Table Of ContentDIOPHANTINE TYPE OF INTERVAL EXCHANGE MAPS
DONGHANKIM
2
1
0 Abstract. Roth type irrational rotation numbers have several equivalent
2
arithmetical characterizations as well as several equivalent characterizations
n
in terms of the dynamics of the corresponding circle rotations. In this paper
a
J we investigate how to generalize Roth-like Diophantine conditions to interval
8
exchange maps. If one considers the dynamics in parameter space one can
1
introduce two nonequivalent Roth-type conditions, the first (condition (Z))
]
S by means of the Zorich cocyle [16], the second (condition (A)) by means of
D a further acceleration of the continued fraction algorithm by Marmi-Moussa-
h. Yoccozintroducedin[9]. Athirdverynaturalcondition(condition(D))arises
t byconsideringthedistancebetweenthediscontinuitypointsoftheiteratesof
a
m themap. Ifoneconsidersthedynamics ofanintervalexchange mapinphase
[ space then one can introduce the notion of Diophantine type by considering
2 theasymptoticscalingofreturntimespointwiseorw.r.t. uniformconvergence
v
(resp. condition (R) and (U)). In the case of circle rotations all the above
9
6 conditions are equivalent. For interval exchange maps of three intervals we
2
show that (D) and (A) are equivalent and imply (Z), (U) and (R) which are
3
. equivalent among them. For maps of four intervals or more we prove several
1
0 results, the only relation which we cannot decide is whether (Z) implies (R)
2
ornot.
1
:
v
i
X
1. Introduction
r
a
Let θ be an irrational number: its type η 1 is defined by
≥
η =sup β :liminfjβ jθ =0
{ j k k }
→∞
where denotes the distance from the nearest integer. An irrational number is
k·k
calledof Roth-type if and only if η =1. This statement is equivalent that for every
2000 Mathematics Subject Classification. 37E05,11J70.
Keywordsandphrases. intervalexchangemap,Rothtype,Diophantinetype,recurrencetime,
Rauzy-Veechcontinuedfractionalgorithm.
Supported by the Korea Research Foundation(KRF) grant funded by the Korea govern-
ment(MEST)(No. 2009-0068804).
1
2 DONGHANKIM
ε>0 there exists a positive constant C such that
ε
p C p
ε
θ for all rationals .
− q ≥ q2+ε q
(cid:12) (cid:12)
(cid:12) (cid:12)
Thesetofirrational(cid:12)numbe(cid:12)rsofRothtypehasLebesguemeasure1andcontains
(cid:12) (cid:12)
all algebraic irrational numbers. Also it is invariant under the modular group
GL(2,Z).
Let (qn)n N be the sequence of the denominators of the principal convergents
∈
of the irrational θ and let (an)n N be the sequence of its partial quotients. Roth
∈
typeirrationalscanalsobedeterminedbythe growthconditionsof(qn)n N: forall
∈
ε>0 there is a constant C such that
ε
q <C q1+ε.
n+1 ε n
This condition is also equivalent to the growth condition of the partial quotients
(an)n N: for all ε>0 there is a constant Cε such that
∈
a <C qε.
n+1 ε n
Rothtypeconditionfortheirrationalθcanalsobegivenintermsofthedynamics
of the associated rotation R : x x+θ on T = R/Z. One arises by considering
θ
7→
the cohomological equation associated to the rotation R (see, e.g., [9]). Another
θ
dynamical characterization of Roth type rotations is obtained by means of the
asymptotic scaling laws of first return times and will be recalled below. Finally,
we consider how evenly an orbit of the rotation is distributed. If the rotation is
of Roth type, then for all ε > 0 there is a constant C such that the minimum
ε
distance between points belonging to a finite segment of orbit made n iterates
should be bigger than C n (1+ε).
ε −
Inthispaperweinvestigatetherelationshipamongseveralnot-necessarilyequiv-
alent generalizations of the definitions of the Roth type given above to interval
exchange maps.
Let r >0 and let τ (x) be the return time to r-neighborhood of x
r
(1) τ (x)=min j 1:d(Tjx,x)<r .
r
{ ≥ }
For an irrational circle rotation ([2]) we have that
logτ (x) 1 logτ (x)
r r
lim = , lim =1.
r→0+ −logr η r→0+ −logr
DIOPHANTINE TYPE OF INTERVAL EXCHANGE MAPS 3
Therefore, the rotation number is of Roth type if and only if
logτ (x)
r
lim =1.
r 0+ logr
→ −
Irrational circle rotations are the prototype of quasiperiodic dynamics and can
be generalized as interval exchange maps. An interval exchange map T on an
interval I is a one-to-onemap to itself which is a translationoneachfinite number
of subinterval partition of I. The map T is an orientation preserving piecewise
isometry and preserves the Lebesgue measure. Let d 2 be the number of the
≥
subintervals on which T is a translation. If d = 2, the interval exchange map T
corresponds the rotation of circle.
A typical interval exchange map is minimal([6]). However, minimality condi-
tion for the interval exchange map does not imply unique ergodicity([7, 8]). But
still almost every interval exchange map is uniquely ergodic([11, 13]) and weakly
mixing([1]).
ThemodulargroupGL(2,Z)playsanimportantroleforthestudyofrotationof
circle with renormalization scheme associated to the continued fraction algorithm.
It was generalized by Rauzy and Veech for interval exchange maps by introducing
the induced map on appropriated subintervals([12, 13]). The continued fraction
algorithm for interval exchange maps is ergodic on the parameter space of interval
exchange maps with respect to an absolutely continuous invariant measure with
infinite mass.
Zorich considered an acceleration scheme to produce an ergodic finite invariant
measure on the parameter space of the interval exchange maps([16]). For the rota-
tional case (d = 2), Zorich’s map indeed corresponds the Gauss map which is an
accelerationoftheFaraymapwhichdoesnothaveaprobabilityabsolutecontinuous
invariant measure.
A further acceleration of the Zorich algorithm was studied in [9] by Marmi,
Moussa and Yoccoz. They considered a more accelerated algorithm which also
preservesanergodicfiniteabsolutecontinuesinvariantmeasureintheinvestigation
oftheregularityofthesolutionsofthecohomologicalequationassociatedtointerval
exchangemaps. BoththeaccelerationsbyZorichandbyMarmi-Moussa-Yoccozare
reduced to the Gauss map for d=2.
4 DONGHANKIM
The notion of Roth type interval exchange map was introduced in [9]: this is
a natural extension of Roth type irrational circle rotations and Roth type interval
exchangemaps forma full measureset inthe parameterspace ofintervalexchange
maps. In[5]itwasprovedthatforRothtypeintervalexchangemapstherecurrence
time has the same scaling behaviour as for irrational rotations, namely
logτ (x)
r
lim =1, a.e. x.
r 0+ logr
→ −
The Roth type condition for the irrational rotation can be generalized to the
interval exchange map in several different ways. We consider arithmetic charac-
terizationusing the Roth type growthconditionfor Marmi-Moussa-Yoccozcocycle
(Condition(A)) andthe Roth type growthconditionfor Zorichcocycle (Condition
(Z)). Uniform return time condition (Condition (U)) and pointwise return time
condition(Condition(R))aredefinedintermsofthedynamicsofthemapinphase
space instead of its evolution in parameter space as is the case for conditions (A)
and (Z). We also consider Roth type condition for the minimal distance between
discontinuities (Condition (D)). In Section 3 these Diophantine conditions for in-
terval exchange maps are given in detail. In this article we show the relations
betweentheDiophantineconditionsespeciallytheequivalenceofRothtypegrowth
condition for Marmi-Moussa-Yoccoz cocycle and Roth type condition for minimal
distance between discontinuities. In [9], it is cited that the relation between them
is not clear. ([9], Sec 1.3.1. Remark 2)
After completing this paper the author noticed the recent work by Marmi,
Moussa, and Yoccoz ([10]). They also considered the equivalence of Condition
(A) and Condition (D) ([10], Proposition C.1).
2. Background on continued fraction algorithms for interval
exchange maps
An interval exchange map is determined by the combinatorial data of the per-
mutation and the length data of subintervals. Let be a finite set for the name
A
of subintervals. We denote the combinatorial data by two bijections (π ,π ) from
t b
onto 1,2,...,d . which indicate the order of the subintervals before and after
A { }
the interval exchange map. The length data, denoted by (λ ) , give the length
α α
∈A
of the corresponding subintervals.
DIOPHANTINE TYPE OF INTERVAL EXCHANGE MAPS 5
We set
λ := λ , I :=[0,λ )
∗ A ∗
α
X∈A
and
p := λ , I :=[p ,p +λ ), I = I .
α β α α α α α
πt(βX)<πt(α) αG∈A
Then the interval exchange map T associated to the combinatorial data (π ,π )
t b
and the length data (λ ) is a bijective map on I given by
α α
∈A
T(x)=x+ λ λ for x I .
β β α
− ∈
πb(βX)<πb(α) πt(βX)<πt(α)
Note that T is discontinuous at p with π (α)>1.
α t
We will consider only combinatorial data (π ,π ) which are admissible, in the
t b
sense that for all k =1,2,...,d 1, we have
−
πt−1({1,...,k})6=πb−1({1,...,k}).
An intervalexchange map T is said to have the Keane property if there exist no
α,β and positive integer m such that Tm(p )=p and π (β)>1. An admis-
α β t
∈A
sibleintervalexchangemapwithrationallyindependent lengthdatahastheKeane
property and an intervalexchange map with Keane’s property is minimal[6]. Thus
Keane’s property corresponds to the notion of irrationality for interval exchange
maps.
ForadmissibleintervalexchangemapswiththeKeanepropertywecanintroduce
the generalization of continued fraction algorithm to interval exchange maps due
to the work of Rauzy [12], Veech [13] and Zorich [16, 17]. We refer to [9, 14, 15]
and references therein for the detailed discussions and proofs.
Let(π ,π )beanadmissiblepair. Wedefinetwonewadmissiblepairs (π ,π )
t b t t b
R
and (π ,π ) as follows: let α and α be the (distinct) elements of such that
b t b t b
R A
π (α )=π (α )=d; one has
t t b b
(π ,π )=(π ,πˆ ),
t t b t b
R
(π ,π )=(πˆ ,π ),
b t b t b
R
6 DONGHANKIM
where
π (α) if π (α) π (α ),
b b b t
≤
πˆb(α)=πb(α)+1 if πb(αt)<πb(α)<d,
π (α )+1 if α=α , (π (α )=d);
b t b b b
π (α) if π (α) π (α ),
t t t b
≤
πˆt(α)=πt(α)+1 if πt(αb)<πt(α)<d,
π (α )+1 if α=α , (π (α )=d).
t b t t t
The Rauzy class of (π ,π) is the set of admissible pairs obtained by saturation
t b
of (π ,π ) under the action of and . The Rauzy diagram has for vertices
t b t b
R R
the elements of the Rauzy class, eachvertex (π ,π ) being the originof two arrows
t b
joining(π ,π )to (π ,π ), (π ,π ). SeeFigure1and3fortheRauzydiagrams
t b t t b b t b
R R
fora3-intervalmapanda4-intervalmap. Foranarrowjoining(π ,π )to (π ,π )
t b t t b
R
(respectively (π ,π )) the element α (respectively α ) is called the
b t b t b
R ∈ A ∈ A
winner and the element α (respectively α ) is called the loser.
b t
∈A ∈A
We say that T is of top type (respectively bottom type) if one has λ > λ
αt αb
(respectively λ >λ ); we then define a new interval exchange map (T) by the
αb αt V
following data: the admissible pair (π ,π ) and the lengths (λˆ ) given by
t t b α α
R ∈A
λˆ =λ if α=α ,
α α t
6
λˆαt =λαt −λαb otherwise
for the top type T; the admissible pair b(πt,πb) and the lengths
R
λˆ =λ if α=α ,
α α b
6
λˆαb =λαb −λαt otherwise
for the bottom type T.
Theintervalexchangemap (T)isthefirstreturnmapofT on 0, λˆ . We
V α α
alsoassociateto T the arrowinthe Rauzydiagramjoining (πt,πb)htoPt(πt,(cid:17)πb)or
R
(π ,π ). Iterating this process, we obtain a sequence of interval exchange map
b t b
R
T(n) = n(T), n 0 and an infinite path in the Rauzy diagram starting from
V ≥
(π ,π ). In fact a further property of irrational interval exchange maps (i.e. with
t b
the Keane property) is that every letter in is taken as a winner infinitely many
A
DIOPHANTINE TYPE OF INTERVAL EXCHANGE MAPS 7
times inthe infinite path (inthe Rauzydiagram)associatedto T. This propertyis
fundamental in order to be able to grouptogether severaliterations of to obtain
V
the accelerated Zorich continued fraction algorithm introduced in [9].
For an arrow γ with winner α and loser β in the Rauzy diagram, let
B =I+E
γ βα
where I is the identity matrix and E is the elementary matrix with the only
βα
nonzero element at (β,α) which is equal to 1. For a finite path γ =(γ ,...,γ ) in
1 n
the Rauzy diagram we have a SL(Z ) matrix with nonnegative entries
A
B =B B .
γ γn··· γ1
Let γT(m,n) = γ(m,n) be the path in the Rauzy diagram from π(m) to π(n) for
m n and denote by
≤
Q(m,n)=B and Q(n)=Q(0,n).
γ(m,n)
Let λ(n) be the length data of T(n). Then we have
(2) λ(m)=λ(n)Q(m,n).
For m n, T(n) is the induced map of T(m) on I(n) = [0,λ (n)), where
∗
≤
λ (n) = λ (n); the return time on I (n) to I(n) under the iteration T(m)
∗ α α β
∈A
is Qβ(m,Pn) := αQβα(m,n) and the time spent in Iα(m) is Qβα(m,n). By (2)
we have P
(3) λ = λ (n)Q (n)= λ (n)Q (n).
∗ β βα β β
αβ β
X X
Moreover,we have
Qα(n)−1
(4) [0,λ )= Ti(I (n)) .
∗ α
α i=0
G∈A G
Zorich’s accelerated continued fraction algorithm is obtained by considering
( nk)k 0 where(nk)k 0 is the followingsequence: n0 =0 andnk+1 >nk is chosen
V ≥ ≥
so as to assure that γ(n ,n ) is the longest path whose arrows have the same
k k+1
winner.
The further acceleration algorithm by Marmi-Moussa-Yoccoz, which was intro-
duced in [9], is obtained by considering ( mk)k 0 where (mk)k 0 is defined as
V ≥ ≥
8 DONGHANKIM
follows: m = 0 and m > m is the largest integer such that not all letters in
0 k+1 k
are taken as winner by arrows in γ(m ,m ).
k k+1
A
Let1
Zorich cocycle Z(k)=Q(0,n ), Z(k,ℓ)=Q(n ,n ),
k k ℓ
Marmi-Moussa-Yoccoz cocycle A(k)=Q(0,m ), A(k,ℓ)=Q(m ,m ).
k k ℓ
ThemostimportantvirtueoftheMarmi-Moussa-Yoccozcocycleisthefollowing:
Lemma 2.1 ([9] Lemma 1.2.4). Let r max(2d 3,2). Then we have
≥ −
A (k,k+r)>0 for all α,β .
βα
∈A
The following inequality follows easily from (3)
λ
∗
(5) minλ (n) maxλ (n),
α α
α ≤ Q(n) ≤ α
∈A k k ∈A
wherethenormofamatrixB issimplythesumoftheabsolutevaluesofitsentries.
This is the norm that we will use for matrices throughout the whole paper. We
assume that λ =1 unless it is specified.
∗
3. Diophantine conditions for interval exchange maps
If one considers the dynamics in parameterspace of interval exchangemaps one
can introduce three slightly different Diophantine conditions:
(A) Roth type growth condition for the Marmi-Moussa-Yoccozcocycle :
For any ε>0 there exist C >0 such that for all k 1 we have
ε
≥
A(k,k+1) C A(k) ε.
ε
k k≤ k k
(Z) Roth type growth condition for the Zorich cocycle :
For any ε>0 there exist C >0 such that for all k 1 we have
ε
≥
Z(k,k+1) C Z(k) ε.
ε
k k≤ k k
Let∆(T)be the minimumdistance betweenthe discontinuitypoints ofT orthe
end points 0 and 1.
1Wewarnthereaderthatournotations areslightlydifferentfromtheonefollowedin[15]: in
this paper the matrices A(k) denote the matrices obtained by the accelerated Zorich algorithm
introduced in [9] and Z(k) are those obtained by the original Zorich algorithm, whereas in [15]
theformerweredenoted Z(n)sincethelatterwereneverusedexplicitely.
DIOPHANTINE TYPE OF INTERVAL EXCHANGE MAPS 9
(D) Roth type condition for the minimal distance between discontinuities :
For any ε>0 there exist C >0 such that for all n 1 we have
ε
≥
C
∆(Tn) ε .
≥ n1+ε
If one considers the dynamics of an interval exchange map in phase space then
one can introduce two slightly different Diophantine conditions:
(R) Pointwise return time condition :
logτ (x)
r
lim =1 for almost every x.
r 0 logr
→ −
(U) Uniform return time condition :
logτ (x)
r
lim =1 uniformly.
r 0 logr
→ −
Here τ (x) be the first return time to r-neighborhood of x defined in (1).
r
Here and in what follows the matrix norm denoted by Q = Q . In
k k αβ| αβ|
the caseofcircle rotations(intervalexchangemapwith d=2)the thPree conditions
in parameter space (namely (A), (Z) and (D)) are equivalent2, as well as the two
conditions in phase space ((R) and (U)). In [2] the equivalence for circle rotations
between the two sets of conditions (Roth type in parameter space and the return
time characterization) was proved. For general interval exchange maps in [5] it is
proved that (A) implies (R).
In this article, we investigate the relation among Condition (A), (Z), (D), (U)
and(R)forgeneralintervalexchangemaps(withd 3). Itisnotdifficulttoverify
≥
that from the definitions one has
(A) (Z) and
⇒
(U) (R).
⇒
2For an irrational rotation, Z(1) = A(1) = 1 0, Z(k −1,k) = A(k −1,k) =
a1−1 1
1 0 1 ak qk−1−pk−1 pk−1 qk−pk pk
or andZ(k)=A(k)= or
ak 1 0 1 qk−pk pk qk−1−pk−1 pk−1
depending on k is odd or even. Therefore, we have kZ(k,k+1)k = kA(k,k+1)k = ak+1+2,
kZ(k)k= kA(k)k=qk+qk−1 and Condition (A) and (D) are equivalent to the statement that
for any ε>0 there is a positive constant Cε such that ak+1 ≤Cεqkε, which just the Roth type
conditionfortheirrationalrotationnumber.
10 DONGHANKIM
In Section 7 we will prove that for 3-interval exchange maps (Z), (U) and (R)
are equivalent and (A) and (D) are equivalent. Moreover, in the same Section, we
construct a family of 3-interval exchange maps which all satisfy Condition (U) but
neither Condition (A) nor (D). For general maps with d 4 we will establish that
≥
(A) (D) : this is proved in Section 4
⇔
(D) (U) : this is proved in Section 5
⇒
(U) (Z) : this is proved in Section 6
⇒
(R) does not imply (Z) : this is proved in Section 8
(Z) does not imply (U) : this is proved in Section 9
The only relation we could not decide is whether (Z) implies (R) or not.
4. Condition (A) is equivalent to Condition (D)
For each α let
∈A
p (n)= λ (n), q (n)= λ (n),
α β α β
πt(n)(βX)<πt(n)(α) πb(n)(βX)<πb(n)(α)
and
I (n)=[p (n),p (n)+λ (n)).
α α α α
Then
T(n)(I (n))=[q (n),q (n)+λ (n)).
α α α α
Denote by D(T) the set of discontinuity points of T. Let
′ = α :πt(α)>1 .
A { ∈A }
Note that we have D(T(n))= p (n):α .
α ′
{ ∈A}
Lemma 4.1. For each α , then we have
′
∈A
Ti(p (n)) D(T)
α
∈
for some i such that 0 i<Q (n). Conversely, if p D(T), then
α
≤ ∈
p=Ti(p (n)) for an α and 0 i<Q (n).
α ′ α
∈A ≤
Proof. We will prove the statement by induction. Both statements are trivial if
n=0. Assume that the lemma holds for n>1. Let α and β be such that
(n) (n)
π (α)=d, π (β)=d.
t b
