) implies (1):

Let $M$ be a graded $\Gamma^{*}$-module. By Theorem 4.4, $M$ has the same
distinguished local cohomology in positive degrees as the graded
$\Gamma^{*}$-module $\operatorname{{\rm conn}}_{0}(M)$ of $M$ generated by all
homogeneous elements of nonnegative degree. Since $\operatorname{{\rm
conn}}_{0}(M)$ is trivial in negative degrees, there exists a connective free
graded $\Gamma^{*}$-module $F$ and an epimorphism
$\epsilon:F\rightarrow\operatorname{{\rm conn}}_{0}(M)$. Since $F$ is
injective, we have $H^{i+1}_{\operatorname{{\rm dist}}}(\ker\epsilon)\cong
H^{i}_{\operatorname{{\rm dist}}}(\operatorname{{\rm conn}}_{0}(M))\cong
H^{i}_{\operatorname{{\rm dist}}}(M)$ for all $i\geq 1$. So if
$H^{i}_{\operatorname{{\rm dist}}}$ is nonzero on some graded
$\Gamma^{*}$-module for some positive $i$, then $H^{i+1}_{\operatorname{{\rm
dist}}}$ is also nonzero on some graded $\Gamma^{*}$-module. Consequently,
under the stated hypotheses, the only way to have a finite cohomological bound
on distinguished local cohomology is for that bound to be zero.

(2) implies (3):

Special case of Theorem 4.3.

(1) implies (4):

Special case of Theorem 4.3.

(4) implies (3):

Immediate, since $AB4^{*}$ is the same condition as $AB4^{*}\mathchar
45\relax(0)$.

(3) implies (2):

Suppose that the category of bounded-above graded $\Gamma$-comodules satisfies
axiom $AB4^{*}\mathchar 45\relax(n)$ for some nonnegative integer $n$. Suppose
that $M$ is a graded $\Gamma^{*}$-module. By Corollary 4.5, $M$ is isomorphic
to the limit of a Mittag-Leffler sequence $\dots\rightarrow N_{2}\rightarrow
N_{1}\rightarrow N_{0}$ of rational, bounded-above graded
$\Gamma^{*}$-modules. Consequently we have a short exact sequence of graded
$\Gamma^{*}$-modules

(23) $0\rightarrow M\rightarrow\prod_{i\in
I}N_{i}\stackrel{{\scriptstyle\operatorname{{\rm
id}}-T}}{{\longrightarrow}}\prod_{i\in I}N_{i}\rightarrow 0$

with each of the modules $N_{0},N_{1},N_{2},\dots$ bounded-above. Since each
of the $N_{i}$ are bounded-above, Theorem 4.3 gives us an isomorphism of
abelian groups

$\displaystyle H^{*}_{\operatorname{{\rm dist}}}\left(\prod_{i\in
I}N_{i}\right)$ $\displaystyle\cong
R^{*}\prod^{\Gamma}_{i}\left(\\{N_{i}:i\in\mathbb{N}\\}\right),$

so the $AB4^{*}\mathchar 45\relax(n)$ assumption on the comodule category
gives us that $H^{j}_{\operatorname{{\rm dist}}}\left(\prod_{i\in
I}N_{i}\right)$ vanishes for all $j>n$. The long exact sequence obtained by
applying $H^{*}_{\operatorname{{\rm dist}}}$ to (23) consequently gives us
that $H^{j}_{\operatorname{{\rm dist}}}(M)$ vanishes for all $j>n+1$. That is,
the category of graded $\Gamma^{*}$-modules has distinguished local
cohomological dimension at most $n+1$.

∎

######  Corollary 6.2.

Suppose that $A$ is a field, and suppose that the $A$-algebra $\Gamma^{*}$ is
a finite-type $\mathcal{P}$-algebra. Then exactly one of the two following
statements is true:

  1. (1)

The category of bounded-above graded $\Gamma$-comodules satisfies condition
$AB4^{*}$. That is, products are exact in the category of graded
$\Gamma$-comodules.

  2. (2)

For each integer $n$, the category of bounded-above graded $\Gamma$-comodules
fails to satisfy condition $AB4^{*}\mathchar 45\relax(n)$. That is, for each
integer $n$, there exists a countably infinite set
$\\{M_{0},M_{1},M_{2},\dots\\}$ of graded $\Gamma$-comodules such that
$R^{m}\prod^{\Gamma}_{i}\left(\\{M_{i}\\}\right)$ is nonzero for some $m>n$.

###### Proof.

By Theorem A.2 from the appendix on Margolis’ results on
$\mathcal{P}$-algebras, every bounded-below free graded $\Gamma^{*}$-module is
injective. Consequently the hypotheses of Theorem 6.1 are satisfied. ∎

######  Corollary 6.3.

Let $p$ be a prime number, and let $\Gamma$ denote the mod $p$ dual Steenrod
algebra. Then, for each integer $n$, the category of bounded-above graded
$\Gamma$-comodules fails to satisfy condition $AB4^{*}\mathchar 45\relax(n)$.
That is, there exists a countably infinite set $M_{0},M_{1},M_{2},\dots$ of
bounded-above graded $\Gamma$-comodules such that the derived product
$R^{m}\prod^{\Gamma}_{i}\left(\left\\{M_{i}:i\in\mathbb{N}\right\\}\right)$ is
nonzero for some $m>n$.

###### Proof.

It is already well-known (but we still give some explanation in the rest of
this proof) that the category of graded comodules over the dual Steenrod
algebra does not have exact products, so the claim is a corollary of Corollary
6.2.

If the category of bounded-above graded comodules over the dual Steenrod
algebra had exact products, this would imply that, for any countable set
$\\{X_{i}\\}$ of bounded-below $H\mathbb{F}_{p}$-nilpotently complete spectra,
the Hovey-Sadofsky spectral sequence

$E_{2}^{*,*}\cong R^{*}\prod_{i\in
I}^{\Gamma}\left(\\{H_{*}(X_{i};\mathbb{F}_{p})\\}\right)\Rightarrow
H_{*}\left(\prod_{i\in I}X_{i};\mathbb{F}_{p}\right)$

collapses on to the $R^{0}\prod^{\Gamma}$-line at the $E_{2}$-page. One can
consult Sadofsky’s unpublished preprint [30] for the original construction of
this spectral sequence (which is given there more generally for sequential
limits, not just countable products), Hovey’s paper [17] for a published
account (which, however, discusses convergence only when
$H_{*}(-;\mathbb{F}_{p})$ is replaced with a Morava $E$-theory), and the
appendix of [26] for a published account which discusses convergence of the
spectral sequence in the case at hand, i.e., the case of mod $p$ homology.
Collapse of the spectral sequence for all $\\{X_{i}:i\in\mathbb{N}\\}$ would
imply that mod $p$ homology commutes with all countable products of bounded-
below $H\mathbb{F}_{p}$-nilpotently complete spectra, and consequently that
mod $p$ homology commutes with all sequential homotopy limits of bounded-below
$H\mathbb{F}_{p}$-nilpotently complete spectra, which is well-known to be
untrue without some additional hypothesis (e.g. that the spectra are uniformly
bounded below). ∎

##  7\. Derived functors of sequential limit in comodule categories.

Let $\Gamma$ be a graded coalgebra over a field $A$. We continue to write
$\iota:\operatorname{{\rm gr}}\operatorname{{\rm
Comod}}(\Gamma)\rightarrow\operatorname{{\rm gr}}\operatorname{{\rm
Mod}}(\Gamma^{*})$ for the inclusion of comodules into modules via the adjoint
$\Gamma^{*}$-action, and we continue to write $\operatorname{{\rm
tr}}:\operatorname{{\rm gr}}\operatorname{{\rm
Mod}}(\Gamma^{*})\rightarrow\operatorname{{\rm gr}}\operatorname{{\rm
Comod}}(\Gamma)$ for the right adjoint of $\iota$.

Let $\mathcal{D}$ be a small category. For any functor
$\mathcal{F}:\mathcal{D}\rightarrow\operatorname{{\rm gr}}\operatorname{{\rm
Comod}}(\Gamma)$, the Bousfield-Kan construction of $\iota\circ\mathcal{F}$ is
the cosimplicial graded $\Gamma^{*}$-module

(24) $\textstyle{\underset{x\in
N\mathcal{D}_{0}}{\prod}\iota\mathcal{F}(\operatorname{{\rm
cod}}x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{x\in
N\mathcal{D}_{1}}{\prod}\iota\mathcal{F}(\operatorname{{\rm
cod}}x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{x\in
N\mathcal{D}_{2}}{\prod}\iota\mathcal{F}(\operatorname{{\rm
cod}}x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The notation is as follows: $N\mathcal{D}_{\bullet}$ is the nerve of
$\mathcal{D}$, so for $i>0$, $N\mathcal{D}_{i}$ is the set of composable
$i$-tuples of morphisms of $\mathcal{D}$, while $N\mathcal{D}_{0}$ is the set
of objects of $\mathcal{D}$. The notation $\operatorname{{\rm cod}}x$ is
supposed to denote the final object appearing in a composable $i$-tuple of
morphisms $x$; in the case $i=1$, this is simply the codomain of a morphism.
We write $C^{\bullet}(\iota\mathcal{F})$ for the Moore complex (i.e.,
alternating sum cochain complex) of (24), so that
$H^{i}(C^{\bullet}(\iota\mathcal{F}))\cong(R^{i}\lim)(\iota\mathcal{F})$ by
Proposition XI.6.2 of [7].

In Theorem 7.1, we write $\lim^{\Gamma}$ for a limit taken in the category of
graded $\Gamma$-comodules.

######  Theorem 7.1.

Suppose that $A$ is a field and that $\Gamma$ is a finite-type $A$-coalgebra
such that $\Gamma^{*}$ is connected. Let

(25) $\dots\rightarrow M_{2}\rightarrow M_{1}\rightarrow M_{0}$

be a sequence of bounded-above181818Again, to be clear: these comodules do not
need to be uniformly bounded above. graded $\Gamma$-comodules such that
$R^{1}\lim$ vanishes on the sequence of graded $\Gamma^{*}$-modules
$\dots\rightarrow\iota M_{2}\rightarrow\iota M_{1}\rightarrow\iota M_{0}$.
Then there is an isomorphism of $\Gamma^{*}$-modules

(26) $\displaystyle\iota R^{s}\lim^{\Gamma}_{i}M_{i}$ $\displaystyle\cong
H^{s}_{\operatorname{{\rm dist}}}(\lim_{i}\iota M_{i})$

for each $s$, and consequently a long exact sequence

(27)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\iota\lim^{\Gamma}_{i}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\operatorname{{\rm
dist}}}\left(\prod_{i}\iota
M_{i}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\operatorname{{\rm
dist}}}\left(\prod_{i}\iota
M_{i}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\iota
R^{1}\lim^{\Gamma}_{i}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{{\rm
dist}}}\left(\prod_{i}\iota
M_{i}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{{\rm
dist}}}\left(\prod_{i}\iota
M_{i}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\iota
R^{2}\lim^{\Gamma}_{i}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{\operatorname{{\rm
dist}}}\left(\prod_{i}\iota
M_{i}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots.}$

###### Proof.

Consider the natural numbers $\mathbb{N}$ under their usual ordering, and let
$\mathbb{N}^{\operatorname{{\rm op}}}$ be the opposite category of the
resulting partially-ordered set. Then the sequence (25) is a functor
$\mathcal{F}:\mathbb{N}^{\operatorname{{\rm op}}}\rightarrow\operatorname{{\rm
gr}}\operatorname{{\rm Comod}}(\Gamma)$. Since $\operatorname{{\rm tr}}$ is a
right adjoint, it is left exact, so we have the hypercohomology spectral
sequences

$\displaystyle{}^{\prime}E_{1}^{s,t}$ $\displaystyle\cong
R^{t}\operatorname{{\rm tr}}(C^{s}\iota\mathcal{F})$
$\displaystyle{}^{\prime\prime}E_{2}^{s,t}$ $\displaystyle\cong
R^{s}\operatorname{{\rm tr}}(H^{t}C^{\bullet}\iota\mathcal{F})$
$\displaystyle\cong R^{s}\operatorname{{\rm tr}}(R^{t}\lim\iota\mathcal{F}),$

each of which converges to the cohomology of the total complex of
$\operatorname{{\rm tr}}$ applied to $C^{\bullet}\iota\mathcal{F}$. Since
$R^{t}\lim\iota\mathcal{F}$ is a derived functor of sequential limit in a
category of modules, it is only capable of being nontrivial in degrees $t=0$
and $t=1$. Furthermore, $R^{1}\lim\iota\mathcal{F}$ vanishes by assumption.
Hence the spectral sequence ${}^{\prime\prime}E_{r}^{*,*}$ collapses at the
${}^{\prime\prime}E_{2}$-page, with ${}^{\prime\prime}E_{2}^{s,0}\cong
R^{s}\operatorname{{\rm tr}}(\lim\iota\mathcal{F})$ and with
${}^{\prime\prime}E_{2}^{s,t}$ trivial for $t>0$.

Meanwhile, ${}^{\prime}E_{1}^{*,t}$ is precisely the Moore complex
$C^{\bullet}\left((R^{t}\operatorname{{\rm tr}})\iota\mathcal{F}\right)$ of
the Bousfield-Kan construction of $(R^{t}\operatorname{{\rm
tr}})\circ\iota\circ\mathcal{F}$, since $\operatorname{{\rm tr}}$ commutes
with products and hence $R^{t}\operatorname{{\rm tr}}$ commutes with products.
So

$\displaystyle{}^{\prime}E_{2}^{s,t}$ $\displaystyle\cong
R^{s}\lim^{\Gamma}(R^{t}\operatorname{{\rm tr}}(\iota\mathcal{F}))$ (28)
$\displaystyle\cong R^{s}\lim^{\Gamma}_{i}(R^{t}\operatorname{{\rm tr}}(\iota
M_{i})),$

with (28) due to exactness of $\iota$. Now ${}^{\prime}E_{2}^{s,t}$ is trivial
for $t>0$, since $\iota$ is exact and faithful and since $\iota
R^{t}\operatorname{{\rm tr}}(\iota M_{i})\cong H^{t}_{\operatorname{{\rm
dist}}}(\iota M_{i})\cong 0$ for $t>0$, as distinguished local cohomology
vanishes on bounded-above modules, by Theorem 4.3. Hence
${}^{\prime}E_{2}^{s,0}\cong R^{s}\lim^{\Gamma}(\operatorname{{\rm
tr}}\iota\mathcal{F})\cong R^{s}\lim^{\Gamma}\mathcal{F}$, and
${}^{\prime}E_{2}^{s,t}$ is trivial for $t\neq 0$.

Since both spectral sequences converge to the cohomology of the same total
complex, we have $R^{n}\lim^{\Gamma}\mathcal{F}\cong R^{n}\operatorname{{\rm
tr}}(\lim\iota\mathcal{F})$ for all $n$, and on $\iota$ and using Theorem 4.3,
we have the desired isomorphism

(29) $\displaystyle\iota R^{n}\lim_{i}^{\Gamma}(M_{i})$ $\displaystyle\cong
H^{n}_{\operatorname{{\rm dist}}}\left(\lim_{i}\iota M_{i}\right).$

Finally, long exact sequence (27) arises from applying
$H^{n}_{\operatorname{{\rm dist}}}$ to the Milnor-type short exact sequence of
graded $\Gamma^{*}$-modules

$0\rightarrow\lim_{i}\iota M_{i}\rightarrow\prod_{i}\iota
M_{i}\rightarrow\prod_{i}\iota M_{i}\rightarrow 0$

and using isomorphism (29). ∎

######  Corollary 7.2.

Suppose that $A$ is a field and that $\Gamma$ is a finite-type Mitchell
coalgebra over $A$. Let $\dots\rightarrow M_{2}\rightarrow M_{1}\rightarrow
M_{0}$ be a sequence of bounded-above graded $\Gamma$-comodules such that
$R^{1}\lim$ vanishes on the sequence of graded $\Gamma^{*}$-modules
$\dots\rightarrow\iota M_{2}\rightarrow\iota M_{1}\rightarrow\iota M_{0}$.
Then we have an isomorphism of abelian groups

$R^{*}\lim^{\Gamma}_{i}M_{i}\cong\underset{j\rightarrow\infty}{\operatorname{{\rm
colim}}}\operatorname{{\rm
Ext}}^{*}_{\Gamma^{*}}\left(\Gamma^{*}/I_{j},\lim_{i}\iota M_{i}\right)$

where $I_{j}$ is the ideal of $\Gamma^{*}$ generated by all homogeneous
elements of degree $\geq j$.

###### Proof.

Consequence of Theorems 7.1 and 5.9. ∎

######  Corollary 7.3.

Suppose that $p$ is a prime number. Write $\Gamma$ for the mod $p$ dual
Steenrod algebra. Let $\dots\rightarrow M_{2}\rightarrow M_{1}\rightarrow
M_{0}$ be a sequence of bounded-above graded $\Gamma$-comodules such that
$R^{1}\lim$ vanishes on the sequence of graded $\Gamma^{*}$-modules
$\dots\rightarrow\iota M_{2}\rightarrow\iota M_{1}\rightarrow\iota M_{0}$.
Then we have an isomorphism of abelian groups between
$R^{*}\lim^{\Gamma}_{i}M_{i}$ and the graded local cohomology
$\underset{j\rightarrow\infty}{\operatorname{{\rm colim}}}\operatorname{{\rm
Ext}}^{*}_{\Gamma^{*}}\left(\Gamma^{*}/I_{j},\lim_{i}\iota M_{i}\right)$ of
the Steenrod algebra $\Gamma^{*}$.

##  Appendix A Review of Margolis’ basic results on $\mathcal{P}$-algebras.

Margolis’ fundamental results on graded module theory over
$\mathcal{P}$-algebras are Theorem 5 from section 13.2 and Theorem 12 from
section 13.3 of Margolis’s book [21]. We state those two results:

######  Theorem A.1.

(Margolis.) Let $B$ be a $\mathcal{P}$-algebra, and let $B(0)\subsetneq
B(1)\subsetneq\dots$ be a sequence of Poincaré subalgebras of $B$, as in
Definition 5.1. Let $M$ be a graded left $B$-module. Then the following are
equivalent:

  1. (1)

The projective dimension of $M$ is $\leq 1$.

  2. (2)

$M$ is flat.

  3. (3)

The injective dimension of $M$ is $\leq 1$.

  4. (4)

$M$ is free over $B(n)$ for each $n$.

Furthermore, if $M$ does not satisfy these (equivalent) conditions, then the
projective dimension, weak dimension, and injective dimension of $M$ are each
infinite.

######  Theorem A.2.

(Margolis.) Let $B$ be a $\mathcal{P}$-algebra, and let $B(0)\subsetneq
B(1)\subsetneq\dots$ be a sequence of Poincaré subalgebras of $B$, as in
Definition 5.1. Let $M$ be a bounded-below graded left $B$-module. Then the
following are equivalent:

  1. (1)

$M$ is free.

  2. (2)

$M$ is projective.

  3. (3)

$M$ is flat.

  4. (4)

$M$ is injective.

  5. (5)

$M$ is free over $B(n)$ for each $n$.

Furthermore, if $M$ does not satisfy these (equivalent) conditions, then the
projective dimension, weak dimension, and injective dimension of $M$ are each
infinite.

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