Table Of ContentSpringer Monographs in Mathematics
Springer-Verlag Berlin Heidelberg GmbH
Grigoriy A. Margulis
On Some Aspects
of the Theory
of Anosov Systems
With a Survey by Richard Sharp:
Periodic Orbits of Hyperbolic Flows
Springer
Grigoriy A. Margulis
Department of Mathematics
Yale University
10 Hillhouse Avenue
New Haven, CT 06520-8283, USA
e-mail: [email protected]
Richard Sharp
Department of Mathematics
University of Manchester
Oxford Road
Manchester M13 9PL, United Kingdom
e-mail: [email protected]
Translated from the Russian
by Szulikowska Valentina Vladimirovna
NIC "Regular and Chaotic Dynamics'~ Izhevsk, Russia
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Mathematics Subject Classification (2000):
37A05, 37AlO, 37BlO, 37ClO, 37C27, 37C30, 37C35, 37C40, 37D20, 37D35, 37D40
ISSN 1439-7382
ISBN 978-3-642-07264-2 ISBN 978-3-662-09070-1 (eBook)
DOI 10.1007/978-3-662-09070-1
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Preface
I am very thankful to Springer-Verlag for publishing a translation of my the
sis, which was defended at Moscow State University in 1970. There have been
only two publications related to the thesis. The first one is a short announce
ment [4], and the second one is [17] which more or less coincides with sections 3
and 4 of the thesis. In the early seventies, I was planning to publish the thesis
in its entirety, but those plans were postponed eventually for more than 30
years. The main reason for the postponement was that I became more involved
in other projects.
About three years ago I wrote to Joachim Heinze with the suggestion
to publish an English translation of my thesis. He agreed, but under the
condition that the publication will come "with a competent commentary on
the development of the field during the last 30 years since the thesis was
written". I am very glad that this translation is published with an excellent
survey "Periodic orbits of hyperbolic flows" by Richard Sharp.
I have updated the terminology from the original. In particular, when I
wrote my thesis, Anosov systems were called in Russian "U-sistemy", where
U is the first letter of "Uslovie" (condition). I also made some small changes
at the beginning of the proof of Lemma 6.2 in order to make the argument
using the Brauer theorem clearer.
I am very thankful to Steve Miller who carefully read this translation and
made numerous corrections.
My thesis advisor was Ya. G. Sinai. His influence on my formation as a
mathematician is hard to overestimate. Now we are good friends and I would
like to take this opportunity to express him my deepest gratitude.
New Haven, CT G. A. Margulis
October, 2003
Table of Contents
G. A. Margulis
On Some Aspects of the Theory of Anosov Systems 1
R. Sharp
Periodic Orbits of Hyperbolic Flows 73
Index 139
On Some Aspects of the Theory
of Anosov Systems
G. A. Margulis
Introduction 3
1 Some Preliminaries on Anosov Flows 8
2 Behavior of Lebesgue Measures on Leaves of 61+1
under the Action of Anosov Flows 12
3 Construction of Special Measures on Leaves
of 61+1, 6 k+1, 61, and 6 k 17
wn
4 Construction of a Special Measure on and the Properties
of the Flow {Tt} with this Measure 26
5 Ergodic Properties of 6 k 29
6 Asymptotics of the Number
of Periodic Trajectories 33
7 Some Asymptotical Properties
of the Anosov Systems 49
8 Appendix 60
References 70
G. A. Margulis, On Some Aspects of the Theory of Anosov Systems
© Springer-Verlag Berlin Heidelberg 2004
Introduction
The theory of Anosov systems is a result of the generalization of certain
properties, which hold on geodesic flows on manifolds of negative curvature.
It turned out that these properties alone are sufficient to ensure ergodicity,
mixing, and, moreover, existence of K-partitions. All above-mentioned prop
erties are connected with the asymptotical behavior of variational equations
along the trajectories of Anosov systems. Therefore, it would be appropriate
to propose that other asymptotical properties of geodesic flows on manifolds
of negative curvature hold for the class of Anosov systems, too. However, it
would be more rational to consider not all of the Anosov flows, but the class L
of Anosov flows that preserve some integral invariant and have no continuous
eigenfunctions.
Let {Tt} be an Anosov flow from L on a Riemannian manifold Wn. Then
there exist four foliations e1+1, ek+l, el, and ek on wn that are invariant
under {Tt}. The leaves of these foliations are called the expanding leaves,
the contracting leaves, the expanding horospheres, and the contracting horo
spheres, respectively. Now, a natural concept of E-equivalency is introduced
for sets belonging to different expanding leaves. Roughly speaking, two sets
belonging to expanding leaves are E-equivalent if one of them may be con
tinuously transformed into the other so that, during the transformation, each
point describes a curve belonging entirely to some contracting horosphere, the
curve's length being less than E (for more details, see Definition 2.1).
In § 2 of the present paper the asymptotical behavior of a Lebesgue mea
sure on expanding leaves under the action of an Anosov flow is investigated.
It turns out that
. f.-tS1+1 (Tt Md
hm = 1, (1)
t __ +oo f.-ts1+1 (Tt M2)
where M1 and M2 are two E-equivalent sets.
wn
In § 2 we consider the class T of functions on whose support is a
compact subset of some expanding leaf. (The leaves may be different for dif
ferent functions.) Every function is continuous on its leaf. In the same way,
we introduce the concept of E-equivalency for these functions and prove that
4 On Some Aspects of the Theory of Anosov Systems
for any two E-equivalent functions hand 12,
11. m ff Tt 1h2 dJ.lel+1 = 1, (2)
t---++oo Tt dJ.lel+1
where f f dJ.lel+1 is the integral of f over the leaf where f does not vanish.
Further, we prove that for any non-negative and non-zero functions hand 12
from T,
O < Cl < t--1-1·+ m+o o ff TTtt 1h2 ddJJ..lleell++11 -< t----1+1m+·o o ff TTtt 1h2 ddJJ..lleell++11 < C2 < 00. (3)
In § 3 we define a family of functionals it (t 2: 0) on T as follows: if f E T,
then
(4)
By (2) and (3), the closure of the convex hull of it (t 2: 0) is compact with
respect to some topology. Hence, by the l'ychonoff theorem on a fixed point
we deduce that there exists a functionali such that
1) if f E T, then
Z(Tt f) = d~I+JU), (5)
2) g-hand 12 are E-equivalent for some 10, then ZUd = Z(h), and
3) i is positive on the set of non-negative and non-zero functions.
After that, by the Riesz-Markov theorem we prove that there exists a class
of count ably additive measures liel+1 on expanding leaves such that
1) liel+1(TtM) = ~el+lliel+l(M),
2) if Ml and M2 are E-equivalent, then liel+1(Md = liel+1(M2), and
3) for any (nonempty) open M, 0 < liel+1(M) < 00.
In the case of Anosov diffeomorphisms, similar measures were constructed by
Ya. G. Sinai in [7], with the help of his theory of Markov partitions.
By using the measures liel+1 and some other measures constructed in § 3,
we construct in § 4 a special measure Ii defined on the whole of wn and
invariant under {Tt}. In the same section we formulate Theorems 2 and 3,
which show that the flow {Tt} with Ii is a K-flow and the foliations connected
with the Anosov flow are metrically transitive with respect to Ii. Theorems 2
and 3 are proved in appendix (§ 8).
The constant del+l is very important. For example, it is proved in § 8 that
log2 del+l is equal to the topological entropy of {Tt}. The constant del+l is
essential when studying the asymptotical behavior of the number of periodic
trajectories, or the volume of a ball with radius R as R --t 00 (for manifolds
with negative curvature), etc.
The closed trajectories of {Tt} are studied in § 6. As it is known, the prob
lem of the closed geodesics on a compact manifold M of negative curvatuve is
Introduction 5
equivalent to the problem of the periodic trajectories of a geodesic flow defined
on the manifold of linear elements of M. In fact, any closed geodesic is natu
rally associated with a closed trajectory, and the length of the closed geodesic
is equal to the period of the corresponding trajectory. (Here we mean that
any closed geodesic is given together with its direction. In the case of closed
geodesics without the given direction, all the estimations made below are to
be divided by two.)
By the well-known Birkhoff theorem we get that for any non-trivial free
homotopy class on M there exists a unique closed geodesicl . So, any free
homotopy class on M corresponds to a conjugacy class in 7rl (M). Therefore we
may say that the asymptotics of the closed geodesics defines the asymptotics
of the conjugacy classes in 7rl (M) (except for the identity class).
It follows from results of Hadamard [11] and Morse [12] that for any two
dimensional compact manifold M with negative curvature, the vectors tangent
to M and tangent to closed geodesics are dense in the space of all tangent
vectors. Anosov has proved a similar theorem for the n-dimensional case.
More precisely, he has proved that periodic trajectories of an arbitrary flow
from L are dense everywhere. This means that an Anosov flow has, in any
case, an infinite number of periodic trajectories. On the other hand, any free
homotopy class of M has a unique closed geodesic; therefore, the number of
closed geodesics on M is at most countable. Moreover, if we denote by v(R)
the number of closed geodesics having length less than R, then v(R) is finite
for any R. It is easy to see that this assumption remains true for arbitrary
Anosov flows.
Since v(R) is finite, we can investigate the asymptotical behavior of v(R)
as R tends to infinity. Ya. G. Sinai proved in [13] that there exist positive
constants Kl and K2 such that
logv(t) - logv(t)
(n - 1)K2 :<::: lim :<::: lim :<::: (n - l)Kl (6)
t t
t--->oo t--->oo
and if {Tt} is a geodesic flow on a compact manifold with negative curvature,
then - K'f is a lower bound and - K? is an upper bound of the curvature over
all two-dimensional directions. (Here v(t) stands for the number of periodic
trajectories of the Anosov flow {Tt} with period that is less than t.)
In § 6 we prove that
v(R) "" d~l+l (7)
R . log drsl+l '
where drsl+l is the constant defined in § 3, or, putting c = log drsl+l,
ecR
v(R) "" -. (8)
cR
1 Existence of a closed geodesic for any non-trivial free homotopy class remains true
for any compact manifold. As for uniqueness, it is a special property of manifolds
with negative curvature (see [11]).
