Table Of ContentNon-robustly intermingled basins for perturbations
6
1
of time-one map of Anosov flows
0
2
n Qianying Xiao∗and Zuohuan Zheng
a
J
February 2, 2016
0
3
]
AcademyofMathematics andSystemsScience, ChineseAcademyofSciences
S
D
Beijing,100190, P.R.China
.
h
t
a Abstract
m
Perturbations of time-one maps of transitive Anosov flows are studied.
[ We show that most perturbations have no intermingled basins of hyperbolic
physical measures.
1
v
0
1 Introduction
8
0
0 Let Md be a C∞ compact Riemannian manifold, f ∈ Diff2(M). To describe the
0 behaviorofalmostallorbitsisagoalofdynamicalsystems. Itismeaningfultoem-
.
2 phasize invariant measures which are physically observable. An invariant measure
0 µ is a physical measure or SRB measure if
6
1 n−1
1
v: B(µ)={x∈M|n→lim+∞n Xδfk(x) =µ}
i k=0
X
r has positive Lebesgue measure. J. Palis [10, 11] conjectured that the existence and
a finitenessofphysicalmeasureswhosebasinscoverLebesguealmosteverypointhold
for a generic map.
On the annulus S1 × [0,1], I. Kan [7] constructed a C2 partially hyperbolic
endomorphism
F :S1×[0,1]→S1×[0,1]
suchthatF(x,t)=(kx mod 0,f (t))withk ≥3. F isrequiredtopreservethetwo
x
boundariesS1×{0}andS1×{1},whoseLebesguemeasuresm andm areergodic
1 2
with negative center exponent and are physical measures consequently. Then one
has Leb(B(m )∪B(m ))=1. Even more interesting is that the Lebesgue density
1 2
points of B(m ) are dense in S1 ×[0,1] for i = 1,2. Therefore m and m have
i 1 2
intermingled basins, by which we mean the closures of Lebesgue density points of
B(m ) and B(m ) coincide, yet B(m ) and B(m ) are disjoint.
1 2 1 2
Uniformly hyperbolic systems have no intermingled basins. According to R.
Bowen[3], any physical measure is supported on an Axiom A attractor. The basin
∗Q.XiaoistheCorrespondingauthor.
1
of the attractor coincides with the open attracting basin of the attractor modulo a
zero Lebesgue measurable set.
One can easily see that the intermingled phenomenon discovered by Kan is C2
robust. Y.Ilyashenko,V.KleptsynandP.Saltykov[8]showintermingledbasinscan
evenbeC1 robustforboundarypreservingpartiallyhyperbolicendormorphismson
the annulus.
Kan’s example is adapted by R. Ures and C. Va´squez in [14], so that on cer-
tain boundaryless manifolds there exists partially hyperbolic diffeomorphisms with
intermingled basins of physical measures.
In the same paper, Ures and Va´squez show that for partially hyperbolic diffeo-
morphisms that are dynamically coherent on T3, intermingled basins of hyperbolic
physicalmeasuresis notrobust. They arereducedto the situationwheneverycen-
terleafiscompact. Bystudyingtheintersectionofau-saturatedsetwiththecenter
leaves, they relate intermingled basins to the existence of tori tangent to Es⊕Eu.
One would ask if non-robustness of intermingled basins holds in other settings.
In this work, we study perturbations of time-one map of transitive Anosov flows.
We give a positive answer to the above question.
Let M support a transitive Anosov flow φ from now on. The time-one map
t
φ is partially hyperbolic with one-dimensional center. That is to say there is a
1
continuous splitting TM =Es⊕Ec⊕Eu, dimEc =1, Es is uniformly contracted,
Eu is uniformly expanded, and the contraction and expansion of Ec is dominated
by Es and Eu respectively.
According to Hirch-Pugh-Shub [6], there is a C1 neighborhood U of φ in
1
Diff2(M) such that any f ∈ U is partially hyperbolic. There exist a foliation
tangent to the center bundle. Moreover any f ∈U is leafwise conjugate to φ .
1
Theorem 1.1. There exists a C1-open and C2-dense subset V ⊂ U such that for
any f ∈V, any hyperbolic physical measures µ and ν of f with disjoint supports do
not have intermingled basins.
A partially hyperbolic diffeomorphism is accessible if any two points can be
connectedbypiecewisedifferentiablecurvewithfinite legs,eachofwhichistangent
tothe stablebundle Es orunstablebundle Eu. K.Burns,F.R.Hertz,J.R.Hertz,
A. Talitskaya and R. Ures [4] prove that for partially hyperbolic diffeomorphisms
withone-dimensionalcenter,the collectionhavingaccessibilitypropertyisCr open
and dense for r ≥1.
By requiring U sufficiently small, the dynamics of f ∈ U at every center leaf is
close to a translationby 1 alongthe positive orientation. Therefore,oneachcenter
leafWc(x)thesegment[x,f(x)] connectingxtof(x)isgeneratinginthefollowing
c
sense:
Wc(x)=∪n∈Zfn([x,f(x)]c).
The ideas of our arguments are inspired by [14]. We study the intersection of
a u-saturated set with every center segment [x,f(x)] . Then we show intermingled
c
basinsofhyperbolicphysicalmeasuresleadstoaC1-laminationtangenttoEs⊕Eu,
which contradicts accessibility. About lamination we will explain later.
We dealwithhyperbolicphysicalmeasuresinthis paper. We donotknowwhat
happens to physicalmeasures with zero center exponent. Moreoverthe supports of
two physical measures is not necessarily disjoint. With the absence of dominated
splitting, I. Melbourene and A. Windsor [9] constructs on T2 a C∞ minimal dif-
feomorphism admitting any given number of absolutely continuous measures. To
2
whatextentpartialhyperbolicityobstructssuchintermingledsupportsisaproblem.
Moreover,the problem of intermingled basins of physical measures whose supports
intersect is not considered in this paper.
Thisworkisorganizedasfollowing: Section2isthepreliminaries. Wewillintro-
duce the relatedproperties of Gibbs u-states and physicalmeasures in this section.
Insection3westateourProposition3.2fromwhichTheorem1.1isdeduced. Proof
of Proposition 3.2 is in Section 4.
2 Preliminaries
Letf beageneralpartiallyhyperbolicdiffeomorphismwithone-dimensionalcenter.
The unstable foliation is denoted by Wuu.
Definition 2.1. An invariant measure µ of f is a Gibbs u-state if the conditional
measures of µ along Wuu are absolutely continuous with respect to the Lebesgue
measures.
Let us list some related facts about Gibbs u-states.
Lemma 2.2. [2, Section 11.2]
1. A physical measure is a Gibbs u-state;
2. An ergodic Gibbs u-state with negative center exponent is a physical measure;
3. The ergodic components of a Gibbs u-state are Gibbs u-states;
4. The support of a Gibbs u-state is u-saturated.
Definition 2.3. A point x∈Md is regular in the sense of Birkhoff if
n−1 n+1
1 1
lim ϕ(fk(x))= lim ϕ(fk(x)).
n→+∞n X n→−∞−n X
k=0 k=0
Apointx∈M isregular in the sense of Lyapunov ifthereexistsasplittingT M =
x
E (x)⊕···⊕E , k(x)≤d, Lyapunov exponents λ (x)<···<λ (x), such that
1 k(x) 1 k
for any v ∈E (x)\{0}, i=1,··· ,k(x),
i
1
lim lnkDfn(v)k=λ (x).
i
n→±∞n
Definition 2.4. For any invariant measure µ of f, the integrated center exponent
λc(µ) is defined by
λc(µ)= λc(x)dµ(x),
Z
with λc(x) denotes the Lyapunovexponents along the one-dimensionalcenter bun-
dle.
Lemma 2.5. If µ is a physical measure such that λc(µ)6=0, then µ is ergodic.
Proof. The proof is divided into two cases.
3
Case 1 : λc(µ)<0.
Since µ is a Gibbs u-state,so is eachofits ergodiccomponentby Lemma 2.2.
The ergodic component with negative center exponent is a physical measure.
Henceµhasatmostcountablymanyergodiccomponentswithnegativecenter
exponent. Let µ be such an component, the µ gives positive weight to µ .
1 1
AccordingtoPesintheory,thereexistsΛ⊂M∩B(µ )suchthatanyx∈Λis
1
regularinthesenseofbothBirkhoffandLyapunov,thePesinstablemanifold
Ws(x) is well-defined, and µ (Λ) = 1. There are compact subsets Λ such
1 n
thatΛ ⊂Λ ,Λ=∪ Λ ,andthesizeofPesinstablemanifoldofx∈Λ
n n+1 n≥1 n n
is uniformly bounded away from 0.
Take Λ such that µ (Λ ) > 0. There exists δ > 0 such that any x ∈ Λ ,
N 1 N N
the size of Ws is greater than δ. Without loss of generality, we can assume
each point of Λ is in the support of µ |Λ .
N 1 N
Let D be a disk in an unstable leaf such that Leb (D∩Λ )>0. Define
D N
R=∪ Ws (x).
x∈D∩ΛN δ/2
Being in the support of µ |Λ , x is in the support of µ. Together with the
1 N
assumption that µ is a Gibbs u-state, one has µ(R)>0. Take y ∈B(µ) such
that y is a Lebesgue density point of B(µ)∩Wuu(y). The positive orbit of
y goes to arbitrarily close to R. By some bounded distortion arguments, one
can easily show B(µ)∩R6=∅. Since R⊂B(µ ), µ=µ .
1 1
Case 2 : λc(µ)>0.
Any x∈B(µ),
n−1
1
lim lnkDf−1|Ec(fk(x))k= lnkDf−1|Ec(x)kdµ(x)<0.
n→+∞n X Z
k=0
ThereforeEc⊕Eu ismostlyexpandinginthesensedefinedbyAlves-Bonatti-
Viana[1]. Takeacu-diskDtransversetoWss suchthatLeb (D∩B(µ))>0.
D
By [1, Lemma 4.5], there exists physical measure ν whose basin is open and
D∩B(µ)∩B(ν)6=∅. Consequently µ=ν.
Combiningthetwocases,wehaveshownthataphysicalmeasureoff withnonzero
integrated center exponent is ergodic.
According to Lemma 2.5, when we say a physical measure with negative (posi-
tive)centerexponent,themeaningisclear,becausethephysicalmeasureisergodic,
the center Lyapunov exponent equals the integrated center exponent almostevery-
where.
3 Proof of Theorem 1.1
Let U be as in the Introduction. For any f ∈U, Es and Eu is uniquely integrable,
theintegratedfoliationsarethestablefoliationWss andtheunstablefoliationWuu.
Moreover, f is dynamically coherent: there are foliations Wcs and Wcu tangent to
Es⊕Ec and Eu⊕Ec, the intersection of Wcs and Wcu is a foliation Wc.
4
AsubsetK ⊂M isu-saturated(s-saturated) ifitistheunionofcompletestrong
unstable leaves (stable leaves). c-saturated set is defined similarly.
We will study certainu-saturatedinvariantcompactset K relatedto a physical
measureinthenextsection. WewillprovesuchkindofK islaminatedbyC1 leaves
tangent to Es⊕Eu.
Definition 3.1. [6, 13] A lamination L of a compact set K ⊂ M is a family of
disjoint submanifolds (leaves of the lamination) whose union is K and which are
assembeld in a C1 continuous fashion. That is, K is covered by lamination boxes,
where a lamination box is a map ϕ:Dc×Y →K, Y is a fixed compact set, ϕ is a
homeomorphismtoarelativelyopensubsetofK,and∂ϕ(x,y)/∂xisnondegenerate
and continuous with respect to (x,y)∈Dc×Y.
Proposition 3.2. For any f ∈ U, any physical measure µ of f with negative
center Lyapunov exponent, if K is a compact invariant u-saturated subset such that
K ⊂B(µ)\supp(µ), then
• there exists k ∈N such that K intersects each center leaf in exactly k orbits,
• K is laminated by C1 leaves tangent to Es ⊕ Eu. In particular, f is not
accessible.
Definition 3.3. Let µ, ν be two physical measures. µ, ν have intermingled basins
if for any openset U inM, Leb(U∩B(µ))>0 implies Leb(U∩B(ν))>0 andvice
versa.
It is obvious from the definition that for two physical measures µ, ν with inter-
mingled basins, supp(ν)⊂B(µ).
Proof of Theorem 1.1. Accordingto [4],thereisaC1 openC2 densesubsetV ⊂U
such that any f ∈V is accessible.
Foranyf ∈V,anyhyperbolicphysicalmeasuresµ,ν whosesupportsaredisjoint
but basins intermingle, one has λc(µ) < 0,λc(ν) < 0. Otherwise, by [1], B(µ) or
B(ν) is openmodulo a zeroLebesgue measurableset,the basinsB(µ) andB(ν) do
not intermingle.
Since B(µ) and B(ν) intermingle, supp(ν) ⊂ B(µ). By assumption, supp(µ)∩
supp(ν)=∅, hence supp(ν)⊂B(µ)\supp(µ).
Apply Proposition3.2toK =supp(ν)andµ, thenf is notaccessible,a contra-
diction.
So any f ∈V has no intermingled basins for hyperbolic physical measures with
disjoint supports.
4 Proof of Proposition 3.2
Let f ∈ U. Assume K is a compact invariant u-saturated subset of f, and K ⊂
B(µ)\supp(µ).
Lemma 4.1. Every center unstable leaf is dense and K intersects each center leaf.
Proof. Take y ∈ M such that Wc(y) is dense(such y exists since f is leafwise
conjugate to φ and φ is a transitive flow).
1 t
5
Any x∈M, Wuu(x)∩Wcs(y)6=∅, let x′ ∈Wuu(x)∩Wcs(y)6=∅, then Wc(x′)
is dense, consequently Wcu(x)⊇Wc(x′) is dense.
SinceKisacompactinvariantu-saturatedsubset,Wc(K)=∪ Wc(x)isboth
x∈K
u-saturatedands-saturated. Moreover,Wc(K)iscompact. ThereforeWc(K)=M,
i.e. K intersects each center leaf.
Let K(M) be the collection of nonempty subsets of M. Define
Φ : M →K(M)
K
x7→[x,f(x)] ∩K
c
with [x,f(x)] denote the closed segment on Wc(x) connecting x and f(x).
c
Since Wc(x)=∪n∈Zfn([x,f(x)]c), K is invariant, and K∩Wc(x)6=∅, one has
K∩[x,f(x)] 6=∅. Therefore Φ is well-defined.
c K
K beingcompact,Φ isuppersemi-continuous,andthe collectionofcontinuity
K
points of Φ is a residual set of M.
K
Let µ be a physical measure with negative center exponent. By Lemma 2.5, µ
is ergodic. There exists Λ⊂B(µ) such that µ(Λ)=1, Λ consists of points that are
regular in the sense of both Birkhoff and Lyapunov.
Lemma 4.2. [14, Lemma 4.1, Lemma 4.3]
(1) B(µ)⊂Ws(Λ), with Ws(Λ)=∪ Ws(x);
x∈Λ
(2) Foranyx∈Ws(Λ),thereexistsy ∈Wc(x)∩supp(µ)suchthat[x,y] ⊂Ws(y).
c
Lemma 4.3. [14, Lemma 4.4] There exists h>0 such that any segment of length
less than h in a center leaf contains at most two points in K.
Proof. Assume for any h >0, there exist x, y, z in a center segment of length less
than h. Let y lie between x and z.
Let k > 0 such that dist(K,supp(µ)) > k. Since y ∈ K ⊂ B(µ) ⊂ Ws(Λ),
there exist q ∈ Ws(Λ) arbitrarily close to y and p ∈ supp(µ)∩Wc(q) such that
[q,p] ⊂Ws(p).
c
h being small enough and q close to y as we like, [q,p] must intersect either
c
Wss(Wuu(x))orWss(Wuu(z)). Takeapointy′ fromtheintersection,thenω(y′)⊂
supp(µ)∩K =∅, which is absurd.
So there exists h > 0 such that any segment of length less than h in a center
leaf contains at most two points in K.
Lemma 4.4. For any x∈M, ♯Φ (x)<+∞.
K
Proof. ItisobviousfromLemma4.3andthefactthat[x,f(x)] iscoveredbyfinitely
c
many segments with length less than h.
Since K isinvariantand♯Φ (x)<∞foranyx∈M,thereexistsk(x)≥1such
K
thatWc(x)∩K consistsofk(x)orbitsoff,and♯Φ (y)=k(x)ify isinWc(x)\K,
K
♯Φ (y)=k(x)+1 if y ∈Wc(x)∩K.
K
k(x)isu-invariantbecauseK isu-saturated. Thereforek(x) is constant on each
center unstable leaf.
Lemma 4.5. There is k ∈N such that k(z)=k for any z ∈M.
6
Proof. Letx be a continuitypoint ofΦ . Thereexists a neighborhoodU ofxsuch
K
that k(y)≥k(x) for any y ∈U.
claim 4.6. k(y) is constant in a small neighborhood V of x.
Proof of the Claim. Assume there exist {y } such that y →x, k(y )>k(x). Let
n n n
x¯∈Φ (x), y1,y2 ∈Φ (y ) such that yi →x¯,i=1,2.
K n n K n n
Let n be sufficient large such that y1,y2 is close to x¯ as we like and d(y1,y2) is
n n n n
small enough. m is much greater than n such that d(y1 ,y2 ) is much smaller than
m m
d(y1,y2).
n n
Take ωi from Wss(Wuu(yi)) ∩ Wc(y ) for i = 1,2. Then at most one of
m n m
{ω1 ,ω2 } lies between y1 and y2 . Applying similar arguments to Lemma 4.3 can
m m m m
lead to a contradiction.
Foranyz ∈M,Wcu(z)isdense,thereforeWcu(z)∩V 6=∅. Sincek(y)isconstant
along each center unstable leaf, k(z)=k(x). So k(z) is constant in M.
Given a compact subset Λ of M, a r dimensional C0 lamination of Λ is a
partition L of Λ such that: there is a collection of closed domains {D}, the union
of whose interiors cover Λ, and for each domain D, there is a compact set Y and
a homeomorphism ϕ : Br ×Y → D ∩ Λ such that ϕ(Br × y) is the connected
component of L(ϕ(0,y))∩D.
Lemma 4.7. K has a codimension-one C0 lamination L.
Proof. Take x outside K, U a small neighborhood of x such that U ∩K = ∅. Let
B a closed codimension-one disk contained in U transverse to Wc and x∈B.
D = ∪ [y,f(y)] is a closed domain. It is obvious that K is covered by the
y∈B c
interiors of such kind of D’s.
Any y ∈ B, Φ (y) = {y ,··· ,y } with k as in Lemma 4.5. Suppose y ’s are
K 1 k i
listed respecting the orientation of Wc, and we can denote y <y <···<y .
1 2 k
Define
ψ : B×{1,··· ,k}→K∩D
(y,i)7→y
i
By taking B small enough we can see that ψ is injective. We are left to show that
ψ is continuous.
Given any (y,i), any y →y, let y¯ be an accumulation point of {(y ) }. Then
n i n i
y¯ =y for some j.
i j
Supposej <i,thenthereexist1≤j′ ≤j,1≤i <i ≤isuchthat(y ) ,(y )
1 2 n i1 n i2
accumulate to yj′. Similar arguments to Lemma 4.3 will lead to a contradiction.
Neither j >i holds.
Therefore j =i. ψ is continuous at any given (y,i).
Since ψ is defined on compact space, ψ is a homeomorphism.
K is partitioned into connected components. The partition is denoted by L.
The above arguments imply L is a codimension-one C0 lamination.
Lemma 4.8. K is s-saturated.
7
Proof. Since if dynamically coherent, We are reduced to prove that for any x∈K,
Wcs(x)∩K is s-saturated. Since Es is uniquely integrable, we only need to show
that Wcs(x)∩K is tangent to Es(x).
Fix a smallcoordinateneighborhoodU ofx inWcs(x)suchthatxis the origin,
Wc (x)={0}×R,Wss(x)=Rs×{0}.
loc loc
Because K is topologically transverse to Wc and by the arguments of Lemma
4.7eachleafisanembedding,thereexistg :Rs →Rsuchthatg(0)=0,graph(g)=
Wcs(x)∩K∩U.
loc
SupposeDg(0)6=0,thenthereexistv ∈Rssuchthatkv k→0,|g(v )|/kv k→
n n n n
2b>0.
Assume g(v )>bkv k for some large n. For 0<a< bkv k,
n n 2 n
b
Wss(0,a)={(x,y)|y =a+ϕ (x),ϕ (0)=0,kDϕ k< }.
loc a a a 2
For 0≤t≤1, define h (t)=g(tv )−ϕ (tv )−a. Then
a n a n
h (1)= g(v )−ϕ (v )−a
a n a n
b b
> bkv k− kv k− kv k
n n n
2 2
> 0,
h (0)=−a<0.
a
There exist 0 < t < 1 such that h (t) = 0. Therefore g(tv ) = ϕ (tv )+a, that
a n a n
is to say graph(g)∩Wss(0,a) ∋ (tv ,g(tv )). So graph(g)∩Wss(0,a) 6= ∅ for
loc n n loc
0<a< bkv k.
2 n
For a fixed a, let p = (v ,ϕ (v )+a), then p ∈ Wss(0,a)\K. There exist a
n a n
neighborhoos V of p such that any y ∈V, Wss(y)∩graph(g)6=∅.
Since (tv ,g(tv )) ∈ graph(g) ⊂ K ⊂ B(µ) ⊂ Ws(Λ), there exist q ∈ Ws(Λ)
n n
suchthatq iscloseto(tv ,g(tv )). Wss(q)intersectsV atsomey bythecontinuity
n n
ofstrongstablefoliation. SinceWss(y)∩graph(g)6=∅onehasWss(q)∩graph(g)6=
∅. Consequently ω(q)⊂K.
On the other hand, ω(q)⊂supp(µ). so supp(µ)∩K 6=∅, a contradictionto the
assumption. Therefore K is s-saturated.
Lemma 4.9. The leaves of K are C1 immersions.
Proof. By [5, Appendix], Wss(Wuu(x)) is once differentiable at x. Being both
u-saturated and s-saturated, every leaf of K is C1.
Proof of Proposition 3.2. Let k be as in Lemma 4.5. Combining Lemma 4.5 and
Lemma4.9,K hasaC0 laminationLwithC1 leaves. Thelaminationbeingtangent
to Es⊕Eu whichis continuous,is C1. The proofofProposition3.2is finished.
Acknowledgement
We would like to express our deep gratitude to Professor Raul Ures for posing the
problem addressed in this paper and for for sharing his ideas generously.
Z.ZhengacknowledgessupportfromNNSFC(♯11071238),theKeyLab. ofRan-
dom Complex Structures and Data Science CAS, and National Center for Mathe-
matics and Interdisplinary Sciences CAS.
8
References
[1] J.Alves,C.BonattiandM.Viana, SRBmeasuresforpartiallyhyperbolicsys-
tems whose central direction is mostly expanding, Invent. Math., 140 (2000),
351–398.
[2] C. Bonatti, L.D´ıaz and M.Viana, Dynamics beyond uniform hyperbolicity, A
globalgeometricandprobabilisticperspective, EncyclopaediaofMathematical
Sciences, 102. Mathematical Physics, III. Springer-Verlag,Berlin, 2005.
[3] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomor-
phisms, Lecture Notes in Mathematics, Vol. 470. Springer-Verlag,Berlin-New
York, 1975.
[4] K. Burns, F. Rodriguez Hertz, J. Rodriguez Hertz, A. Talitskaya and R.
Ures, Density of accessibility for partially hyperbolic diffeomorphisms with
one-dimensional center, Disc. Cont. Dynam. Syst., 22 (2008), 75–88.
[5] A. Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory Dynam.
Systems, 33 (2013), 896–933.
[6] M. Hirsch and C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in
Mathematics, Vol. 583. Springer-Verlag,Berlin-New York, 1977.
[7] I. Kan, Open sets of diffeomorphisms having two attractors, each with an
everywhere dense basin, Bull. Amer. Math. Soc. (N.S.), 31 (1994), 68–74.
[8] Y. Ilyashenko, V. Kleptsyn and P. Saltykov, Openness of the set of boundary
preserving maps of an annulus with intermingled attracting basins. J. Fixed
Point Theory Appl., 3 (2008), 449–463.
[9] I. Melbourene and A. Windsor, A C∞ diffeomorphism with infinitely many
intermingled basins, Ergodic Theory Dynam. Systems, 25 (2005), 1951–1959.
[10] J. Palis, A global view of dynamics and a conjecture on the denseness of
finitudeofattractors,G´eom´etriecomplexeetsystemesdynamiques, Ast´erisque,
261 (2000), 335–347.
[11] J. Palis, A global perspective for non-conservative dynamics, Ann. I. H.
Poincar´e, 22 (2005), 485–507.
[12] R.Potrie, PartiallyhyperbolicityandfoliationsinT3, J. Mod. Dyn.,9(2015),
81–121.
[13] C. Pugh, M. Shub and A. Wilkinson, Holder foliations, revisited, J. Mod.
Dyn., 6(2012), 79–120.
[14] R. Ures and C. Va´squez, On the non-robustness of intermingled basins,
arXiv1503.07155v2.
9
