ON ALMOST AUTOMORPHIC OSCILLATIONS 1. Introduction Almost automorphic functions ...

26 Pages·2012·0.22 MB·English

ON ALMOST AUTOMORPHIC OSCILLATIONS
YINGFEIYI
Abstract. Almostautomorphicdynamicshavebeengivenanotableamount
of attention in recent years with respect to the study of almost periodically
forced monotone systems. There are solid evidences that these types of dy-
namicsshouldalsobeoffundamentalimportanceinnon-monotoneespecially
conservative systems due to the interaction of multi-frequencies. Thisarticle
will give a preliminary discussion in this regard by reviewing certain known
casesandraisingsomeproblemsforpotentialfuturestudies.
1. Introduction
Almostautomorphicfunctions,generatingthe(Bohr)almostperiodicones,were
(cid:12)rst introduced by Bochner ([12]) in 1955 in a di(cid:11)erential geometry context, and,
withaminormodi(cid:12)cation,theywereshownin[27,94]tocoincidewiththeLevitan
classofN-almostperiodicfunctions([56]). LetT bealocallycompact,(cid:27)-compact,
Abelian, (cid:12)rstcountable topologicalgroupandlet E beacompletemetricspace. A
functionf C(T;E)issaidtobealmostautomorphicifwhenever t isasequence
n
2 f g
such that f(t +t) g(t) C(T;E) uniformly on compact sets, then g(t t )
n n
! 2 (cid:0) !
f(t) uniformly on compact sets, as n . f becomes almost periodic if any
! 1
sequence tn admitsasubsequence tn0 suchthatf(tn0+t)convergesuniformlyon
f g f g
T, as n . In the literature, the abovenotion of almost automorphyis referred
!1
to as sequential or continuous almost automorphy. In fact, almost automorphic
functions(and(cid:13)ows)canbede(cid:12)nedinanabstractfashiononanytopologicalgroup
with respect to pointwise net convergence (see [27, 94, 98] for details). A function
is sequential almost automorphic i(cid:11) it is net almost automorphic and uniformly
continuous ([98]). We choose the sequential notion in this paper because of its
convenience in the applications to di(cid:11)erential equations.
An almost periodic function is necessarily almost automorphic, but not vice
versa. One can de(cid:12)ne Fourier series for both almost periodic and almost auto-
morphic functions valued in a Banach space but the one for an almost periodic
function is unique and converges uniformly in terms of Bochner-Fejer summation,
while the one for an almost automorphic function is in general non-unique and its
Bochner-Fejer sum only converges pointwise ([98]). Although there can be many
Fourier series associated with a given almost automorphic function, one can de(cid:12)ne
the frequency module of an almost automorphic function in the usual way as the
smallestAbeliangroupcontainingaFourierspectrum-thesetofFourierexponents
associatedwithaFourierseries,and,ithasbeenshownthatsuchafrequencymod-
uleisuniquelyde(cid:12)ned([106]). Intheabovesense,bothalmostperiodicandalmost
1991 Mathematics Subject Classi(cid:12)cation. Primary 34C27, 34C28, 37B55; Secondary 54H20,
58F14,58F05,58F30.
Keywordsandphrases. Almostautomorphicdynamics,multi-frequency,nonlinearoscillations.
PartiallysupportedbyNSFgrantDMS0204119.
1
2 YINGFEIYI
automorphicfunctionscanbeviewedasnaturalgeneralizationstotheperiodicones
in the strongest and the weakest sense respectively.
Almost automorphic minimal (cid:13)ows were (cid:12)rst introduced and studied by Veech
([97]-[100]). A (cid:13)ow (E;T) is called almost automorphic (almost periodic) min-
imal if E is the closure of an almost automorphic (almost periodic) orbit. An
almost automorphic minimal (cid:13)ow contains residually many almost automorphic
points and becomes almost periodic only if every point in E is almost automor-
phic ([98, 100]). Typical examples of almost automorphicminimal sets include the
well known Toeplitz minimal sets in symbolic dynamics ([19, 61]), the Denjoy set
([17]) on the circle, and the Aubry-Mather sets ([5, 62]) on an annulus. Unlike
an almost periodic minimal (cid:13)ow, an almost automorphic one can be non-uniquely
ergodic and can admit positive topological entropy ([61]), and its general measure
theoretical characterization is completely random ([31]). Topologically, while an
almost periodic minimal set is always a compact topological group ([22]), a non-
almostperiodic, almostautomorphicone is only an almost 1-coverof a topological
group ([98]) and can be topologically complicated ([46]). Hence, on one hand, an
almost automorphic (cid:13)ow resembles an almost periodic one harmonically, but on
the other hand, it presents certain complicated dynamical, topological and mea-
sure theoretical features which are signi(cid:12)cantly di(cid:11)erent from an almost periodic
one.
Systematicstudiesofalmostautomorphicdynamicsindi(cid:11)erentialequationswere
made in a seriesof recent worksof the authorwith Shen ([89]-[93]) with respect to
almostperiodicallyforcedmonotonesystemswhichareroughlythoseadmittingno
internalfrequencies. Consideraskew-product semi-(cid:13)ow(cid:25) overanalmost periodic
t
minimal base (cid:13)ow. Loosely speaking, it has been shown that if (cid:25) is (cid:12)ber-wise
t
totallymonotone (e.g., skew-product (cid:13)ows and semi-(cid:13)ows generatedby almostpe-
riodicallyforcedscalarODEsandalmostperiodicallyforcedparabolicPDEsinone
space dimension, respectively), then all its minimal sets are almost automorphic
with frequencymodules respondingtothatof the base(cid:13)owharmonically, and,if it
is (cid:12)ber-wise strongly monotone (e.g., skew-product (cid:13)ows and semi-(cid:13)ows generated
by cooperative almost periodic systems of ODEs and almost periodically forced
parabolicPDEsin higherspacedimensions,respectively),then eachlinearlystable
minimalsetisalmostautomorphicwithfrequencymodulerespondingtothatofthe
base (cid:13)ow sub-harmonically,and moreover,a minimal set in the (cid:12)ber-wise strongly
monotone skew-product semi-(cid:13)ow (cid:25) over an almost periodic minimal base (cid:13)ow
t
becomes almost periodic if it is uniformly stable. It is well known that even in the
simplest almost periodically forced monotone system such as a quasi-periodically
forcedscalarODEwith onlytwofrequenciesandalinearscalarODEwith limiting
periodic coe(cid:14)cients, almost periodic motions need not exist ([26, 46, 56, 75, 93]).
Thus,the(cid:12)ndingofalmostautomorphicdynamicsin[89]-[93]actuallyshowsafun-
damentalphenomenoninalmostperiodicallyforcedmonotonesystems,i.e.,almost
automorphic solutions largely exist but almost periodic ones need not. We refer
the readersto [85, 93] for referenceson the study of almost automorphicdynamics
in almost periodically forced di(cid:11)erential systems and to [3, 41, 42, 74, 76, 86, 88]
for recent developments in the subject.
Comparing with periodically forced monotone systems in which periodic solu-
tions are generically expected, the existence of almost automorphic solutions in
almost periodically forced monotone systems re(cid:13)ects a general harmonic nature of
ON ALMOST AUTOMORPHIC OSCILLATIONS 3
the systems due to the interactions of several frequencies especially when they are
close to the resonance. It is therefore well expected that almost automorphic dy-
namicsshouldalsolargelyexistinalmostperiodicallyforcednon-monotonesystems
(i.e., systems with self-excitations or containing internal frequencies) and even in
autonomous conservative systems. Some examples of periodically forced nonlinear
oscillators and Hamiltonian systems have already supported this assertion but no
systematic investigation is made to the existence of almost automorphic dynamics
in general almost periodically forced non-monotone systems yet. In this article,
we will review some known cases of the existence of almost automorphic dynamics
in non-monotone systems and formulate some general open problems as potential
starting points in the subject. We are also attempting to link the study of almost
automorphic oscillations with other active areas in dynamical systems such as the
almostperiodicFloquettheory,Aubry-Mathertheory,chaos,Hamiltoniansystems,
non-chaotic strange attractors, nonlinear oscillations, and toral (cid:13)ows. Problems
arising in multi-frequency, non-monotone systems are far more complicated and
challengingthanthoseinmonotonesystems. Atthisstage,wehavemorequestions
than answers. Therefore, instead of an expository article as it was supposed to
be, the present article is rather a research note, aiming at making some prelimi-
nary discussions in this interesting subject, which is by no means complete or well
thought out.
Throughout the paper, we let (Y;R) be an almost periodic minimal (cid:13)ow and
denote y t as the orbit of the (cid:13)ow passing through a point y Y. We will mainly
(cid:1) 2
considerODEs with almostperiodic forcingof the formf(x;y t), wherex Rn is
(cid:1) 2
a state variable (the global existence of solutions of the ODEs is always assumed).
This is without loss of generality. Suppose that an ODE
(1.1) x0 =F(x;t); x Rn;
2
is considered with F being uniformly Lipschitz in x and almost periodic in t
uniformly with respect to x. Let Y = H(F) = cl F (cid:28) R be the hull of
(cid:28)
f j 2 g
y = F under the compact open topology (hence Y is compact metric), where
0
F (x;t) F(x;t+(cid:28)). De(cid:12)ne f : Rn Y Rn: f(x;y) = y(x;0), y Y. Then
(cid:28)
(cid:17) (cid:2) ! 2
the time translationy t=y de(cid:12)nesanaturalalmostperiodicminimal (cid:13)ow(Y;R)
t
(cid:1)
and F(x;t) f(x;y t). Thus instead of studying the single equation (1.1), we
0
(cid:17) (cid:1)
will consider a family of equations
(1.2) x0 =f(x;y t); x Rn;
(cid:1) 2
which consist of the translated and limiting equations of (1.1). To study the dy-
namical behaviors of solutions of (1.1), we note that the equations (1.2) give rise
to a skew-product (cid:13)ow (Rn Y;(cid:25) ):
t
(cid:2)
(cid:25) (x;y)=(X(x;y;t);y t);
t
(cid:1)
where X(x;y;t) denotes the solution of (1.2) with the initial value x. Such a
formulation was originated in [67, 84] in studying dynamics of non-autonomous
di(cid:11)erential equations (see also [48, 105]). We note that if F is quasi-periodic in
t with k frequencies, then Y Tk and (Y;R) is topologically conjugated to the
’
standard quasi-periodic (cid:13)ow on the k-torus.
Part of the material in this article has been lectured at the 7th International
Conferenceon Di(cid:11)erence Equationsand Applications (ICDEA), Changsha, China,
4 YINGFEIYI
2002, the International Conference on Dynamical Methods for Di(cid:11)erential Equa-
tions, Medina, Spain, 2002, and the International Workshop on Global Analysis
of Dynamical Systems, Leiden, The Netherlands, 2001. The author would like to
thank the organizers of these conferences for opportunities to present part of the
resultscontainedin this article, in particularthe organizersforthe 7th ICDEA for
their invitation to write this article.
2. Almost automorphic functions and flows
In this section, unless speci(cid:12)ed otherwise, we let T be a locally compact, (cid:27)-
compact, Abelian, (cid:12)rst countable group, and E be a complete metric space.
2.1. Harmonicproperties. Fouriertheorywas(cid:12)rstdevelopedin[98]forcomplex
valued almost automorphic functions. ParallelFourier theory can be also found in
[56] for N-almost periodic functions.
Theorem 2.1. Letf be an almost automorphic function on T valued in a Banach
space. Then f admits Fourier series (not necessarily unique) whose Bochner-Fejer
sums converge to f pointwise (in fact, uniformly on compact sets).
The theorem was (cid:12)rst proved in [98] for complex valued functions and was gen-
eralized in [106] for almost automorphic functions on R valued in a Banach space.
The same proof can be carried over for general T.
Let f be an almost automorphicfunction on T. Then the topologicalhull H(f)
isalmostautomorphicminimalunderthe‘time’translation(cid:13)owwhichisanalmost
1-coverof its maximal almost periodic factor (Y;T) (see Theorem 2.3 below). The
almost periodic factor is unique up to (cid:13)ow isomorphism and admits a natural
compact Abelian topological group structure inherited from the (cid:13)ow ([22]), e.g.,
when T = R, Y is a solenoidal group ([2]). Hence the dual group Y0 is discrete.
We de(cid:12)ne the frequency module (f) of f asthe subgroupof Y0 generatedby the
M
Fourierspectrum associatedwith aFourierseriesof f. Thisiswellde(cid:12)nedbecause
of the following theorem which was originally shown in [106] for T = R but also
holds in general.
Theorem 2.2. With respect to any Fourier series of f, (f) Y0.
M ’
The following module containment result can be shown similarly as in [106] for
functions de(cid:12)ned on R.
Proposition 2.1. For two almost automorphic functions f;g C(T;E), (g)
2 M (cid:26)
(f) i(cid:11) whenever f(t+t ) f(t) for some sequence t T then g(t+t )
n n n
M ! f g(cid:26) !
g(t), uniformly on compact sets.
2.2. Structural properties. The following result shown in [98] is known as the
Veech almost automorphic structure theorem.
Theorem 2.3. A compact minimal (cid:13)ow (E;T) is almost automorphic i(cid:11) it is an
almost 1-1 extension of its maximal almost periodic factor (Y;T), i.e., there is a
residual subset Y Y such that each (cid:12)ber over Y is a singleton.
0 0
(cid:26)
ON ALMOST AUTOMORPHIC OSCILLATIONS 5
Let (E;T), (Y;T) be as in the above. It is shown in [100] that as long as there
is a y Y corresponding to a singleton (cid:12)ber, then there are residually many of
0
2
them. The set of points in E lying in the singleton (cid:12)bers consists of precisely
the almost automorphic points, and, E becomes almost periodic i(cid:11) all points in E
are almost automorphic. Again, as (Y;T) is almost periodic minimal, Y admits a
compact Abelian group structure inherited from the (cid:13)ow whose dual group Y0 is
discrete. We thus de(cid:12)ne the frequency module (E) of E as the dual group Y0.
M
Thisde(cid:12)nitionisinconsistentwith thatforanalmostautomorphicfunction which
we introduced above.
Theorem 2.4. 1) A function f C(T;E) is almost automorphic i(cid:11) it is a
2
pointwise limit of a sequence of jointly almost automorphic, almost periodic
functions.
2) f C(T;E) is almost automorphic i(cid:11) there exists a dense subgroup G
2
of a compact group G , a continuous homomorphism (cid:30) : T G, and a
(cid:3)
!
function g : G E which is continuous on G such that f = g (cid:30) is
(cid:3)
! (cid:14)
uniformly continuous.
3) If (cid:30):T D, where D is complete metric, is almost automorphic, g:D
! !
E is continuous on R((cid:30)), then f =g (cid:30) C(T;E) is almost automorphic
(cid:14) 2
if it is uniformly continuous.
Part1)of the theoremwasshownin[98] forcomplexvalued(sequential)almost
automorphic functions. Part 2) of the theorem was stated and shown in [96] for
complex valued function f as follows: f is (net) almost automorphic i(cid:11) there is a
continuous homomorphism (cid:30):T G, where G is a totally bounded group, and a
!
complex valued function g which is continuous on G such that f = g (cid:30). Part 3)
(cid:14)
of the theorem was stated and shown in [99] for complex valued functions f;g as
follows: if (cid:30):T C is (net) almost automorphic and g :R((cid:30)) C is continuous,
! !
then f = g (cid:30) is (net) almost automorphic. With proper modi(cid:12)cations as the
(cid:14)
above,all these resultscan be extended to the onesstated in the theoremby using
similar arguments as originally used in [96, 98, 99].
Parts2)3)oftheabovetheoremisparticularlyusefulinconstructingnon-quasi-
periodic, almost automorphic functions with (cid:12)nite many frequencies. Let ! =
(! ;! ; ;! )
1 2 k
(cid:1)(cid:1)(cid:1)
Rk be a given non-resonant vector and f : Tk Rn be a measurable func-
2 !
tion which is continuouson an orbit (cid:18) +!t:t T of the quasi-periodic(cid:13)ow on
0
f 2 g
Tk with the frequency !, here T = R or Z. As !t : t R is embedded into Tk
f 2 g
as a dense subgroup (or the map R Tk : t (cid:18) +!t is almost periodic hence
0
! 7!
almostautomorphic),animmediateapplicationofpart2)or3)ofTheorem2.4and
Proposition 2.1 yields the following.
Corollary 2.1. Let f;!;(cid:18) be as in the above. Then F(t) =f((cid:18) +!t), t R or
0 0
2
Z, is almost automorphic whose frequency module (F) is contained in (!) -
M M
the additive subgroup of R generated by ! ;! ; ;! .
1 2 k
(cid:1)(cid:1)(cid:1)
Remark 2.1. For each (cid:18) Tk, the function f((cid:18)+!t), where t R or Z, may be
2 2
regarded as a weak quasi-periodic function. Weak quasi-periodic orbits are known
to existence in monotone twist maps on an annulus as orbits lying in the Aubry-
Mather sets ([62]) and in scalar ODEs with quasi-periodic time dependence ([76]).
6 YINGFEIYI
The existence of such orbits in these dynamical systems were shown under certain
monotonicity and properties that f : Tk R1 is semi-continuous and admits
!
bounded directional derivative along !. Indeed, the semi-continuity implies that f
admits a residual set Y of points of continuity, and the directional di(cid:11)erentiability
0
together with the continuous dependence on initial conditions imply the continuity
of f on each orbit (cid:18) +!t if (cid:18) Y . It is clear that a weak quasi-periodic orbit
0 0 0
f g 2
f((cid:18)+!t) becomes almost automorphic precisely when (cid:18) is a point of continuity of
f, and becomes quasi-periodic only if f itself is a continuous function on the torus.
It is important to note that the family of weak quasi-periodic orbits such de-
(cid:12)ned always contains almost automorphic ones. This is because of the harmonic
similarity between almost automorphic functions with (cid:12)nite many frequencies and
quasi-periodic ones. Indeed, if (cid:18) is not a point of continuity of f, then the function
f((cid:18)+!t)needs not even be recurrentand needsnot admit any Fourier series whose
Bochner-Fejer sum is pointwise convergent, due to the lack of continuity of f on
the set (cid:18)+!t .
f g
2.3. Inherit properties. Similar to almost periodic minimal (cid:13)ows ([22]), it was
shownin[9]thatanalmostautomorphicminimal(cid:13)ow(X;T)alsoadmitsaninher-
itance property as follows.
Theorem 2.5. Let S be a syndetic subgroup of T. Then (X;T) is almost auto-
morphic minimal with maximal almost periodic factor (Y;T) i(cid:11) (X;S) is almost
automorphic minimal with maximal almost periodic factor (Y;S).
Theaboveinheritpropertycanbe easilyextendedtoaPoincar(cid:19)emapassociated
witharealcompact(cid:13)ow(X;R). RecallthataclosedsubsetZ X isaglobal cross
(cid:26)
section of (X;R) if i) all orbits meet Z; ii) there is a positive continuous function
T : Z R, called (cid:12)rst return time, such that z T(z) Z and z t Z for all
! (cid:1) 2 (cid:1) 62
z Z and 0<t<T(z). In case a global cross section Z exists, one can de(cid:12)ne the
2
Poincar(cid:19)e map P : Z Z : z z T(z), which is a homeomorphism on Z. The
! 7! (cid:1)
following can be shown similarly to a special case considered in [9].
Proposition 2.2. Suppose that (X;R) admits a global cross section Z and let
P : Z Z be the associated Poincar(cid:19)e map. Then (X;R) is almost automorphic
!
minimal i(cid:11) (Z;P) is.
Let P : S1 S1 be an orientation preserving homeomorphism with irrational
!
rotationnumber. Then P hasa uniqueminimal set E whichis either topologically
conjugatedtoapurerotation([17])orisanalmost1-1extensionofapurerotation
onS1 ([60]),bothwithzerotopologicalentropy. Inanycase,E is(discrete)almost
automorphic.
Now considera C1 (cid:12)xed-point-free (cid:13)ow (cid:25) on the 2-torusT2. Then (cid:25) admits a
t t
Poincar(cid:19)esection and its rotation vector
(cid:25)~ (y)
t
! = lim
t!1 t
exists and is independent of y T2, where (cid:25)~ : R2 R2 denotes a continuous
t
2 !
lift of (cid:25) . We assume further that ! is non-resonant, or equivalently, (cid:25) admits no
t t
periodic orbits. An easy application of the Denjoy theory implies that (cid:25) admits
t
a unique minimal set E which is topologically equivalent to a suspension of either
ON ALMOST AUTOMORPHIC OSCILLATIONS 7
a pure rotation or a Denjoy Cantor set on the circle (hence E is uniquely ergodic
with zerotopologicalentropy). ApplyingProposition2.2,wefurtherconcludethat
E is actually almost automorphic.
Similarly, let f be an area preserving, orientation preserving, boundary compo-
nents preserving, twist homeomorphism of an annulus A. Let (cid:26) < (cid:26) denote the
0 1
rotation numbers of f restricted to the lower and upper boundaries of A, respec-
tively. Then it has been shown in [62] that for any (cid:26) [(cid:26) ;(cid:26) ] irrational,f admits
0 1
2
a minimal set M which is topologically conjugated to either a pure rotation of
(cid:26)
the circle or a Denjoy set of the circle, with the rotation number (cid:26). Hence, M
(cid:26)
is an almost automorphic minimal set. In fact, Fourier series and their pointwise
convergencefor the almostautomorphic orbits(referredto asquasi-periodicorbits
by Mather) in M were also discussed in [62] which (cid:12)t in the harmonic nature of
(cid:26)
general almost automorphic functions described in Theorem 2.1.
To summarize, we have the following.
Theorem 2.6. A Denjoy minimal set of a circle homeomorphism, a Denjoy mini-
malsetof atoral(cid:13)ow, andAubry-Mather setsof anareapreservingmonotonetwist
map on an annulus are all almost automorphic minimal sets with zero topological
entropy.
3. Almost periodically forced circle flows
Consider an almost periodically forced circle (cid:13)ow generated by the following
almost periodically forced scalarODE:
(3.1) (cid:30)0 =f((cid:30);y t);
(cid:1)
where(cid:30) R1,f :R1 Y R1 isLipschitzcontinuous,periodicinthe(cid:12)rstvariable
2 (cid:2) !
with period 1. Using the identi(cid:12)cation (cid:30) (cid:30)0 (mod1), (3.1) clearly generates a
(cid:17)
skew-product (cid:13)ow (cid:3) :S1 Y S1 Y:
t
(cid:2) ! (cid:2)
(3.2) (cid:3) ((cid:30) ;y )=((cid:30)((cid:30) ;y ;t);y t)
t 0 0 0 0 0
(cid:1)
when both (cid:30)((cid:30) ;y ;t) and (cid:30) are identi(cid:12)ed modulo 1, where (cid:30)(t) = (cid:30)((cid:30) ;y ;t) is
0 0 0 0 0
the solution of (3.1) with y=y , (cid:30)(0)=(cid:30) .
0 0
3.1. Rotation number and mean motion. Using arguments of [49] and the
Birkho(cid:11) ergodic theorem, it is easily seen that the equation (3.1), or equivalently,
the skew-product (cid:13)ow (3.2) admits a well de(cid:12)ned rotation number
(cid:30)((cid:30) ;y ;t)
0 0
(cid:26)= lim
t!1 t
i.e., the limit exists and is independent of initial values ((cid:30) ;y ) R1 Y.
0 0
2 (cid:2)
We say that the equation (3.1), or equivalently, the skew-product (cid:13)ow (3.2)
admits mean motion if
sup (cid:30)((cid:30) ;y ;t) (cid:30) (cid:26)t <
0 0 0
t2R1j (cid:0) (cid:0) j 1
for all ((cid:30) ;y ) R1 Y.
0 0
2 (cid:2)
8 YINGFEIYI
Remark 3.1. 1) It is well known that if either (3.1) is periodically dependent
on time ([38]), i.e., (Y;R) = (S1;R) is a pure rotation of the circle, or almost
periodically dependent on time but admits an almost periodic solution ([25]), then
it always admits mean motion. This is however not the case for general almost
periodic time dependence. As an extreme example, let Y =Tk, f f(!t) be quasi-
(cid:17)
periodic in (3.1), where ! Rk is a non-resonant frequency vector. Then (cid:26)=[f] -
2 t
themeanvalueof f, and, itiswellknownthat(cid:30)((cid:30) ;t) (cid:30) (cid:26)t= (f(!s) [f])ds
0 (cid:0) 0(cid:0) 0 (cid:0)
can be unbounded if k >1 unless some additional conditions are aRssumed on f and
! (e.g., f is su(cid:14)ciently smooth and ! is Diophantine).
2) A natural question here is that under what conditions (3.1) can admit mean
motion. Another question is that whether the mean motion property is generic,
e.g., among the class of real analytic functions f : S1 Tk R1 endowed with
(cid:2) !
sup-norm, with (Y;R)=(Tk;R) being a (cid:12)xed Diophantine, quasi-periodic (cid:13)ow.
Wedonothaveanswerstothesequestionsbutconjecturethattheanswertothe
second question should be a(cid:14)rmative. Below, we give some equivalent conditions
for mean motion to hold in (3.1).
Proposition 3.1. The equation (3.1) admits mean motion i(cid:11) there is a ((cid:30) ;y )
0 0
2
R1 Y such that
(cid:2)
(cid:30)((cid:30) ;y ;t) (cid:30) (cid:26)t
0 0 0
j (cid:0) (cid:0) j
is bounded for either t 0 or t 0.
(cid:21) (cid:20)
Proof. We note that, by the periodicity of f in (cid:30), for any t R1, y Y, if
2 2
(cid:30)(cid:3) (cid:30)(cid:3) <l for some positive integer l, then also
j 1(cid:0) 2j
(3.3) (cid:30)((cid:30)(cid:3);y;t) (cid:30)((cid:30)(cid:3);y;t) <l:
j 1 (cid:0) 2 j
Without loss of generality, assume that ((cid:30) ;y ) R1 Y is such that
0 0
2 (cid:2)
sup (cid:30)((cid:30) ;y ;t) (cid:30) (cid:26)t < :
0 0 0
j (cid:0) (cid:0) j 1
t(cid:21)0
Then it follows from the (cid:13)ow property that
sup (cid:30)((cid:30) ;y ;t) (cid:30) (cid:26)t <
(cid:3) (cid:3) (cid:3)
t2R1j (cid:0) (cid:0) j 1
forany((cid:30) ;y ) !((cid:30) ;y )withrespecttothe(cid:13)ow(3.2). Hence(3.1)admitsmean
(cid:3) (cid:3) 0 0
motion by (3.3)2. (cid:3)
Lemma 3.1. Consider (3.1). Then
(cid:30)((cid:30) ;y;t)dy (cid:30) (cid:26)t 4
0 0
jZ (cid:0) (cid:0) j(cid:20)
Y
for any (cid:30) R1.
0
2
Proof. Again, by the periodicity of f in (cid:30), for any t R1, y Y, if (cid:30)(cid:3) (cid:30)(cid:3)
2 2 1 (cid:17) 2
(mod1), then
(3.4) (cid:30)((cid:30)(cid:3);y;t) (cid:30)(cid:3) =(cid:30)((cid:30)(cid:3);y;t) (cid:30)(cid:3);
1 (cid:0) 1 2 (cid:0) 2
and, if (cid:30)(cid:3) (cid:30)(cid:3) <l for some positive integer l, then (3.3) holds.
j 1(cid:0) 2j
Now, (cid:12)x t;(cid:30) R1. Forany y Y, 0 s t, let 0 (cid:30) ;(cid:30) <1 be such that
0 1 2
2 2 (cid:20)j j(cid:20)j j (cid:20)
(cid:30) (cid:30) ; (cid:30) (cid:30)((cid:30) ;y;s); (mod1):
1 0 2 0
(cid:17) (cid:17)
ON ALMOST AUTOMORPHIC OSCILLATIONS 9
It follows from (3.4) that
(3.5) (cid:30)((cid:30) ;y s;t) (cid:30) = (cid:30)((cid:30) ;y s;t) (cid:30) ;
0 0 1 1
(cid:1) (cid:0) (cid:1) (cid:0)
(3.6) (cid:30)((cid:30) ;y;t+s) (cid:30)((cid:30) ;y;s) = (cid:30)((cid:30) ;y s;t) (cid:30) :
0 0 2 2
(cid:0) (cid:1) (cid:0)
Hence, by (3.3)-(3.6),
(cid:30)((cid:30) ;y;t+s) (cid:30)((cid:30) ;y;s) (cid:30)((cid:30) ;y s;t)+(cid:30)
0 0 0 0
j (cid:0) (cid:0) (cid:1) j
= (cid:30)((cid:30) ;y s;t) (cid:30)((cid:30) ;y s;t)+(cid:30) (cid:30)
2 1 2 1
j (cid:1) (cid:0) (cid:1) (cid:0) j
(cid:30)((cid:30) ;y s;t) (cid:30)((cid:30) ;y s;t) + (cid:30) (cid:30) 4;
2 1 2 1
(cid:20)j (cid:1) (cid:0) (cid:1) j j (cid:0) j(cid:20)
i.e.,
t+s
(3.7) (cid:30)0((cid:30) ;y;(cid:21))d(cid:21) (cid:30)((cid:30) ;y s;t)+(cid:30) 4:
0 0 0
jZ (cid:0) (cid:1) j(cid:20)
s
Let T =0. An application of Frobine’s theorem yields
6
t 1 t (cid:30)((cid:30) ;y;T +s) 1 t
0
(cid:30) (cid:30)((cid:30) ;y;(cid:21))d(cid:21)+ ds (cid:30)((cid:30) ;y s;t)ds
0 0 0
j (cid:0)Z T Z T (cid:0) T Z (cid:1) j
0 0 0
1 T t+s 1 T
(3.8) = (cid:30)0((cid:30) ;y;(cid:21))d(cid:21)ds (cid:30)((cid:30) ;y s;t)ds+(cid:30) 4:
0 0 0
jT Z Z (cid:0) T Z (cid:1) j(cid:20)
0 s 0
Since
(cid:30)((cid:30) ;y;T +s)
0
lim =(cid:26)
T!1 T
uniformly in y Y, s [0;t], and, by the Birkho(cid:11) ergodic theorem,
2 2
1 T
lim (cid:30)((cid:30) ;y s;t)ds= (cid:30)((cid:30) ;y;t)dy;
0 0
T!1T Z0 (cid:1) ZY
the lemma is proved by passing limit T in (3.8). (cid:3)
!1
Theorem 3.1. (3.1) admits mean motion i(cid:11) there is a ((cid:30) ;y ) R1 Y such that
0 0
2 (cid:2)
(cid:30)((cid:30) ;y ;t) (cid:30)((cid:30) ;y;t)dy
0 0 0
j (cid:0)Z j
Y
is bounded for t 0 or t 0.
(cid:21) (cid:20)
Proof. This follows immediately from Proposition 3.1 and Lemma 3.1. (cid:3)
3.2. Mean motion and almost automorphic dynamics. One signi(cid:12)cance for
havingmeanmotionisthatitactuallyimpliesthe existenceofalmostautomorphic
dynamics in the skew-product (cid:13)ow (3.2).
First, we consider the following almost periodically forced scalar ODE
(3.9) x0 =f(x;y t);
(cid:1)
wherex R1,f :R1 Y R1 isLipschitzcontinuous. Let(cid:25) :R1 Y R1 Y:
t
2 (cid:2) ! (cid:2) ! (cid:2)
(3.10) (cid:25) (x ;y )=(x(x ;y ;t);y t)
t 0 0 0 0 0
(cid:1)
be the skew-product (cid:13)ow generated by (3.9), where x(t) = x(x ;y ;t) denotes
0 0
the solution of (3.9) with y = y and x(0) = x . As (3.9) forms a special class of
0 0
almostperiodicallyforced,1-dimensional,scalarparabolicPDEswiththeNeumann
boundary condition, the following lemma follows from the corresponding result in
10 YINGFEIYI
[91, 93]. As the proofs for the scalar ODE case is much less involved than the
parabolic PDE case, we include them here for the readers’ convenience.
Lemma 3.2. Let E be a minimal set of (3.10). Then the following holds.
1) (E;R) isalmost automorphic and infactanalmost 1-1extensionof (Y;R);
2) E is uniquely ergodic i(cid:11) Y admits full Haar measure, where Y Y is the
0 0
(cid:26)
residual set corresponding to the singelton (cid:12)bers.
3) Letx(x ;y ;t)be an almost automorphic solution of (3.9)for some y Y.
0 0 0
2
Then (x) (f).
M (cid:26)M
Proof. Let p:R1 Y Y be the natural projection and de(cid:12)ne
(cid:2) !
a (y)=max x:(x;y) E p(cid:0)1(y) ; a (y)=min x:(x;y) E p(cid:0)1(y) :
M m
f 2 \ g f 2 \ g
1) Let 2E be furnished with Hausdor(cid:11) metric. Since the map h : Y 2E:
!
y E p(cid:0)1(y) is upper semi-continuous, the points of continuity of h form a
7! \
residualsubsetY Y. Fixay Y andlett besuchthat(cid:25) (a (y);y)
0 (cid:26) 2 0 n !1 tn M !
(a (y);y). By the continuity of h at y, there is a sequence of points (x ;y)
m n
2
E p(cid:0)1(y) such that (cid:25) (x ;y) (a (y);y). Now, by taking limits to both side
\ tn n ! M
of the inequalities
x(x ;y;t ) x(a (y);y;t );
n n M n
(cid:20)
weobtainthata (y) a (y). Hencea (y)=a (y)andE p(cid:0)1(y)isasingleton.
M m M m
(cid:20) \
2)Let(cid:22)denotethe HaarmeasureonY. If(cid:22)(Y )=1,then E isclearlyuniquely
0
ergodic. Now suppose that (cid:22)(Y ) = 0 (note that Y is invariant). Then the func-
0 0
tionals l :C(E) R,
M;m
!
l (f)= f(a (y);y)d(cid:22)
M;m M;m
Z
Y
would de(cid:12)ne two distinct invariant measures on E.
3) follows immediately from 1) and Proposition 2.1. (cid:3)
Remark 3.2. 1) If (3.9) is periodically dependent on time, then each minimal set
of its associated skew-product (cid:13)ow becomes periodic (hence is uniquely ergodic and
admits zero topological entropy). We conjecture that in the almost periodic time
dependent case, the skew-product (cid:13)ow associated with (3.9) can have almost auto-
morphic minimal sets which are non-uniquelyergodic and admit positive topological
entropy.
2) Assume that (3.9) is quasi-periodically dependent on time, i.e., (Y;R) =
(Tk;R) : y t = y +!t is a quasi-periodic (cid:13)ow on the k-torus for some k with
(cid:1)
the toral frequency ! Rk. It was shown in [76] that if (3.9) admits a bounded
2
solution x(t) for some y = y , then it has two families of weak quasi-periodic so-
0
lutions x (y +!t);x (y +!t), y Tk, in the sense of Remark 2.1, such that
(cid:0) +
2
x ;x :Tk R1 are lower and upper semi-continuous respectively, and,
(cid:0) +
!
(3.11) infx(t) x (y) x (y) supx(t); y Tk;
(cid:0) +
(cid:20) (cid:20) (cid:20) 2
(3.12) x (y +!t) (cid:30)(t) x (y +!t); t R:
(cid:0) 0 + 0
(cid:20) (cid:20) 2
Moreover, if y Tk is a point of continuity for x (x resp.), then x (y +!t)
0 (cid:0) + (cid:0) 0
2
(x (y +!t) resp.) becomes almost automorphic.
+ 0