A Mathematical Journal
Vol. 7, No 3, (49 - 63). December 2005.

The ergodic measures related with
nonautonomous hamiltonian systems and

their homology structure. Part 1 1

Denis L.Blackmore
Dept. of Mathematical Sciences at the NJIT, Newark, NJ 07102, USA

deblac@m.njit.edu

Yarema A.Prykarpatsky
The AGH University of Science and Technology, Department of Applied

Mathematics, Krakow 30059 Poland,
and Brookhaven Nat. Lab., CDIC, Upton, NY, 11973 USA

yarchyk@imath.kiev.ua, yarpry@bnl.gov

Anatoliy M.Samoilenko
The Institute of Mathematics, NAS, Kyiv 01601, Ukraine

Anatoliy K.Prykarpatsky 2

Department of Applied Mathematics, The AGH University of Science and
Technology

Applied Mathematics, Krakow 30059 Poland
pryk.anat@ua.fm, prykanat@cybergal.com

ABSTRACT
There is developed an approach to studying ergodic properties of time-dependent

periodic Hamiltonian flows on symplectic metric manifolds having applications
in mechanics and mathematical physics. Based both on J. Mather’s [9] results
about homology of probability invariant measures minimizing some Lagrangian

1The authors are cordially indebted to Profs. Anthoni Rosato (NJIT, NJ,USA) and Alexander
S. Mishchenko (Moscow State University, Russia) for useful comments on the article. They are also
thankful to participants of the Seminar ” Nonlinear Analysis” at the Dept. of Applied Mathematics
of the AGH University of Science and Technology of Krakow for valuable discussions.

2The fourth author was supported in part by a local AGH grant.



50 D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)

functionals and on the symplectic field theory devised by A. Floer and others
[3-8,12,15] for investigating symplectic actions and Lagrangian submanifold inter-
sections, an analog of Mather’s β-function is constructed subject to a Hamiltonian
flow reduced invariantly upon some compact neighborhood of a Lagrangian sub-
manifold. Some results on stable and unstable manifolds to hyperbolic periodic
orbits having applications in the theory of adiabatic invariants of slowly perturbed
integrable Hamiltonian systems are stated within the Gromov-Salamon-Zehnder
[3,5,12] elliptic techniques in symplectic geometry.

RESUMEN

Un método para estudiar propiedades ergódicas de flujos Hamiltonianos que
dependen del tiempo sobre variedades simplécticas es desarrollado. Basados tanto
en un trabajo de J. Mather [9] sobre homoloǵıa de medidas invariantes de prob-
abilidad que minimizan algunos funcionales lagrangianos, como en la teoŕıa de
campos simplécticos, desarrollada por A. Floer y otros [3-8,12,15] para investigar
acciones simplécticas e intersecciones de subvariedades lagrangianas, se construye
un análogo de la función β de Mather sujeto a un flujo hamiltoniano reducido
invariantemente sobre una vecindad compacta de una subvariedad Lagrangiana.
Se plantean algunos resultados sobre variedades estables e intestables de órbitas
hiperbólicas periódicas. Estas tienen aplicaciones en la teoŕıa de sistemas hamilto-
nianos integrables con perturbaciones lentas, en el marco de las técnicas eĺıpticas
de Gromov-Salamon-Zehnder [3,5,12] en geometŕıa simpléctica.

Key words: Ergodic measures, Holonomy groups, Dynamical systems,
Quasi-complex structures, Symplectic field theory

Math. Subj. Class.: 37A05, 37B35, 37C40, 37C60, 37J10, 37J40, 37J45

Introduction

The past years have given rise to several exciting developments in the field of sym-
plectic geometry and dynamical systems [3-12], which introduced new mathematical
tools and concepts suitable for solving many before too hard problems. When study-
ing periodic solutions to non-autonomous Hamiltonian systems Salamon & Zehnder
[3] developed a proper Morse theory for infinite dimensional loop manifolds based on
previous results on symplectic geometry of Lagrangian submanifolds of Floer [4, 6].
Investigating at the same time ergodic measures related with Lagrangian dynamical
systems on tangent spaces to configuration manifolds, Mather [9] devised a new ap-
proach to studying the correspondingly related invariant probabilistic measures based
on a so called β-function. The latter made it possible to describe effectively the so
called homology of these invariant probabilistic measures minimizing the correspond-



7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ... 51

ing Lagrangian action functional.
As one can easily see, the Mather approach doesn’t allow any its direct applica-

tion to the problem of describing the ergodic measures related naturally with a given
periodic non-autonomous Hamiltonian system on a closed symplectic space. Thereby,
to overcome constraints to this task we suggest in the present work some new way to
imbedding the non-autonomous Hamiltonian case into the Mather β-function theory
picture, making use of the mentioned above Salamon & Zehneder and Floer [3, 4, 6]
loop space homology structures. Based further on the Gromov elliptic techniques
in symplectic geometry, the latters make it possible to construct the invariant sub-
manifolds of our Hamiltonian system, naturally related with corresponding compact
Lagrangian submanifolds, and the related on them a β-function analog.

1 Symplectic and analytic problem setting

Let (M2n,ω(2)) be a closed symplectic manifold of dimension 2n with a symplec-
tic structure ω(2) ∈ Λ(M2n) being weakly exact, that is ω(2)(π2(M2n)) = 0. Every
smooth enough time-dependent 2π-periodic function H : M2n × S1 → R gives rise to
the non-autonomous vector field XH : M2n × S1 → T(M2n) defined by the equality

iXH ω
(2) = −dH, (1)

where as usually [1], the operation ” iXH ” denotes the intrinsic derivation of the
Grassmann algebra Λ(M2n) along the vector field XH. The corresponding flow on
M2n × S1 takes the form:

du/ds = XH (u; t), dt/ds = 1, (2)

where u : R →M2n is an orbit, t ∈ R/2πZ ' S1 and s ∈ R is an evolution parameter.
We shall assume that solutions to (2) are complete and determine a one-parametric
ψ-flow of diffeomorphisms ψs : M2n × S1 → M2n × S1 for all s ∈ R which are due
to (1) evidently symplectic, that is ψs∗t0 ω

(2) = ω(2) where ψst0 := ψ
s|M 2n at any fixed

t0 ∈ R/2πZ ' S1. Take now an (n + 1)-dimensional submanifold Ln+1 ⊂ M2n × R,
such that for any closed contractible curve γ with γ ⊂ Ln+1 the following integral
equality ∮

γ

(α(1) − H(t)dt) = 0 (3)

holds, where α(1) ∈ Λ1(M 2n) is such a 1-form on M2n which satisfies the condition∫
D2

(ω(2) − dα(1)) = 0 for any compact two-dimensional disk D2 ⊂ M2n due to the
weak exactness of the symplectic structure ω(2) ∈ Λ2(M2n) and existing globally on
Ln+1 due to Floer results [4, 6]. Assume now also that for the flow of symplectomor-
phisms ψst0 : M

2n → M2n, s ∈ R, the condition

{(ψst0L
n
t0
, t0 + s) : s ∈ R} ⊂ Ln+1 (4)



52 D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)

holds for some compact Lagrangian submanifold Lnt0 ⊂ M
2n upon which ω(2)

∣∣
Lnt0

= 0.

The condition (4) in particular means [2] that the following expression

α(1) − H(t)dt = dA(t), (5)

t = t0 + s(mod2π) ∈ R/2πZ, holds in some vicinity of the Lagrangian submanifold
Lnt0 ⊂ M

2n, where a mapping A : R/2πZ → R is the so called [1, 2] generating
function for the defined above continuous set of diffeomorphisms ψst0 ∈Diff(M

2n),
s ∈ R. The expression (5) makes it possible to define naturally the following Poincare-
Cartan type functional on a set of almost everywhere differentiable curves γ : [0,τ] →
M 2n × S1

A(τ )t0 (γ) :=
1
τ

∫
γ

(α(1) − H(t)dt), (6)

with end points { γ(τ) = ψτ (γ(0)) }, supp γ ⊂ U(Lnt0 ) × S
1 for all τ ∈ R and U(Lnt0 )

is some compact neighborhood of the Lagrangian submanifold Lnt0 ⊂ M
2n satisfying

the condition ψst0U(L
n
t0

) ⊂ U(Lnt0 ) for all s ∈ R.
Let us denote by Σt0 (H) the subset of curves γ with support in U(L

n
t0

) × S1 and
fixed end-points as before minimizing the functional (6). If the infimum is realized,
one easily shows that any such curve γ ∈ Σt0 (H) solves the system (2). For the above
set of curves Σt0 (H) to be specified more suitably, choose, following Floer’s ideas
[3-8,12], an almost complex structure J : M2n → End(T(M2n)) on the symplectic
manifold M2n, where by definition J2 = −I, compatible with the symplectic structure
ω(2) ∈ Λ2(M2n). Then the expexpression

< ξ,η >:= ω(2)(ξ,Jη), (7)

where ξ,η ∈ T(M2n), naturally defines a Riemannian metric on M2n. Subject to the
metric (7) our Hamiltonian vector field XH : M2n ×S1 → T(M2n) is now represented
as XH = J∇H, where ∇ : D(M2n) → T(M2n) denotes the usual gradient mapping
with respect to this metric.

Consider now the space Ω := Ω(M2n × S1) of all continuous curves in M2n × S1
with fixed end-points. Then one can similarly define the gradient mapping grad
A(τ )t0 : Ω → T(Ω) as follows:

(grad A(τ )t0 (γ),ξ) :=
1
τ

∫ τ
0

ds < J(γt0 )γ̇t0 (s) + ∇H(γt0 ; s + t0),ξ >, (8)

where γ = {(γt0 (s); t0 + s( mod 2π)) : s ∈ [0,τ]} ∈ Ω as before, and ξ ∈ T(Ω).
Since all critical curves γ ∈ Σt0 (H) minimizing the functional (6) solve (2), this fact
motivates a way of construction of an invariant subset ΩH ⊂ Ω, such that ΩH :=
Ω(U(Lnt0 ) × S

1). Namely, define a curve γ ∈ ΩH (γ(−)) ⊂ ΩH as satisfying [3] the
following gradient flow in U(Lnt0 ) × S

1 :

∂ut0/∂z = −grad A
(τ )
t0

(u), ∂t/∂z = 0 (9)



7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ... 53

for all z ∈ R and any τ ∈ R under the asymptotic conditions

lim
z→−∞

ut0 (s; z) = γ
(−)
t0

(s), lim
z→∞

γt0 (s; z) = γt0 (s) (10)

with the corresponding curves γ(−)t0 , γt0 : R →M
2n satisfying the system (2), and

moreover, with the curve γ(−)t0 : R →M
2n being taken to be hyperbolic [1, 2] with supp

γ
(−)
t0

⊂ Lnt0. Now we can construct a so called [1] unstable manifold W
u(γ(−)t0 ) to this

hyperbolic curve γ(−)t0 defined for all τ ∈ R. Thus due to the above construction, the
functional manifold W u(γ(−)t0 ) when compact can be imbedded as a point submanifold
into M2n thereby interpreting supports of all curves solving (9) and (10) where supp
γt0 ⊂ Lnt0, as a compact neighborhood L

(−)
t0

(H) ⊂ U(Lnt0 ) of the compact Lagrangian
submanifold Lnt0 ⊂ M

2n looked for above.
The same construction can be done evidently for the case when the conditions

(10) are changed either by

lim
z→+∞

γt0 (s; z) = γ
(+)
t0

(s), lim
z→−∞

γt0 (s; z) = γt0 (s), (10a)

or by
lim

z→−∞
γt0 (s; z) = γ

(−)
t0

(s), lim
z→∞

γt0 (s; z) = γ
(+)
t0

(s), (10b)

where γ(−)t0 : R →M
2n and γ(+)t0 : R →M

2n are some strictly different hyperbolic
curves on M2n with supp γ(±)t0 ⊂ L

n
t0

and solving (2). Based on (10a) one constructs
similarly the stable manifold W s(γ(+)t0 (s)) to a hyperbolic curve γ

(+)
t0

and further the
corresponding compact neighborhood L(+)t0 (H) ⊂ U(L

n
t0

) of the compact Lagrangian
submanifold Lnt0 ⊂ M

2n which is of crucial importance when studying intersection
properties of stable W s(γ(+)t0 ) and unstable W

u(γ(−)t0 ) manifolds. Based similarly on
(10b), one constructs the neighborhood Lt0 (H) ⊂ U(Lnt0 ) of the compact Lagrangian
submanifold Lnt0 ⊂ M

2n being of interest when investigating so called adiabatic
perturbations of integrable autonomous Hamiltonian flows on the symplectic manifold
M2n.

Now we make use of some statements [3, 5, 12] about the properties of the set ΩH
constructed above. For a generic choice of the Hamiltonian function H : M2n×S1 → R
the functional space of curves ΩH is proved to be finite-dimensional what gives rise
right away to hereditary finite-dimensionality of the neighborhood L(−)t0 (H) with the
compact manifold structure. To see this linearize equation (9) in the direction of a
vector field ξ ∈ T(ΩH ). This leads to the linearized first-order differential operator:

Ft0 (u)ξ := ∇zξ + J(u)∇sξ + ∇ξJ(u)∂u/∂s + ∇ξ∇H(u; t0 + s), (11)

where u ∈ ΩH satisfies the following equation stemming from (9) :

∂u/∂z + J(u)∂u/∂s + ∇H(u; s + t0) = 0 (12)



54 D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)

and ∇z, ∇s and ∇ξ denote here the corresponding covariant derivatives with
respect to the metric (7) on M2n. If u ∈ ΩH satisfies (12), the curve γt0 in M2n

has supp γt0 ⊂ Lnt0 and a curve γ
(−)
t0

in Lnt0 is hyperbolic and nondegenerate [3],
then the operator Ft0 (u) : T(ΩH ) → T(ΩH ) defined by (11) is a Fredholm operator
[12] between appropriate Sobolev spaces. The corresponding pair (H,J) with J :
M2n → End(T(M2n)) satisfying (7) is called regular [3] if every hyperbolic solution
to (2) is nondegenerate [1, 3] and the operator Ft0 (u) is onto for u ∈ ΩH. In general
one can prove that the space (H,J )reg ⊂ (H,J ) of regular pairs (H,J) ∈ (H,J ) is
dense with respect to the C∞-topology. Thus, for the regular pairs it follows from
an implicit function theorem [1] that the space ΩH (γ

(−)
t0

) is indeed for any curve γt0
with supp γt0 ⊂ Lnt0 a finite-dimensional compact functional submanifold whose local
dimension near u ∈ ΩH (γ

(−)
t0

) is exactly the Fredholm index of the operator Ft0 (u).
As a simple inference from the finite-dimensionality of the set ΩH (γ

(−)
t0

) and its com-
pactness one gets that the corresponding point set L(−)t0 (H) is finite-dimensional and
compact submanifold smoothly imbedded into M2n. The same is evidently true for
the point manifolds L(+)t0 (H) and Lt0 (H) supplying us with compact neighborhoods
of the compact Lagrangian submanifold Lnt0 ⊂ M

2n. Let us specify the structure of
the manifold L(−)t0 (H) more exactly making use of the Floer type analytical results
[3, 8, 12] about the space of solutions to the problem (9) and (10). One has that for
any two curves γ(−), γ : [0,τ] → Lnt0 × S

1 satisfying the system (2), the following
functional

Φ(τ )t0 (u) :=
1
τ

∫ τ
0

ds

∫
R
dz(|∂u/∂z|2 + |∂u/∂s − XH (u; s + t0)|

2) (13)

if bounded satisfies the characteristic equality

Φ(τ )t0 (u) = A
(τ )
t0

(γ(−)) − A(τ )t0 (γ) (14)

for any τ ∈ R. Thereby, in the case when the right hand side of (14) doesn’t vanish, the
functional space ΩH (γ(−)) will be a priori nontrivial. Similarly, for any u ∈ L

(+)
t0

(H)
one finds that

Φ(τ )t0 (u) = A
(τ )
t0

(γ) − A(τ )t0 (γ
(+)), (14a)

where the corresponding curve γ(+)t0 : [0,τ] → M
2n satisfies the system (2), is hy-

perbolic having supp γ(+)t0 ⊂ L
n
t0
, and the curve γt0 : [0,τ] → M2n also satisfies the

system (2) having supp γt0 ⊂ Lnt0, and at last, for u ∈ Lt0 (H)

Φ(τ )t0 (u) = A
(τ )
t0

(γ(−)) − A(τ )t0 (γ
(+)), (14b)

where γ(±) : [0,τ] → M2n × S1, τ ∈ R, are taken to be strictly different, hyperbolic
and having supp γ(±) ⊂ Lnt0. The case when γ

(+)
t0

= γ(−)t0 needs some modification
of the construction presented above on which we shall not dwell here. Thus we
have constructed the corresponding neighborhoods L(±)t0 (H) and Lt0 (H) of the com-
pact Lagrangian submanifold Lnt0 ⊂ M

2n consisting of all bounded solutions to the



7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ... 55

corresponding equations (9), (10) and (10a,b). Based now on this fact and the ana-
lytical expressions (14) and (14a,b) one derives the following important lemma.

Lemma 1.1. All neighborhoods L(±)t0 (H) and Lt0 (H) constructed via the scheme
presented above are compact and invariant with respect to the Hamiltonian flow of
diffeomorphisms ψs ∈Diff(M2n) × S1, s ∈ R.

Let us consider below the case of the neighborhood Lt0 (H) ⊂ M2n. The preceding
characterization of the space of curves ΩH leads us following Mather’s approach [9] to
another important for applications description of the compact neighborhood Lt0 (H)
by means of the space of normalized probability measures Mt0 (H) := M(T(Lt0 (H))×
S) with compact support and invariant with respect to our Hamiltonian ψ-flow of
diffeomorphisms ψs ∈Diff(M2n) × S1, s ∈ R, naturally extended on T(Lt0 (H)) × S.
The Hamiltonian ψ-flow due to Lemma 1.1 can be reduced invariantly upon the
compact submanifold Lt0 (H) × S ⊂ M2n × S. For the behavior of this reduced ψ-
flow upon Lt0 (H) × S to be studied in more detail let us assume that our extended
Hamiltonian ψ∗-flow on T(Lt0 (H))× S is ergodic, that is the limτ→∞ A

(τ )
t0

(γ) doesn’t
depend on initial points (u0, u̇0; t0) ∈ T(Lt0 (H)) × S.

Recall now that the basic result [13] in functional analysis (the Riesz representation
theorem) states that the set of Borel probability measures on a compact metric space
X is a subset of the dual space C(X)∗ of the Banach space C(X) of continuous
functions on X. It is obviously a convex set and it is well known [13] to be metrizable
and compact with respect to the weak topology on C(X)∗ defined by C(X), also
called the weak (∗)-topology. The restriction of this topology to the set of Borel
measures is frequently called the vague topology on measures [9]. Since the space
Pt0 := T(Lt0 (H))×S is metrizable and can be as well compactified, it follows that the
set of Borel probability measures on Pt0 is a metrizable, compact and convex subset
of the dual to the Banach space of continuous functions on Pt0. The corresponding
set Mt0 (H) is then evidently a compact, convex subset of this set. The well known
result of the Kryloff and Bogoliuboff [14] states that any ψ-flow on a compact metric
space X has an invariant probability measure. This result one can suitably adapt [9]
to our metric compactified space Pt0 := T(Lt0 (H)) × S as follows. Take a trajectory
γ ∈ ΩH of the extended ψ∗-flow on Pt0 with supp γ ⊂ Lt0 (H) × S defined on a time
interval [0,τ] ⊂ R and let a measure µτ on T(Lt0 (H))×S be evenly distributed along
the orbit γ. Then evidently ||ψs∗µτ −µτ|| ≤ 2s/τ for s ∈ [0,τ]. Denote by µ a point of
accumulation of the set {µτ : τ ∈ R+} as τ → ∞ with respect to the before mentioned
vague topology. For any continuous function f ∈ C(Pt0 ), any s ∈ R and any τ0,ε > 0
there exists τ > τ0 such that |

∫
Pt0

f ◦ψs̄∗dµ−
∫
Pt0

f ◦ψs̄∗dµτ| < ε for s̄ ∈ {0,s}. Then
it follows from the above estimations

|
∫
Pt0

f ◦ ψs∗dµ −
∫
Pt0

fdµ| ≤ |
∫
Pt0

f ◦ ψs∗dµ −∫
Pt0

f ◦ ψs∗dµτ| + |
∫
Pt0

f ◦ ψs∗dµτ −
∫
Pt0

fdµτ| + |
∫
Pt0

fdµτ −∫
Pt0

fdµ| ≤ 2ε + ||f|| ||ψs∗µτ − µτ|| ≤ 2ε + 2s||f||/τ,



56 D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)

that is |
∫
Pt0

f ◦ ψs∗dµ −
∫
Pt0

fdµ| = 0 since ε > 0 can be taken arbitrarily small and
τ0 > 0 arbitrarily large. Thereby one sees that the constructed measure µ ∈ Mt0 (H),
that is it is normalized and invariant with respect to the extended Hamiltonian ψ∗-flow
on Pt0.

Thus, in the case of ergodicity of the ψ∗-flow on T(Lt0 (H)) × S the mentioned
above limit

lim
τ→∞

A(τ )t0 (γ) =
∫
Pt0

(α(1) − H)dµ, (15)

with 1-form α(1) ∈ Λ1(M2n) being considered above as a function α(1) : Pt0→ R,
since the submanifold Lt0 (H) by construction is compact and invariantly imbedded
into M2n due to Lemma 1.1. So, it is natural to study properties of the functional

At0 (µ) :=
∫
Pt0

(α(1) − H)dµ (16)

on the space Mt0 (H), where we omitted for brevity the natural pullback of the 1-form
α(1) ∈ Λ1(M2n) upon the invariant compact submanifold Lt0 (H) ⊂ M2n. Being in-
terested namely in ergodic properties of ψ∗-orbits on T(Lt0 (H))×S), we shall develop
below an analog of the J. Mather Lagrangian measure homology technique [9, 10] to a
more general and complicated case of the reduced Hamiltonian ψ-flow on the invariant
compact submanifold Lt0 (H) ⊂ M2n. In particular, we shall construct an analog of
the so called Mather β-function [9] on the homology group H1(Lt0 (H); R) whose linear
domains generate exactly ergodic components of a measure µ ∈ Mt0 (H) minimizing
the functional (16), being of great importance for studying regularity properties of
ψ∗-orbits on T(Lt0 (H))×S. The results can be extended further to adiabatically per-
turbed integrable Hamiltonian systems depending on a small parameter ε ↓ 0 via the
continuous dependence H(t) := H̃(εt), where H̃(τ + 2π) = H̃(τ) for all τ ∈ [0, 2π].
It makes also possible to state the existence of so called adiabatic invariants with
compact supports in Lt0 (H) having many applications in mathematical physics and
mechanics. Some of the results can be also applied to investigating the problem of
transversal intersections of corresponding stable and unstable manifolds to hyperbolic
curves or singular points, related closely with existence of highly irregular motions in
a periodic time-dependent Hamiltonian dynamical system under regard.

2 Invariant measures and mather’s type β-function

Before studying the average functional (16) on the measure space Mt0 (H), let us first
analyze properties of the functional∮

σ

a(1) :=≺ a(1),σ � (17)

on H1(Lt0 (H); R) at a fixed σ ∈ H1(Lt0 (H); R). Since the 1-form a(1) ∈ H1(Lt0 (H); R)
in (17) can be considered as a function a(1) : Pt0→ R, in virtue of the Riesz theorem



7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ... 57

[13] there exists a Borel measure µ : Pt0 → R+ (still not necessary ψ-invariant), such
that

≺ a(1),σ �=
∫
Pt0

a(1)dµ. (18)

The following lemma characterizing the right hand side of (18) holds.
Lemma 2.1. Let a 1-form a(1) = dλ(0) ∈ Λ1(Lt0 (H)) be exact, that is the cohomology
class [dλ(0)] = 0 ∈ H1(Lt0 (H); R). Then for any µ ∈ Mt0 (H)∮

σ

a(1) = 0. (19)

C Really, for a(1) = dλ(0), where λ(0) : Lt0 (H) → R is an absolutely continuous
mapping, the following holds due to The Fubini theorem for any τ ∈ R+ :

|
∫
Pt0

dλ(0)dµ.| = |1
τ

∫ τ
0
ds

∫
Pt0

dλ(0)(ψs∗dµ)| =
|1
τ

∫
Pt0

dµ
∫ τ
0
dsd(λ(0) ◦ ψs∗)/ds|

= |1
τ

∫
Pt0

dµ[λ(0) ◦ ψτ∗ − λ(0) ◦ ψ0∗]| ≤ 2||λ(0)||/τ.
(20)

The latter inequality as τ → ∞ gives rise to the wanted equality (19), that proves
the lemma.B

Thus, the right hand side of (18) defines a true functional

H1(Lt0 (H); R) 3 a
(1) →

∫
Pt0

a(1)dµ ∈ R (21)

on the cohomology space H1(Lt0 (H); R). All the above can be formulated as the
following theorem.

Theorem 2.2. Let an element σ ∈ H1(Lt0 (H); R) be fixed. Then there exists a ψ-
invariant probability measure (not unique) µ ∈ Mt0 (H), such that the representation
(18) holds and vice versa, for any measure µ ∈ Mt0 (H) there exists the homology
class σ := ρt0 (µ) ∈ H1(Lt0 (H); R), such that

≺ a(1),ρt0 (µ) �=
∫
Pt0

a(1)dµ (22)

for all a(1) ∈ H1(Lt0 (H); R).
Definition 2.3. ([10]) For any measure µ ∈ Mt0 (H) the homology class ρt0 (µ) ∈

H1(Lt0 (H); R) is called its homology.
Corollary 2.4. The homology mapping ρt0 : Mt0 (H) → H1(Lt0 (H); R) defined

within Theorem 2.2 is surjective.
C Sketch of a proof of Theorem 2.2. The fact that for each µ ∈ Mt0 (H)

there exists the unique homology class σ := ρt0 (µ) ∈ H1(Lt0 (H); R) is based on
the well known Poincare duality theorem [1]. The inverse statement is about the
surjectivity of the mapping ρt0 : Mt0 (H) → H1(Lt0 (H); R). For it to be stated, con-
sider following [8-10] a covering space Lt0 (H) over Lt0 (H) defined by the condition



58 D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)

that π1(Lt0 (H)) = ker ht0, where ht0 : π1(Lt0 (H)) → H1(Lt0 (H); R) denotes the
Hurewicz homomorphism [10]. Since in reality the functional (22) is defined on the
covering space Lt0 (H), it is necessary to lift all curves γ ∈ ΩH on Lt0 (H)×S to curves
γ̃ ∈∈ Ω̃H on Lt0 (H)×S. In the case when the homotopy group π1(Lt0 (H)) is abelian,
the covering space L̃t0 (H) becomes universal, but in general it is obtained as some
universal covering of L̃t0 (H) quotioned further with respect to the action of the kernel
of the corresponding Hurewicz homomorphism ht0 : π1(Lt0 (H)) → H1(Lt0 (H); R).

Take now any element σ ∈ H1(Lt0 (H); R) and construct a set of approximating
it so called Deck transformations τ−1στ ∈ im ht0 ⊂ H1(Lt0 (H); R), τ ∈ R+, such
that weakly limτ→∞ τ−1στ = σ holds. Put further x̃τ := στ ◦ x̃0 ∈ Lt0 (H) × S,
τ ∈ R+, where x̃0 ∈ Lt0 (H) × S is taken arbitrary and consider such a curve γ̃ :
[0,τ] → Lt0 (H) × S with end-points γ̃(0) = x̃0, γ̃(τ) = x̃τ whose projection on
Lt0 (H)×S is the curve γ ∈ Σt0 (H), minimizing the functional (6). Consider also a set
{µτ : τ ∈ R+} of probability measures on Pt0 evenly distributed along corresponding
curves γ ∈ Σt0 (H) for each τ ∈ R+ and denote by µ a point of its accumulation
as τ → ∞. Due to the uniform distribution of measures µτ, τ ∈ R+, along curves
γ ∈ Σt0 (H) having the end-points agreed with chosen above Deck transformations
στ ∈ H1(Lt0 (H); R), τ ∈ R+, one gets right away from the Birkhoff-Khinchin ergodic
theorem [1, 2] that ∫

Pt0
a(1)dµτ =≺ a(1),τ−1στ ) � (23)

for any a(1) ∈ H1(Lt0 (H); R). Passing now to the limit in (23) as τ → ∞ and taking
into account that weakly limτ→∞ τ−1στ = σ, one gets right away that the equality
(22) holds for some measure µ ∈ Mt0 (H), such that ρt0 (µ) = σ ∈ H1(Lt0 (H); R),
thereby giving rise to the surjectivity of the mapping ρt0 : Mt0 (H) → H1(Lt0 (H); R)
and proving the theorem. B

Return now to treating the average functional (16) subject to the space of all
invariant measures Mt0 (H). Namely, consider the following β-function
βt0 : H1(Lt0 (H); R) → R defined as

βt0 (σ) := inf
µ
{At0 (µ) : ρt0 (µ) = σ ∈ H1(Lt0 (H); R)} (24)

It will be further called a Mather type β-function due to its analogy to the definition
given in [9,10]. The following lemma holds.

Lemma 2.5. Let a 1-form a(1) ∈ H1(Lt0 (H); R) be taken arbitrary. Then the
Mather type β-function

β
(a)
t0

(σ) := inf
µ
{A(a)t0 (µ) : ρt0 (µ) = σ ∈ H1(Lt0 (H); R)}, (25)

where by definition

A(a)t0 (µ) :=
∫
Pt0

(α(1) + a(1) − H)dµ, (26)

satisfies the following equation:

β
(a)
t0

(σ) = βt0 (σ)+ ≺ a
(1),σ) � . (27)



7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ... 59

C The proof easily stems from the definition (25) and the equality (22). B
Assume now that the infimum in (24) is attained at a measure µ(σ) ∈ Mt0 (H).

Then evidently, ρt0 (µ(σ)) = σ for any homology class σ ∈ H1(Lt0 (H); R). Denote by
M(σ)t0 (H) the set of all minimizing the functional (24) measures of Mt0 (H). In the
next chapter we shall proceed on study its ergodic and homology properties.

3 Ergodic measures and their homologies

Consider the introduced above Mather type β-function β(a)t0 : H1(Lt0 (H); R) → R for
any a(1) ∈ H1(Lt0 (H); R). It is evidently a convex function on H1(Lt0 (H); R), that
is for any λ1,λ2 ∈ [0, 1], λ1 + λ2 = 1, and σ1,σ2 ∈ H1(Lt0 (H); R) there holds the
inequality

β
(a)
t0

(λ1σ1 + λ2σ2) ≤ λ1β
(a)
t0

(σ1) + λ2β
(a)
t0

(σ2). (28)

As usually dealing with convex functions, one says that an element σ ∈ H1(Lt0 (H); R)
is extremal point [13] if β(a)t0 (λ1σ1 + λ2σ2) < λ1β

(a)
t0

(σ1) + λ2β
(a)
t0

(σ2) for all λ1,λ2 ∈
(0, 1), λ1 + λ2 = 1, and σ = λ1σ1 + λ2σ2. Correspondingly, we shall call a convex set
Zt0 (H) ⊂ H1(Lt0 (H); R) by a linear domain of the Mather type function (25) if

β
(a)
t0

(λ1σ1 + λ2σ2) = λ1β
(a)
t0

(σ1) + λ2β
(a)
t0

(σ2) (29)

for any σ1,σ2 ∈ Zt0 (H) and λ1,λ2 ∈ R. It is easy to see now that if σ ∈ H1(Lt0 (H); R)
is extremal, then the set M(σ)t0 (H) contains [15] ergodic minimizing measure compo-
nents. Namely, following [9, 10] one states that if Zt0 (H) is a linear domain and
P(σ)t0 ⊂ Pt0 is the closure of the union of the supports of measures µ(σ) ∈ M

(σ)
t0

(H)
with σ ∈ Zt0 (H), then the set P

(σ)
t0

is compact and the inverse mapping (pt0|P(σ)t0
)−1 :

pt0 (P
(σ)
t0

) → P(σ)t0 is Lipschitzian, where pt0 : Pt0 → Lt0 (H) × S is the standard pro-
jection, being injective upon P(σ)t0 . Moreover, one can show [9] that if a measure
µ ∈ M(σ)t0 (H) is minimizing the functional (26), then its support supp µ ⊂ P

(σ)
t0

and
all its ergodic components {µ̄} are minimizing this functional too, and the convex
hull of the corresponding homologies conv{ρt0 (µ̄)} is a linear domain Z

(σ)
t0

(H) of the
Mather type β-function (25). These results are of very interest concerning many ap-
plications in dynamics. Especially, the ergodic measures, as is well known, possess
the crucial property that every invariant Borel set has measure either 0 or 1, giving
rise to the following important equality:

lim
τ→∞

A(τ )t0 (γ) = At0 (µ̄)) (30)

uniformly on (γt0,(0), γ̇t0 (0); t0) ∈ Pt0∩ supp µ̄, where γ ∈ Σt0 (H). All of the prop-
erties formulated above are inferred from the following theorem modeling the similar
one in [10].

Theorem 3.1. Let a measure µ ∈ Mt0 (H) be minimizing the functional (26)
satisfying the condition β(a)t0 (ρt0 (µ)) = At0 (µ). Then supp µ ⊂ Σt0 (H) and the convex



60 D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)

hull of the set of homologies ρt0 (µ̄) ∈ H1(Lt0 (H); R), where {µ̄} ⊂ Mt0 (H) are the
corresponding ergodic components of the measure µ ∈ Mt0 (H), is a linear domain
Zt0 (H) of the Mather type β-function (25).

C Sketch of a proof. Let ht0 : π1(Lt0 (H)) → H1(Lt0 (H); R) be the corresponding
Hurewicz homomorphism and take some basis σk ∈ im ht0 ⊂ H1(Lt0 (H); R), k = 1,r,
where r = dim im ht0, being its dual basis a

(1)
j ∈ H

1(Lt0 (H); R), j = 1,r. Then for
any points x̃, ỹ ∈ Lt0 (H) × S one can define an element ξ(τ )(x̃, ỹ|γ̃) ∈ H1(Lt0 (H); R)
as the sum

ξ(τ )(x̃, ỹ|γ̃) :=
1
τ

r∑
j=1

σj

∫ τ
0

ã
(1)
j (γ̃), (31)

where γ : [0,τ] → Lt0 (H) × S is any continuous arc joining these two chosen points
x̃, ỹ ∈ Lt0 (H) × S, and ã

(1)
j ∈ H

1(Lt0 (H); R) are the corresponding liftings to Lt0 (H)
of 1-forms a(1)j ∈ H

1(Lt0 (H); R), j = 1,r. One can show then that if µ ∈ Mt0 (H)
is ergodic and supp µ ⊂ Σt0 (H), then the measure µ is minimizing the functional
(26). Put σ := ρt0 (µ) and let a set Zt0 (H) ⊂ H1(Lt0 (H); R) be a supporting
domain containing this homology class σ ∈ H1(Lt0 (H); R). Thus, one can see that the
extremal points of the convex set Zt0 (H) are extremal points also of the Mather type
β-function (25). Next expand the homology class σ = ρt0 (µ) as a convex combination
of extremal points σ̄j ∈ Zt0 (H), j = 1,m, for some m ∈ Z+. Then, since elements
σ̄j ∈ Zt0 (H), j = 1,m, are extremal, there exist ergodic measures µ̄j ∈ M

(σj )
t0

(H),
j = 1,m, such that ρt0 (µ̄j ) = σ̄j, j = 1,m. Moreover, since Z

(σ)
t0

(H) is a linear
domain, one easily brings about that

β
(a)
t0

(σ) =
m∑

j=1

cjβ
(a)
t0

(σ̄j ) =
m∑

j=1

cjA
(a)
t0

(µ̄j ), (32)

where σ =
∑m

j=1 cjσ̄j with some real coefficients cj ∈ R, j = 1,m. Due to the
ergodicity of the measure µ ∈ Mt0 (H) from the Birkhoff-Khinchin ergodic theorem
[1] one derives that there exists an orbit γ̃ : [0,τ} → Lt0 (H) × S with the supp γ ⊂
supp µ, such that the property (30) together with the equality

σ := ρt0 (µ) = lim
τ→∞

ξ(τ )(x̃, ỹ|γ̃) (33)

hold. Further, there exist curves γ̃j ∈ Σt0 (H), supp γj ⊂ supp µ̄j, j = 1,m, such the
expressions

σ̄j := ρt0 (µ̄j ) = lim
τ→∞

ξ(τ )(x̃, ỹ|γ̃j ) (34)

as well as β(a)t0 (σ̄j ) = A
(a)
t0

(µ̄j ) = limτ→∞ A
(τ )
t0

(γ̃j ) hold for every j = 1,m. Under the
conditions (14b) involved on the invariant neighborhood Lt0 (H) one shows that for
any measure µ ∈ Mt0 (H) such that ρt0 (µ) = σ, the inequality A

(a)
t0

(µ) ≤ β(a)t0 (ρt0 (µ))
holds thereby proving its minimality. Suppose now that the measure µ ∈ Mt0 (H)
has all its ergodic components with supports contained in Σt0 (H) and the convex



7, 3(2005)
The ergodic measures related with nonautonomous hamiltonian ... 61

hull of its homologies is a linear domain of the Mather type function (25). One can
approximate (in the weak topology) a measure µ ∈ Mt0 (H) by means of a convex
combination µ̂ :=

∑m
j=1 ĉjµ̄j, where ĉj ∈ R and µ̄j ∈ Mt0 (H), j = 1,m, are ergodic

components of the measure µ ∈ Mt0 (H). Then supp µ̄j ⊂ Σt0 (H) implying that all
µ̄j ∈ Mt0 (H), j = 1,m, are minimizing (26), that is are minimal. Therefore, since
the convex hull of homologies {ρt0 (µ̄j ) ∈ H1(Lt0 (H); R) : j = 1,m} is a linear domain
due to its minimality, one gets that

A(a)t0 (µ̂) =
∑m

j=1 ĉjA
(a)
t0

(µ̄j ) =
∑m

j=1 ĉjβ
(a)
t0

(ρt0 (µ̄j ))
= β(a)t0 (ρt0 (

∑m
j=1 ĉjµ̄j )) = β

(a)
t0

(ρt0 (µ),
(35)

meaning evidently that the measure µ̂ ∈ Mt0 (H) is minimal too. Making use now
of the fact that limits of minimizing measures are minimizing too, one obtains finally
that the measure µ ∈ Mt0 (H) is minimizing the functional (26), thereby proving the
theorem. B

Consider some properties of a so called [10] supporting domain

Z
(a)
t0

(H) := {σ ∈ H1(Lt0 (H); R) : β
(a)
t0

(σ) =≺ a(1),σ � +c(a)t0 } (36)

for the Mather type β-function (25) at some fixed a(1) ∈ H1(Lt0 (H); R) with c
(a)
t0

∈
R properly defined by (27). Define also by P(a)t0 := ∪σ∈Z(a)t0 (H)

supp µ(σ), where

µ(σ) ∈ Mt0 (H) and ρt0 (µ(σ)) = σ ∈ Z
(a)
t0

(H). Present now a supporting domain
Z

(a)
t0

(H) ⊂ H1(Lt0 (H); R) due to the expression (27) as follows:

Z
(a)
t0

(H) = {σ ∈ H1(Lt0 (H); R) : β
(0)
t0

(σ) = c(a)t0 }, (37)

where the function β(0)t0 : H1(Lt0 (H); R) being bounded from below is chosen in such
a way that β(0)t0 (σ) ≥ c

(a)
t0

for all σ ∈ H1(Lt0 (H); R). Take now a measure µ ∈ Mt0 (H)
and suppose that supp µ ⊂ Σt0 (H). Since β

(0)
t0

(σ) ≥ c(a)t0 for all σ ∈ H1(Lt0 (H); R)
and due to (37) Z(a)t0 (H) = (β

(0)
t0

)−1{c(a)t0 } at some fixed a
(1) ∈ H1(Lt0 (H); R), this

evidently implies that the measure µ ∈ Mt0 (H) is minimizing the functional (26) and
ρt0 (µ) ∈ Z

(a)
t0

(H). Thereby the following theorem is stated.
Theorem 3.2. Suppose that Z(a)t0 (H) ⊂ H1(Lt0 (H); R) is a supporting domain of

the Mather type function (27) and a measure µ ∈ Mt0 (H) satisfies the condition supp
µ ⊂ Σt0 (H). Then this measure µ ∈ Mt0 (H) is minimizing and ρt0 (µ) ∈ Z

(a)
t0

(H).
The following corollaries from the Theorem 3.2 as in [10] hold.
Corollary 3.3. The minimizing measure µ ⊂ Mt0 (H) with supp µ ⊂ Σt0 (H)

satisfies the condition A(0)t0 (µ) = c
(a)
t0
. By means of choosing the element a(1) ∈

H1(Lt0 (H); R) one can make the value c
(a)
t0

be zero, that is one can put c(a)t0 = 0.
Corollary 3.4. For any strictly extremal closed curve σ ∈ H1(Lt0 (H); R) the

following properties take place:
i) there exists an ergodic measure µ̄(σ) ∈ Mt0 (H) whose support is a minimal

set and ρt0 (µ̄(σ)) = σ;



62 D.L.Blackmore, Y.A.Prykarpatsky, A.M.Samoilenko & A.K.Prykarpatsky
7, 3(2005)

ii) for every closed 1-form a(1) ∈ H1(Lt0 (H); R) the equality ≺ a(1),σ �=
limτ→∞ 1τ

∫ t0+τ
t0

a(1)(γ̇)ds holds uniformly for all (γt0(0), γ̇t0(0); t0) ∈ Pt0∩ supp µ̄(σ),
ρt0 (µ̄(σ)) = σ and γ ∈ Σt0 (H);

iii) if (γt0(0), γ̇t0(0); t0) ∈ Pt0∩ supp µ̄(σ), ρt0 (µ̄(σ)) = σ and γ ∈ Σt0 (H) is the
corresponding orbit in Lt0 (H) × S, then β

(a)
t0

(σ) = limτ→∞ A
(τ )
t0

(γ) uniformly.
The statements formulated above can be effectively used for studying dynamics

of many perturbed integrable Hamiltonian flows and their regularity properties. As
it is well known, they are strongly based on the intersection theory of stable and
unstable manifolds related with hyperbolic either closed orbits or singular points of
a Hamiltonian system under regard. These aspects of our study of ergodic measure
and homology properties of such Hamiltonian flows are supposed to be treated in a
proceeding article under preparation.

Received: June 2004. Revised: January 2005.

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