ON MIXING PROPERTIES OF COMPACT GROUP
EXTENSIONS OF HYPERBOLIC SYSTEMS.
DMITRY DOLGOPYAT
Abstract. We study compact group extensions of hyperbolic dif-
feomorphisms. We relate mixing properties of such extensions with
accessibility properties of their stable and unstable laminations.
We show that generically the corre lations decay faster than any
power of time. In particular, this is always the case for ergodic
semisimple extensions as well as for stably ergodic extensions of
Anosov diffeomorphisms of infranilmanifolds.
1. Introduction
1.1. Overview. This paper treats compact group extensions of hy-
perbolic systems. These systems have attracted much attention in the
past because they provide one of the simplest examples of weakly hy-
perbolic systems. Due to the major developments in 60 ’ and 70’ the
theory of uniformly hyperbolic systems ( i.e., Anosov and Axiom A
diffeomorphisms) is quite well understood (see [3, 7]). It is also now
generally accepted that the hyperbolic structure is the main cause of
the chaotic behavior in deterministic systems. Thus it is importa nt to
understand how much the assumptions of uniform hyperbolicity can
be weakened so that the same conclusions remain valid. One direction
of research which experiences a new wave of interest now is the theory
of partially hyperbolic or slightly less generally transversely hyperbolic
systems. In this case our diffeomorphism preserves some foliation and
is hyperbolic in the transverse direction, at least, when restricted to
the non-wandering set. The systems we deal with can be specified
by the requirement that the foliation involved has compact leaves and
the maps between leaves are isometries. If G is a compact group the
diffeomorphisms with this property form a n open set in the space of
G–equivariant dynamical systems and they play the same role in t he
equivariant theory as Axiom A play in the space of all diffeomorphisms.
Thus the systems under consideration are the simplest partially hy-
perbolic systems since we have very strong control over what happ ens
in the center. Besides harmonic analysis can be used to study such
1
2 DMITRY DOLGOPYAT
systems. These reasons make compact group extensions over hyper-
bolic systems an attractive object of investigation. In fact qualitative
properties of these systems are well understood now. The progress
here can be summarized as follows. First, Brin in a series of papers
[10, 11, 12] applied the general theory of partially hyperbolic systems
[13] to show that, in the volume–preserving case, such systems are
generically ergodic and weak mixing. It then follows from the general
theory of compact group extensions [43] that they are also Bernoulli.
Quite recently Burns and Wilkinson [17] used new adva nces in par-
tially hyperbolic theory [28, 40, 41] to show that generically ergodicity
of such systems persists under small not necessary equivariant pertur-
bations. In another direction Field, Parr y and Pollicott generalized
Brin’s theory to the non-volume preserving context. By contrast not
much is known a bout quantitative properties of such systems. This
paper is a first step in this direction.
To explain our results we need to introduce some notation. Let
F be a topologically mixing Axiom A diffeomorphism on a compact
manifold Y. L et f be a Holder continuous function and µ
f
be a Gibbs
measure with potential f. Also, let G be a compact connected and
simply connected Lie group and X be a transitive G–space. Write
M = Y × X. Let τ : Y → G be a smooth function. Consider the skew
action
T (y, x) = (F (y), τ(y)x). (1)
It preserves measure µ = µ
f
×Haar. If A and B are functions on M let
ρ
A,B
(n) =
Z
A(y, x)B(T
n
(y, x))dµ(y, x).
Denote by
¯ρ
A,B
(n) = ρ
A,B
(n) −
Z
A(y, x)dµ(y, x)
Z
B(y, x)dµ(y, x)
the correlation function. Call T rapidly mixing (T ∈ RM) if ¯ρ is
a continuous map from C
∞
(M) × C
∞
(M) to rapidly decreasing se-
quences, that is given k there a r e constants C, r such that
|¯ρ
A,B
| ≤ C||A||
C
r
(M)
||B||
C
r
(M)
n
−k
. (2)
On the first glance this definition depends also on the Gibbs potential
f but we will show that it is not the case. One may think that better
bounds should hold for generic extensions. However the decay of cor-
relation this definition requires is fast enough to imply good stochastic
behavior. As an example in Subsection 6.1 we derive the Central Limit
COMPACT GROUP EXTENSIONS OF HYPERBOLIC SYSTEMS 3
Theorem from it. On the other hand (2) is mild enough so that it can
be verified in many cases.
As in qualitative theory, accessibility properties o f the system under
consideration play impor t ant role in our analysis. Let Ω
F
be the non–
wandering set of F and Ω = Ω
F
× Y be the non-wandering set of T.
Let m
′
, m
′′
be points in Ω. We say that m
′′
is accessible from m
′
if
there is a chain of points m
′
= m
0
, m
1
. . . m
n
= m
′′
such that m
j+ 1
belongs to either stable or unstable manifold of m
j
. (We call such a
chain n-legged.) Given m, the set of points in the same fiber which
are accessible from m lie on an orbit of a group Γ
t
which we call Brin
transitivity group. As usual different choices of reference point give
conjugated gro ups. The Brin transitivity group can be obtained as
follows. Let
¯
T be the principal extension associated to T (that is
¯
T
acts by (1) on Y × G) . Let Γ(n, R) be the set of points which can be
accessed from (y, id) by n-legged chains such that the distance between
m
j+ 1
and m
j
inside the corresponding stable (unstable) manifold is at
most R. Then if n, R a r e large enough, Γ(n, R) generates Γ
t
. It was
shown by Brin that T is mixing if and only if Γ
t
acts ergodically on X.
Here we prove the following refinement.
Theorem 1.1. Le t n, R be so large that Γ(n, R) ge nerates Γ
t
. Then
T ∈ RM if and only if Γ(n, R) is Di ophantine.
Here as usual Diophantine condition means the absence of reso-
nances. More exactly we call a subset S ⊂ G Diophantine for the
action of G on X if for large k, S does not have non–constant almost
invariant vectors in C
k
(X). See Appendix A for details.
It can be shown that a generic pair of element s of G is Diophantine.
(The exceptional set is a union of a counta ble number of positive codi-
mension submanifolds. In case G is semisimple it is a finite union of
algebraic subvarieties. See [25].) From this we can deduce that in a
generic f amily, the condition of Theorem 1.1 is satisfied on the set of
full measure. A drawback of this result is that it does not tell how the
constant C from (2) varies along the family. Thus one may wonder
how large the interior of RM is. This question is easier if Ω is large
[38, 27] (since then Γ
t
is also large) or if G is semisiple.
Let ERG be the set of ergodic group extensions.
Corollary 1.2. If f is an Anosov diffeomorphism of an infranil mani-
fold then Int(RM) = Int(ERG).
This is a direct consequence of Theorem 1.1 and [17]. This result is
quite satisfying because one would not expect good mixing properties
4 DMITRY DOLGOPYAT
from a diffeomorphism which can be well–approximated by non–ergodic
ones.
Corollary 1.3. If G is semisimple then Int(RM) = Int(ERG) =
ERG.
In general we can reduce the problem to an Abelian extension. Let
T
a
be the factor of T on Y × (X/[G, G]).
Corollary 1.4. If T ∈ ERG then T ∈ RM if and only i f T
a
∈ RM.
Still in the general case o f compact extensions of Axiom A diffeo-
morphism we do not know how large the interior of rapidly mixing
diffeomorphisms is. To get some insight into this we study two related
classes of dynamical systems. These are compact group extensions of
subshifts of a finite typ e and of expanding maps of Riemannian mani-
folds. Heuristically the subshifts of finite type are less rigid than Axiom
A diffeos because any subshift of a finite type has an Axiom A real-
ization but small perturbations of the subshift correspond to piecewise
Holder p erturbations of diffeomorphisms. Similarly natural extensions
of expanding maps have Axiom A realizations but the unstable foliation
will be more smooth than in the general case. So they are more rigid.
Nonetheless, in bot h cases we show that the interior of rapidly mixing
maps is dense. In the second case even the interior of the exponentially
mixing maps is dense. This suggests that the same result might be true
in the context of compact extensions of Axiom A diffeomorphisms.
1.2. Organization of the paper. Let us describe the structure of the
paper. Section 2 is preliminary. Here we recall necessary facts about
Axiom A diffeomorphisms and symbolic dynamics. We also present
Brin’s theory of compact extensions a nd its generalization by Field,
Parry and Pollicott. In Section 3 we study compact group extensions
of expanding maps. First, we describe the Lie algebra of Brin tran-
sitivity group. We then proceed to show that if this algebra equals
the whole Lie algebra of G (infinitesimal complete non- integrability)
then the system is exponentially mixing. Under some technical as-
sumptions we establish the converse of this statement. Also we show
that if this condition is not satisfied t he map can be made non-ergodic
by an arbitra r y small perturbation. We conclude Section 3 by show-
ing that infinitesimal complete non-integrability is generic. Section 4
treats symbolic dynamical systems. We show that, in the absence of
resonances, our skew extension is ra pidly mixing. (See Appendix A for
the detailed discussion of the notion of resonances we use). We also
describe the reduction of a general extension t o the semisimple and
COMPACT GROUP EXTENSIONS OF HYPERBOLIC SYSTEMS 5
abelian cases. We conclude Section 4 by showing that rapid mixing is
generic. In Section 5 we apply the results of the previous section to
study extensions of Axiom A diffeomorphisms and prove Theorem 1.1
and Corollaries 1.2–1.4. Section 6 contains some applications of our
estimates. Some open questions are collected in Section 7.
For the reader familiar with the concepts of Section 2, Sections 3
and 4–6 constitute blocks which could be read separately. Roughly
speaking the difference between Section 3 and Section 4 is that in the
former we work with Lie algebras while in the latter we work with Lie
groups. The unavailability of the differential calculus accounts fo r the
fact that results of Section 4 are weaker t han results of Section 3.
Some of the arguments of this paper are similar to [20]–[22]. The
main difference which appear here a s compared to [20]– [2 2] is that we
have to work with arbitrary finite dimensional representations rather
than one-dimensional ones. Still we show that most of t he results of
[20]–[22] can be generalized to the setting of the present paper.
Notation. if W is a subset of G we denote by < W > the smallest
Lie subgroup of G containing W.
Acknowledgment. It is a pleasure for me to thank W. Parry, M. Pol-
licott, M. Ratner, K. Schmidt and A. Wilkinson for useful discussions.
This work is supported by the Miller Institute of Basic Research in
Science. I am also grateful for the referee who found 3 41 errors and
misprints in the original version of this paper.
2. Preliminaries.
2.1. Subshifts of finite type. In this section we recall how t o reduce
the study of Axiom A diffeomorphisms to symbolic systems. First, we
recall some facts about subshifts of finite type we. For proofs and more
information on the subject see [7, 3 7].
For a n×n matrix A whose entries ar e zeroes and ones we denote by
Σ
A
= {{ω
i
}
+∞
i=−∞
: A
ω
i
ω
i+1
= 1} the configuration space of a subshift of
a finite type. Usually we omit A and write Σ instead of Σ
A
. The shift
σ acts on Σ by (σω)
i
= ω
i+1
. The one-sided shift (Σ
+
A
, σ) is defined in
the same way but the index set is the set of non-negative integers. For
θ < 1 we consider the distance d
θ
(ω
1
, ω
2
) = θ
k
where k = max{j : ω
1
i
=
ω
2
i
for |i| ≤ j}. If X is a metric space we denote by C
θ
(Σ, X ) the space
of d
θ
−Lipschitz functions from Σ to X . C
+
θ
(Σ, X ) is defined similarly to
Σ
+
instead of Σ. There is a natural embedding of C
+
θ
(Σ, X ) to C
θ
(Σ, X )
corresponding to the projection Σ → Σ
+
. We use the no tation L(h) for
the Lipschitz constant of h. If X is a Banach space we write h
n
(ω) =