Dynamical properties of singular-hyperbolic attractors

DISCRETE AND CONTINUOUS Website: http://aimSciences.orgDYNAMICAL SYSTEMSVolume 19, Number 1, September 2007 pp. 67–87

DYNAMICAL PROPERTIES OF SINGULAR-HYPERBOLIC

ATTRACTORS

Aubin Arroyo1

UNAM - Instituto de Matematicas, U. CuernavacaA.P. 273 Admon. de correos # 3

Cuernavaca, Morelos 62251, Mexico

Enrique R. Pujals2

IMPA - Instituto de Matematica Pura e AplicadaEst. D. Castorina 110

22460-320. Rio de Janeiro, Brazil

(Communicated by Marcelo Viana)

Abstract. We provide a dynamical portrait of singular-hyperbolic transitiveattractors of a flow on a 3-manifold. Our Main Theorem establishes the ex-

istence of unstable manifolds for a subset of the attractor which is visitedinfinitely many times by a residual subset. As a consequence, we prove thatthe set of periodic orbits is dense, that it is the closure of a unique homoclinicclass of some periodic orbit, and that there is an SRB-measure supported onthe attractor.

1. Introduction. Hyperbolicity is the paradigm of stability for dynamical systemsin which some conditions on the behavior of the derivative constrain the dynam-ics to a robust scenario. Time evolution laws that fit into the setting of uniformhyperbolicity, also called Axiom A, are well understood after the work of Smale[16] and others. In particular, a useful dynamical portrait is obtained by this prop-erty: uniform hyperbolic systems are structurally stable, the set of periodic orbitsis dense in the non-wandering set, and they have a spectral decomposition into afinite number of disjoint basic pieces. This notion might be enough to understanddiscrete dynamical systems, but in the realm of continuous ones, something weakeris needed. In fact, uniform hyperbolicity for flows cannot allow the accumulation ofnon-wandering regular orbits on equilibrium points. This is the situation depictedin the so called Lorenz Attractor.

In the mid-sixties, numerical experiments performed by E. Lorenz, in [10], on thefollowing differential equation on R

3

x = α(y − x), y = βx − y − xz, z = xy − γz (1)

suggested the existence of a strange attractor for some parameters close to α = 10,β = 28 and γ = 8/3; that is, the existence of an indecomposable invariant set whichtraps the positive orbit of solutions of all initial conditions in a full neighborhood ofthe origin of R

3. Such an invariant set is nowadays known as the Lorenz Attractor.

2000 Mathematics Subject Classification. Primary: 37D10 37D30.Key words and phrases. Singular-Hyperbolicity, 3-dimensional Flows, Attractors.

67

68 A. ARROYO AND E. R. PUJALS

Notice that the origin is an equilibrium point of (1) and its eigenvalues satisfy thefollowing relations:

λss < λs < 0 < −λs < λu. (2)

Quantitative analysis of equations in (1) did not provide enough informationto develop its dynamical nature, and only after the introduction of a geometricalmodel of the Lorenz Attractor, in the mid-seventies ([6], [1]), did it become possibleto prove the existence of a non-trivial transitive attractor with singularities. Recallthat an invariant set is transitive if it contains a dense orbit. Notably, only threeand a half decades after the discovers of Lorenz, it was proved by Tucker in [17] thatthe solutions of (1) behave in the same way as the geometrical model, for values α,β and γ near the ones originally considered.

The structure of the geometrical model of the Lorenz Attractor relies on somecharacteristics always present among C1-robustly transitive sets (see next para-graph) with singularities, at least in dimension 3, and they are at the core of thenotion now called singular-hyperbolicity. These characteristics are precisely thoseobtained in [13], where they prove that any C1-robustly transitive set with singu-larities on a closed 3-manifold verify that: it is either an attractor or a repeller set;all equilibrium points contained in are Lorenz-like, i.e., its eigenvalues satisfy (2);and the tangent bundle splits into a partially hyperbolic invariant splitting withone direction (2-dimensional) volume expanding and the other vector contracting.

Let us state some definitions and fix notation we shall use. A continuous timedynamical system Φ : M × R → M is a C1-action of the group R on a completeriemannian manifold M ; such action is also called a flow. Equivalently, a flow isgiven by the integration curves of a vector field X ∈ X r(M). Let Sing(Φ) := {p ∈M |Φt(p) = p, ∀t} = {p ∈ M |X(p) = 0} be the set of singular points of Φ. Thecomplement of singular points is the set of regular orbits. Denote by Cl(U) theclosure of a set U ⊂ M . A closed invariant set Λ ⊂ M of Φt = Φ(·, t) is an attractor

if there is an open neighborhood U ⊃ Λ such that Λ = ∩t>0Φt(Cl(U)); such U iscalled its basin of attraction of Λ. An invariant set of Φt is C1-robustly transitive ifthere is a C1-neighborhood U of Φ for which the set ∩t∈RΨt(Cl(U)) is transitive forall Ψ ∈ U . For now and on, let M be a 3-dimensional riemaniann closed manifold.

Definition 1.1. Let Φt be a flow on M . A compact Φt-invariant set Λ is singular-hyperbolic if: any singular point in Λ is hyperbolic, the tangent bundle of Λ splitsinto two invariant sub-bundles, TΛM = Es⊕Ecu, and there are two constants λ < 0and c > 0 such that the following properties hold, either for Φt or Φ−t.

1. The splitting Es ⊕ Ecu is dominated by λ;2. ||DΦt|Es || < c exp(λt); that is, Es is uniformly contracting;3. det(DΦt|Ecu) > c exp(−λt); that is, Ecu is volume expanding.

The invariant splitting: TΛM = E ⊕ F , is dominated by λ < 0 if there exists c′ > 0such that ||DΦt|E || < c′ exp(λt)m(DΦt|F ), for any t > 0; here, m(·) denotes themininorm.

The notion of singular-hyperbolicity not only encloses Axiom A systems butgeometric Lorenz Attractors, Singular Horseshoes [9], and others as well. Singular-hyperbolic systems deserve a complete theory relying on this weaker notion of hy-perbolicity – like the one already constructed for uniform hyperbolic systems, andwhere local stable and unstable manifolds of uniform size are fundamental. Let usfocus on their definition: A compact invariant set Λ of a flow Φt is hyperbolic if there

SINGULAR-HYPERBOLIC ATTRACTORS 69

is a DΦt-invariant splitting TΛM = Es ⊕ [X ] ⊕ Eu, of the tangent bundle on Λ,where DΦt contracts vectors on Es(u), exponentially fast, for t → ±∞, respectively.Denote by Bε(x) ⊂ M the open ball of radius ε centered in x. If an invariant set Λis hyperbolic, there exist ε > 0 such that the sets

W s(u)ε (x) = {y ∈ Bε(x)|d(Φt(x), Φt(y)) → 0 as t → ±∞},

for any point x ∈ Λ, are smooth disks embedded on M , of the same dimensions asEs and Eu, respectively. These are the local stable and local unstable manifoldsfor Λ, respectively.

The goal of this work is the Main Theorem, stated below, that establishes theexistence of unstable manifolds of uniform size for a certain subset of any singular-hyperbolic transitive attractor in a 3-manifold. Since this set is visited infinitelymany times by a residual set of the whole attractor, this is enough to derive sev-eral dynamical properties of them. Before we state it properly, let us describe itsconsequences.

Let Per(Λ) be the set of periodic orbits in Λ ⊂ M .

Theorem A. If Λ is a singular-hyperbolic transitive attractor for a flow Φt, then

the set of periodic orbits in Λ is dense in Λ; that is, Cl(Per(Λ)) = Λ.

Of course, there are examples of singular-hyperbolic sets which do not have anyperiodic orbit at all (see [11]). However, such sets are neither attractors nor robust.Also, it has been proved recently that any transitive singular-hyperbolic attractorhas at least one periodic orbit; see [3].

The homoclinic class of a periodic point p, denoted by H(p), is the closure ofthe transversal intersections of the stable and unstable manifolds of p. Given amaximal invariant transitive hyperbolic set, it always coincides with the homoclinicclass of certain hyperbolic periodic point. This is not true in general for singular-hyperbolic maximal invariant (see [11]). Theorem B provide a complete descriptionof the spectral decomposition of a singular-hyperbolic transitive attractor.

Theorem B. If Λ is a singular-hyperbolic transitive attractor for a flow Φt, then

there exists a periodic orbit p such that H(p) = Λ.

In [4], they prove that a general singular-hyperbolic invariant set does not allowa decomposition into a finite number of disjoint homoclinic classes. In [14], thereare examples of singular-hyperbolic transitive attractors in which there is a periodicorbit in the closure of an homoclinic class which is not homoclinically related toit. Observe that this cannot happen for a hyberbolic basic piece, since any pairof periodic orbit is homoclinically related. Such systems are examples of singular-hyperbolic transitive attractors that are not C1-robustly transitive sets.

A flow Φ is C1+α, for α > 0, if DΦt is α-Holder continuous, for any t > 0.Regarding at the statistical properties of singular-hyperbolic attractors, in [5] theyprove the following statement: if Λ is a singular-hyperbolic transitive attractor of

a C1+α flow, α > 0, and Per(Λ) is dense in Λ, then it has an SRB measure. As aconsequence of this and Theorem A we obtain the following Corollary.

Corollary: If Λ is a singular-hyperbolic transitive attractor of a C1+α flow, α > 0,then it has an SRB measure supported on it.


1.1. Main theorem. Let Λ ⊂ M be a singular-hyperbolic attractor for a flow Φt

defined on M . Assume that Es is one dimensional. The problem in finding unstablemanifolds on points of Λ is in its definition: when integral curves come arbitrarilyclose to an equilibrium point, they are arbitrarily “slowed down.” Hence, mosttransversal C1-intervals on points of Λ asymptotically lay over the orbit. This isthe case except for the stable direction Es. The strong stable bundle is integrableand it contracts exponentially vectors within. In fact, this implies the existence oflocal stable manifolds of uniform size εs > 0, on any point x ∈ U , where U is aneighborhood of Λ contained in its basin of attraction. Denote them by W s

loc(x).For singular points such interval corresponds to the stable manifold associated to itsstrongest contracting eigenvalue, and we denote them by W ss

loc(σ). However, for theunstable direction a new definition for unstable sets is needed; this new definitiondeals with holonomy maps between 2-dimensional transversal sections over pointsof the same regular orbit. Let us introduce transitions.

1.2. Transitions. Let U be some neighborhood of Λ contained in its basin of at-traction. Denote by S = {σ1, . . . , σk} = Λ ∩ Sing(Φt), the set of singular points ofΦt contained in Λ. Of course, S is not empty, otherwise Λ would be hyperbolic.

On the set of regular points U := U − S, the normal bundle N ⊂ TM is definedpointwise: Let Np be the 2-dimensional subspace, orthogonal to the space spanned

by X(p) ∈ TpM . On each point p ∈ U , consider some transversal section Np,tangent to N (p) (in Section 2 we shall construct precisely these sections using theexponential map).

Let x ∈ U and t ∈ R. Denote by xt := Φt(x), to simplify notation. If t > 0, theImplicit Function Theorem asserts that there is an open connected subset D ⊂ Nx,containing x, and a continuous function τ : D ⊂ Nx −→R such that τ(x) = t andΦτ(y)(y) ∈ Nxt

, for any y ∈ D. The map τ induces a difeomorphism:

Gtx : dom(Gt

x) ⊂ Nx −→ Nxt

Gtx(y) := Φτ(y)(y) ∈ Nxt

where dom(Gtx) is the maximal connected component of Nx, containing x, where τ

is defined. Notice that x ∈ int(dom(Gtx)). The map Gt

x represents the holonomybetween Nx and Nxt

of time t > 0. Holonomy maps are defined for the past, aswell by

G−tx : img(Gt

x−t) ⊂ Nx−→dom(Gt

x−t) ⊂ Nx−t

,

where G−tx = (Gt

x−t)−1. Transitions are restrictions of these holonomy maps to

smaller domains of definition. We shall revisit transitions in §2. Now let us statethe following definition of unstable set according to transitions:

Definition 1.2. The local unstable set of a regular point x ∈ Λ is the set Wuε (x) ⊂

Nx, defined as the set of the points y ∈ Bε(x) ∩ Nx such that y ∈ Dom(G−tx ), for

all t > 0, and d(G−tx (y), x−t) → 0 as t → +∞.

In order to state the Main Theorem we need to recall the notion of center unstablemanifolds of [7]. Given any point x ∈ U , the set W cu

ε (x) is called the center unstable

manifold of x if Tx(W cuε (x)) = Ecu(x); and for any ε1 > 0 and t > 0, there is ε2 > 0

such that if y ∈ W cuε2

(x) and d(Φ−t(x), Φ−t(y)) < ε1 then Φ−t(y) ∈ W cuε1

(Φ−t(x)).

Main Theorem. Let U be an open neighborhood of a singular-hyperbolic transitive

attractor Λ, contained in its basin. There is a subset K ⊂ U and constants εu > 0,λu < 0 and c > 0 such that:


1. Wuεu

(y) = W cuεu

(y) ∩ Ny, for any y ∈ K;

2. if y ∈ K, then for any z ∈ Wuεu

(y) there is a unbounded sequence {ti} of

positive numbers such that d(G−tiy (z), y−ti

) < C exp(tiλu);3. the set Λ ∩

⋃t>t0

Φ−t(K) is open and dense in Λ, for any t0 > 0.

The proof of the Main Theorem is based on the construction of a family of returnmaps on a certain collection (finite) of small transversal sections, with markovianproperties. In general, a global transversal section (connected) does not necessarilyexist.

A sketch of the argument to find this family is the following: We consider afinite collection of small transversal sections, inside some neighborhood of S, whichintersect a local stable manifold of some σ ∈ S. This collection is called a system of

transversal sections. The first return time of points on these sections is arbitrarilylarge (perhaps infinite), provided the point is close to the stable manifold of σ. Us-ing the volume expanding property of singular-hyperbolicity (in particular Lemma3.4 and Corollary 2), we can deduce that transition G, induced by the first returnmap, expands vectors along the direction defined by the intersection of center un-stable disks with transversal sections, on any point of its domain of definition. Thisdirection we call unstable direction and is what we care about most. In fact, thestable foliation induced by the stable bundle allows us to assume that transversalsections are actually foliated by one dimensional stable manifolds (Lemma 2.2).Hence, shrinking the system of transversal sections enough, we can assert that anybox B contained in the domain of definition of G is stretched along the unstabledirection, by G, up to a factor at least 2; in particular, the image G(B) has unsta-ble length two times larger than the unstable length of B. Once we have obtainedexpansion, we have to care about definition domains, since every time the image ofa box intersects a stable manifold of a singularity the next transition map may bediscontinuous.

So, the image of the box G(B), contained in the target section, either intersectsthe local stable manifold of the singularity or not (Lemma 3.6). If this intersectionis empty, we consider the transition induced by the next return of B. On the otherhand, G(B) is split into two connected components by the the local stable manifoldof the singularity. In particular, one of them has unstable length larger than theunstable length of B. We keep this component and we iterate it up to the nextreturn (which is proved that it exists; see Lemma 3.5). In this description of theargument, we are implicitly assuming that the box B “did not get large beforeit returned;” actually, if that is the case, then the argument is simplified: a sub-box contained in B has the property that its image covers one of the connectedcomponents of the section (see Lemma 3.6). This process eventually finishes witha sub-box whose image covers one connected component of the transversal sectionsplit by the local stable manifold of the singularity. This argument allows us toselect a collection of returns with markovian properties (Corollary 1).

Techniques developed in this work present an adequate framework to continuea more detailed study on ergodic and topological properties of singular-hyperbolicattractors (for instance, exponential decay of correlation functions, or a useful gen-eralization of the notion of local product structure), and they may have deeperconsequences on the behavior on the differentiability of their corresponding SRBstates, according to the work of Ruelle in [15].


G1G2

σ

σ′

BG1(B)G2 ◦ G1(B)

Figure 1. Sketch of the argument

In [8], they prove a theorem of existence of stable manifolds for what they callsmooth maps with singularities, from a measure theoretical point of view. They usean invariant measure and some hypothesis on the growth of the second derivativeof the map to constrain the recurrence to the singular region. It is important toremark that our strong hypothesis is only about the hyperbolic behavior of thederivative.

Remark 1. All results we prove here are true on any finite dimensional compactmanifold regarding that the codimension of the strong stable bundle is 2. That is,the splitting on the attractor is TΛM = Es ⊕ Ecu, where dim(Es) = dim(M) − 2.

The structure of this paper is the following: In §2 we define transitions, normalsections, boxes and systems of transversal sections. In §3 we state and prove aMain Lemma; the proof of such a lemma will be obtained after we develop severaldynamical properties of transitions, summarized in Proposition 1 and 2. In §4 wegive a proof for the Main Theorem. Finally, in §5, we give proofs for Theorem Aand Theorem B, stated in the Introduction.

2. Boxes, transversal sections and transitions. Recall that each fiber Nx ofthe normal bundle N ⊂ TM on the set of regular points U is two dimensionaland has a coordinate system induced by the singular splitting in the following way:Es

x := Π(Esx) and Eu

x := Ecux ∩ Nx, where Π : TM → N denotes the orthogonal

projection. Notice that the angle between these two directions is uniformly boundedaway from zero in U . Denote by Np(η) the standard rectangle (−η, η)2 ⊂ Np,according to this coordinate system.

For any point p ∈ U there is a number η(p) > 0 such that the exponential mapExpp :Np(η(p))−→M is an isometry. Observe that for any δ > 0 there is ηδ > 0

such that if d(p, S) > δ, then η(p) > ηδ. Consider η∗ : U −→ R+ the function that

η∗(p) = min{η(p), ηδ}. Denote by Np := Expp(Np(η∗(p))) ⊂ M , a 2-dimensional

transversal section on p of size η∗(p). Of course, the function η∗ depends on δ.Later, in §2.3, we shall fix δ, and hence η∗.

After the definition of transition maps given in the Introduction, we are left todefine precisely their domains that we are considering.


Definition 2.1. Let D(p, t) ⊂ Np be the interior of the connected component of∩s∈[0,t]dom(Gs

p) that contains p. The map Gtp|D(p,t) is a transition map between

Np and Npt.

Observe that if t > 0, then D(p, t) ⊂ D(p, r), for all r ∈ [0, t]. Hence, the imageGs

p(D(p, t)) ⊂ int(Nps) for any 0 < s < t. Also, for any point x ∈ D(p, t) and any

s ∈ [0, τ(x)] there exists s ∈ [0, t] such that xs ∈ Nps(η∗(p)).

2.1. Center manifolds. Let Ds(cu) stand for the disks of radius 1 in R and R2,

respectively. Let Emb1(D, M) be the set of C1-embeddings of D into the manifoldM .

Lemma 2.2. Let Λ be a singular-hyperbolic invariant set for the flow Φt. There

exists two continuous functions: Ψs (cu) : U → Emb1(Ds (cu), M) such that, denoting

by Ws (cu)ε (x) := Ψs (cu)(x)(D

s (cu)ε ), for any x ∈ U , verify the following:

1. Tx(W sε (x)) = Es(x) and Tx(W cu

ε (x)) = Ecu(x).2. There exists εs > 0 and λs < 0 that Φt(W

sεs

(x)) ⊂ W sexp(tλs)εs

(Φt(x)), for any

t > 0.3. Given t > 0 and ε1 > 0, there is ε2 > 0 such that if y ∈ W cu

ε2(x) and

d(Φ−t(y), Φ−t(x)) < ε1, then Φ−t(y) ∈ W cuε1

(Φ−t(x)).

Proof. The proof follows a classic argument of partial hyperbolic systems. Considerthe time-one map Φ1 induced by the flow and observe that it is a partial hyperbolicsystem with a one-dimensional contractive sub-bundle and a two-dimensional sub-bundle and then apply the classical results of invariant manifolds for Φ1, [7]. Inparticular, the singular-hyperbolic splitting on Λ allows us to define an invariantcone-field in all U . Such invariant cone-fields allow us to fit into the frameworkof [7] to guarantee the existence of central manifolds in all U . The exponentialcontracting property in Es proves item 2.

In terms of transitions, the last item of the previous lemma implies that: given

t > 0 and any ε1 > 0, there is ε2 > 0 such that if y ∈ W cuε2

(x) ∩ Nx, then G−tx (y) ∈

W cuε1

(x−t) ∩ Nx−t. Therefore, it follows that: Wu

ε (x) ⊂ Nx ∩ W cuε (x).

2.2. Boxes. For a regular point p, the coordinates induced by Esp and Eu

p on eachNp allow us to define boxes with respect to these axis. Let B be a box on Np.

We shall use the projection πs : Np → Eup , along the Es-axis (respectively, the

projection πu : Np → Esp, along the Eu-axis), in order to define the stable (or

unstable) length of B, say:

|B|s (u) = length[πu (s)(B)].

If it is necessary to be explicit on both dimensions of a box, we shall use a vectorv = (εu, εs) to denote the box: B[v]. The first coordinate of v stands for theunstable length and the second for the stable length. When we care only about theunstable length we denote only by B(εu). In the same way, the boundary of a box issplit into the unstable and stable boundaries, according to the induced coordinates:

∂uB = π−1s (∂πs(B)) and ∂sB = π−1

u (∂πu(B)).

Given a box B on p, a sub-box B ⊂ B is a box that contains p and that πu(B) =

πu(B). Given two boxes B ⊂ B, we say that B covers B if πs(B) = πs(B); in

such a case: ∂uB ⊂ ∂uB. A semi-box on p is a box containing p in the unstableboundary.


The exponential map allow us to define boxes in Np ⊂ M , by the image of a boxB ⊂ Np(η). To avoid confusing notation we shall not make any difference betweenthe boxes in Np and their image by Expp in M . In fact, without lose of generalitywe can assume that the transversal foliations Fs and Fcu in M , given in Lemma2.2, correspond to the previous coordinates in Np. Actually, Lemma 2.2 impliesthat Fs are actual stable manifolds of size εs.

2.3. System of transversal sections. Any equilibrium point of a singular-hyper-bolic set is hyperbolic. In [12] it is proved that all singular points are Lorenz-like;that is, the eigenvalues, λss

σ , λsσ and λu

σ, of σ ∈ S satisfy relations in equation(2). Locally, each singular point σ is of saddle type. On the same paper, theyprove: W ss(σ) ∩ Λ = {σ}, for any σ ∈ S. Let us concentrate on a neighborhood ofeach singular point σ ∈ S. Hartman-Grobman’s Theorem imply the existence of aneighborhood Oσ of σ, where the flow is C0-conjugated to its linear part. That is,for any point (x0, y0, z0) ∈ Oσ the flow is given in these coordinates by:

Φt(x0, y0, z0) = (x0 exp(tλssσ ), y0 exp(tλu

σ), z0 exp(tλsσ)). (3)

Inside Oσ, the local stable, local strong-stable and local unstable manifolds of σare given in these coordinates by: W s

loc(σ) = [y = 0], W ssloc(σ) = [y = z = 0] and

Wuloc(σ) = [x = z = 0], respectively.The flow is not transversal to ∂Oσ, since σ is a saddle. Moreover, there are

two positive numbers cz and cy for which the intersection of Oσ with the fourplanes [z = ±cz] and [y = ±cy] are transversal to the flow. Notice that on theformer sections the vector field points inwards to the region containing the origin,and on the latter it points outwards. Denote by p± each point in the intersectionWu

loc(σ) ∩ [y = ±cy], respectively.Given two numbers hu(s) > 0, they define a rectangle contained in the plane

[z = cz], say

Σ+[hu, hs] = {(x, y, cz)| y ∈ (−hu, hu), x ∈ (−hs, hs)}.

If hs and hu are small then Σ+[hu, hs] is contained in Oσ. In the same way,Σ−[hu, hs] ⊂ Oσ is a rectangle contained in the plane [z = −cz]. Moreover, ifhs and hu are small enough, then there exist two disks ∆± contained in the planes[y = ±cy] such that for any point x ∈ Σ±[hu, hs] − W s(σ) there is t > 0 suchthat Φt(x) ∈ ∆+ ∪ ∆−, and Φs(x) ∈ Oσ, for any s ∈ (0, t). Notice that the vectorfield points outwards in both disks ∆±.

It is important to notice that the flow is transversal on the intersection of Oσ

and two more planes [x = ±hs], and there, the flow points inwards.Observe also that if ε > 0 is small enough and if x ∈ Σ+ ∪ Σ−, then we can

assume without lose of generality that Nx(ε) ⊂ [z = cz] ∪ [z = −cz], since theactual transversal section is C0-close to these planes.

Lemma 2.3. Let σ ∈ S, and let Oσ be the neighborhood defined above. There exist

positive numbers cz, hu, hs and δ such that the following holds:

1. cz, hu and hs define two rectangles Σ±[hu, hs] ⊂ Oσ that if x ∈ Λ and σ ∈ω(x), then either x ∈ W s

loc(σ) or O+(x) ∩ Σ±[hu, hs] 6= ∅;2. Bδ(σ) ⊂ Oσ, and for any point x ∈ Bδ(σ), there are two numbers t− < 0 < t+

such that Φt−(x) ∈ Σ+ ∪ Σ− and Φt+(x) ∈ ∆+ ∪ ∆−; moreover, Φt(x) ∈ Oσ,

for any t ∈ (t−, t+).


Σ+Q

Σ−

∆− ∆+

Wuloc(σ)

σ

Figure 2. Transversal sections inside a neighborhood of σ ∈ S

Proof. Let B be a tubular neighborhood of W sloc(σ) ∩ Oσ of height 2hs > 0. The

flow is transversal to B. Label the points of the intersection of W ssloc(σ) with B by

a0 and a1, respectively. There are two open sets V0 and V1 such that ai ∈ Vi, andsuch that Λ∩ (V0 ∪ V1) = ∅, since Λ is closed and W ss

loc(σ) ∩Λ = {σ}. If hs is smallenough, B − (V0 ∪ V1) has two connected components.

Hence there is cz > 0 such that each plane [z = ±cz] contains the image of theprojection along orbits of each connected component of B − (V0 ∪V1); furthermore,each image is contained in some rectangle Σ±[hu, hs], for some hu > 0. Notice thatany orbit accumulating on σ cannot cross V0 ∪ V1, proving item 1.

Choosing δ > 0 small enough, one obtains directly item 2, since inside Oσ thedynamic is linear.

Notice that constants obtained in Lemma 2.3 depend on each σ ∈ S. Fix δ =min{δσ|σ ∈ S} once and for all, and hence the function η∗ defined on the beginningof §2. Doing this, we also fix the stable length of sections we shall use hereafter.Denote by Vδ := U − Cl(∪σ∈SBδ(σ)). The next step is to determine an adequateunstable length for each transversal section.

Consider the set Γ of all transversal sections we have defined, two by each σ ∈ S.Give some index to the set Γ, say Γ = {Σ1, . . . , Σ2k}. Each Σi ∈ Γ is dividedinto two connected components by the interval Qi := Σi ∩ W s

loc(σ), where σ is thecorresponding singular point. Given a pair of positive numbers (ε−i , ε+

i ), they definea subsection Σi(ε

−i , ε+

i ) ⊂ Σi such that

πs(Σi(ε−i , ε+

i )) = (πs(Qi) − ε−i , πs(Qi) + ε+i ).

Definition 2.4. Let E be a set of 2k positive vectors (ε−i , ε+i ) ∈ R

2. A system oftransversal sections of size E is the set Γ(E) = {Σi(ε

−i , ε+

i ) ⊂ Σi} of subsections ofΓ.

Given a system of transversal sections Γ(E), denote by Q := {Qi} and Γ(E) =

{Σi := Σi − Qi|Σi ∈ Γ}. As a convention, when we say x ∈ Γ(E) or x ∈ Q, forsome point x ∈ M , we agree that this means “x ∈ Σ for some Σ ∈ Γ(E),” and thesame for Q. Also we write

max(E) = max{ε−i , ε+i | i = 1, . . . , 2k}.


Γ(E)

Σ−1 Σ+

1

Q1

Σ−2 Σ+

2

Q2

GB(B) B

Figure 3. Subset of a Markov Collection A.

Definition 2.5. A connected subset B ⊂ Σi for some Σi ∈ Γ(E) is a band of Γ(E)if both connected components of ∂sB, the stable boundary of B, are contained in∂sΣ.

3. Markovian induced map. Exploiting the condition of volume expansion alongthe center unstable direction, it is possible to obtain an induced map, defined oncertain system of transversal sections, with some markovian properties.

Definition 3.1. A collection A of bands of Γ(E) is called a Markov collection ifthere is µ > 1 such that, for any B ∈ A, there is a transition G : B → Γ(E) suchthat G(B) covers either Σ−

j or Σ+j , for some j, and ||DG|Eu || > µ.

The existence of a Markov collection for some system of transversal sections isa consequence of the following Main Lemma, which summarizes all properties weshall obtain for transition maps. The Main Lemma will be proved in §3.2.

Main Lemma. If Λ is a singular-hyperbolic transitive attractor, then there exist a

system of transversal sections Γ(E) and µ > 1 such that: for any band B there is

a sub-band B ⊂ B and a transition GB :B−→Γ(E) such that GB(B) covers either

Σ+j or Σ−

j , for some j ∈ {1, . . . , k}, and ||DxGB|Eu || > µ, for any x ∈ B.

Corollary 1. There is a Markov collection A for Γ(E) and µ > 1 such that ∪B∈ABis open and dense in Γ(E).

Proof of the Corollary. Consider Γ(E) and µ > 1 obtained in the Main Lemma.The first consequence of the Main Lemma is that the set A of all bands B ⊂ Γ(E)such that GB(B) covers Σ+

j or Σ−j , for some j, is not empty. It holds that ∪B∈AB

is an open set in Γ(E), since each band B is open. To prove that is a dense subset,consider any x ∈ Γ(E) and U a neighborhood of x in Γ(E). The set B = π−1

s (πs(U))

is a band in Γ(E) and the Main Lemma implies there is B ⊂ B that B ∈ A. Since

U covers B, B ∩ U 6= ∅. Hence, ∪B∈AB is dense in Γ(E).

3.1. Dynamical properties of transitions. The following two properties sum-marize the dynamical behavior of transitions. We devote the rest of this section toprove them.


Np

Np

Npt

Npt

Bp(ε1)

Bp(ε1)

Bpt(ε2)

Bpt(ε2)

C(p, t)

C(p, t)

D(p, t)

D(p, t)

σ

Figure 4. Items 2 & 3 of Proposition 2.

Proposition 1. There is λu < 0 and C > 0 such that, given any point p ∈ U and

t > 0 such that Φs(p) ∈ Vδ for s = 0 and s = t, and given any box B ⊂ D(p, t),then:

|Gtp(B)|u > C exp(−λut)|B|u.

Proposition 2. Let t > 0 and any point p ∈ Vδ such that Φt(p) ∈ Vδ. Denote by

G = Gtp the corresponding transition. Given two boxes B0 ⊂ Np that ∂B0∩∂Np = ∅,

and B1 ⊂ Nptof stable length εs, such that p ∈ B0 and pt ∈ B1, then there is a

maximal sub-box B ⊂ B0 ∩ D(p, t), p ∈ B for which one of the following holds:

1. G(B) ⊂ B1;

2. G(B) covers B1, i.e., πs(G(B)) = πs(B1);

3. There is a point y ∈ ∂u(B ∩ D(p, t)) such that y ∈ W s(σ), for some σ ∈ S.

See Figure 4 for a schematic statement of items 2 and 3, of Proposition 2. Toprove these two propositions several lemmas are needed. The first one translatesthe contraction property of the strong stable direction, of the singular-hyperbolicsplitting, to the local action of transitions between normal sections. Recall thenotation introduced in §2.

Lemma 3.2. Let p ∈ U and t > 0. Consider G = Gtp : D(p, t) → Npt

, the

corresponding transition. For any x ∈ D(p, t) we have:

1. DxG(Esx) = Es

G(x) and DxG(Eux ) = Eu

G(x);

2. There is λs < 0 such that ||DxG|Esx|| < exp(λsτ(x)).

Therefore, if x ∈ D(p, t) then ∪{t∈(−ε,ε)}Wsloc(Φt(x))∩Np ⊂ D(p, t); that is, D(p, t)

is a box foliated by local stable manifolds.

Proof. The invariance of the splitting under the derivative of G is immediate afterthe definition of the splitting. Recall λs < 0 of Lemma 2.2. The contractionproperty is a consequence of the existence of the local stable foliation in U ; seeLemma 2.2.


The next lemma describes the boundary of domains of transitions and, roughlyspeaking, states that points in the boundary of the domain either hit the boundaryof the target section or their ω-limit is a singularity of the flow.

Lemma 3.3. Let p ∈ U and t > 0. If x ∈ ∂uD(p, t) − ∂Np then either there is a

singular point σ and x ∈ W s(σ), or there is a point z ∈ ∂uNptand r < +∞ such

that Φr(x) = z.

Proof. Consider p ∈ U and t > 0. Recall that D(p, s) ⊂ D(p, s′), if s > s′. For anypoint x ∈ Np let sx := inf{s > 0|x /∈ D(p, s)}. Clearly, if x ∈ D(p, t) then sx ≥ tand sp = +∞. Also sx > 0 for all x ∈ Np because Np is a transversal section. Fixa point x ∈ Np and for each s < sx denote by tx(s) > 0 that Φtx(s)(x) ∈ Nps

. Thefunction tx(s) is continuous and strictly increasing on the variable s ∈ (0, sx).

Let x ∈ ∂uD(p, t), then sx < t; in fact, if sx ≥ t then the tubular flow-boxtheorem implies x ∈ D(p, t). Observe that:

l := lims→sx

tx(s) ∈ R+ ∪ {+∞}.

Suppose l = +∞. We claim that

limt→+∞

X(Φt(x)) = lims→sx

X(Φtx(s)(x)) = 0;

this clearly implies that the ω-limit of x is some σ ∈ S. Hence, x ∈ W s(σ) aswe required. To get the claim assume, by contradiction, there is a > 0 such that||X(Φtx(s)(x))|| > a for all s ∈ (s0, sx), for some s0 near sx. If ε > 0 is small

enough, sx − ε > s0, and the C0-distance between the two normal sections, Npsx−ε

and Npsx+ε, is smaller than, say a/10. Then Φtx(sx−ε)(x) ∈ Npsx−ε

. Because||X(Φtx(sx−ε)(x))|| > a, one can conclude that the orbit of x hits Npsx+ε

in finitetime and hence also Npsx

, contradicting the definition of sx. Hence, we obtain theclaim.

On the other hand, if l < +∞ then we can take z = Φl(x). We claim thatz ∈ ∂uNpsx

. If it is not the case, z ∈ int(Npsx) and the tubular flow-box theorem

implies x ∈ D(p, sx), since x /∈ ∂uNp. This is a contradiction. To finish theargument it is enough to observe the following: If the time needed to flow z toreach Npt

is finite, then we are done. If not, a similar argument to the previous onestates the existence of σ ∈ S such that z = Φl(x) ∈ W s(σ).

The next lemma gives information about the expansion rate of transitions alongthe unstable direction, in finite time orbits. One requirement is needed: bothextremal points of the orbit lay outside a neighborhood of the singular points. Recallλu < 0, the rate of volume expansion from the definition of the singular-hyperbolicsplitting.

Lemma 3.4. Consider t > 0 and any point p ∈ Vδ such that Φt(p) ∈ Vδ; then for

any vector v ∈ Eup we have: ||DpG

tp(v)|| > C exp(−λut)||v||, where C > 0 depends

only on δ.

Proof. Consider t > 0 and p ∈ Vδ as in the hypothesis. Take any vector hv ∈ Eup ,

where h > 0 and ||v|| = 1. The area of the parallelogram defined by v and X(p)in Ecu

p is a(0) = h||X(p)||. Denote by a(t) the area of the parallelogram defined bythe image under DpΦt of these two vectors. Observe DpΦt(X(p)) = X(pt). Thevolume expansion condition implies that

a(t) > a(0) exp(−λut).


Notice that a(t) = ||X(pt)||||Π(DpΦt(v))||, where Π is the projection onto the normalsection. However, Π(DpΦt(v)) = DpG

tp(v). Hence,

||DpGtp(v)|| >

h||X(p)||

||X(pt)||exp(−λut) > C exp(−λut)||v||,

where C = inf{||(X(x)||||X(y)||−1|x, y ∈ Vδ}.

Corollary 2. Under the same hypothesis of Lemma 3.4, there is C′ > 0 such

that, for any x ∈ D(p, t), ||DxGtp|Eu

x|| > C′ exp(−λut). Hence, ||DxGt

p|Eux|| >

C′ exp(−λuT0) if t > T0.

This fact provide us with information on the unstable length of the image of abox, contained in D(p, t), and gives us a proof of Proposition 1.

Proof of Proposition 1: Denote by Cux (γ) the affine cone in Np around direction Eu

x

of angle γ > 0. If β : [0, 1] → B is a curve that (d/dt)β(t)|t=s ∈ Cuβ(s)(γ) for any

s ∈ [0, 1], joining the two connected components of ∂u(B), then

|B|u > C′ length(β).

On the other hand, there is C′′ > 0, depending on γ, such that:

|Gtp(B)|u > C′′

∫ 1

0

||DxGtp(β

′(t))||dt > C exp(−λut)|B|u.

The last inequality follows from Lemma 3.4.

Proof of Proposition 2: Let t > 0, p ∈ Vδ, and two boxes B0, B1 be as in thehypothesis of Proposition 2. Lemma 3.2 focuses the proof only on the unstablelength of boxes. First, B0 ∩ D(p, t) 6= ∅, since p ∈ B0. Moreover, W s

loc(p) splitsB0 into two semi-boxes, A and A′, having W s

loc(p) on their unstable boundary.Consider the semi-box A, for instance. The argument applies identically to A′.

Denote by l = ∂uA − W s(p). If l ⊂ D(p, t) then G(l) ⊂ Npt, and we have three

options:

G(l) ⊂ B1, G(l) ⊂ ∂uB1, G(l) ∩ B1 = ∅.

In the first case, G(A) ⊂ B1. In the second, G(A) covers one side of B1. In the

third case there is a box A = G−1(B1 ∩ G(A)) ⊂ A, and G(A) covers one side ofB1. These three cases satisfy the thesis of Proposition 2 for one side of the box.

Now suppose l ∩ D(p, t) = ∅. There is a point y ∈ ∂uD(p, t) ∩ (A ∪ l). Noticethat l ⊂ Np; in fact, by hypothesis ∂B0 ∩ ∂Np = ∅. Lemma 3.3 implies that either:∃σ ∈ S such that y ∈ W s(σ), and then satisfy item 3; or ∃ r < +∞ such thatΦr(y) ∈ ∂uNpt

. In the last case the corresponding semi-box of B1 is contained in

G(A). Then, A = G−1(B1) ∩ A satisfy item 2. Notice that in any case, semi-boxesobtained are maximal. Repeating the argument on A′, the other side of B0, weobtain the statement of Proposition 2.

Now we seek an appropriate scale for a system of transversal sections where wecan find expansion properties. Consider a system of transversal sections Γ(E). Foreach point x ∈ Γ(E), denote by 0 < t(x) ≤ +∞ the minimum number such thatΦt(x) ∈ Γ(E). Of course if Φt(x) /∈ Γ(E), ∀t > 0 then t(x) = +∞; this is the case,for instance, when x ∈ Q.


Lemma 3.5. For any band B in Γ(E) there is a point x ∈ B such that ω(x)∩S 6= ∅,and there is a sub-band B ⊂ B, x ∈ B, such that t : B → R is continuous and finite.

Moreover, for any T > 0 there is β > 0 such that if max(E) < β, then the function

t|B is also bounded from below by T .

Proof. Recall Γ(E) is contained in the basin of attraction of Λ. Take a band B ofΓ(E) and assume, by contradiction, that ω(x)∩S = ∅, ∀x ∈ B. Lemma 2.3 impliesthat there is t+(x) > 0 such that Φt(x) ∈ Vδ for any t > t+(x). Hence, the set

A := Cl(∪x∈Bω(x)) ⊂ Λ

is a singular-hyperbolic set that does not contain singularities, and so it is hyperbolic(see [13]). In fact, the saturation by the orbits of the flow of B is an open set in Mwhere each orbit of it converges to A. Therefore, Λ is not transitive: a contradiction.This argument proves that there exists a point x ∈ B such that ω(x) ∩ S 6= ∅.

The Implicit Function Theorem guarantees then the existence of a neighborhoodU ⊂ Γ(E), x ∈ U such that t|U is continuous. Hence, there is a sub-band B ⊂ B

where t|B is continuous; in fact, Lemma 3.2 implies that π−1s (πs(U)) ⊂ B. Of

course, t|B is finite because x ∈ B.For the second part observe that the time needed to escape the corresponding

neighborhood Oσ is uniformly bounded from below, depending on β = max(E),and this bound goes to +∞ when β goes to 0. Hence, we are done.

Remark 2. Let B be a band of Γ(E). If y ∈ W s(σ) ∩ B, for some σ ∈ S,then W s

loc(y) ⊂ W s(σ); furthermore, if y′ ∈ W sloc(y) and Φt(y) ∈ Qj, for some

t > 0, then Φt(y′) ∈ Qj, since Qj is an actual stable manifold.

The previous Remark implies that for any band B, the set

R = {(z, t)| z ∈ πs(B), t ∈ R+ s. t. Φt(π

−1s (z)) ⊂ Qj for some j}

is well ordered according to the following relation: (z, t) < (z′, t′) if and only ift < t′. This is true because all points of any stable manifold hit Qj at the sametime. Hence, if R is not empty, then there exists a first time t > 0 and a firstelement y ∈ B such that W s

loc(y) hits Σj .The next lemma gathers together all information we have collected on transitions

and allows us to find a system of transversal sections with some expansion proper-ties. We need to introduce more notations. Given a box B and εx > 0, denote byB(x, εx) ⊂ B a sub-box such that π(B(x, εx)) = (π(x) − εx, π(x) + εx).

Lemma 3.6. Given K > 2, there are ε∗ > 0, ε > 0 and a system of transversal

sections Γ(E), with max(E) < ε∗, such that: for any band B of Γ(E) one of the

following happens:

1. there is a sub-band B ⊂ B, a point x ∈ B such that ω(x) ∩ S 6= ∅ and t > 0,

such that Gtx : B → B(Φt(x), ε) is continuous and πs ◦G|B covers the interval

πs(B(Φt(x), ε));2. there is y ∈ B and t > 0 such that Φt(y) ∈ Qj, for some j ∈ {1, . . . , k}; and

the transition G :B−→Σj is continuous and |G(B)|u > K|B|u.

Proof. Take K > 2 and set T0 = | lnK/ lnλu|. For such T0, consider ε∗ > 0 givenin Lemma 3.5 in a way that: if max(E) < ε∗, for any point x in any system oftransversal sections of size < ε∗, then t(x) > T0. Consider Σ(E′) a system oftransversal sections such that max(E′) < ε∗. Fix now ε = min(E′)/4 and consider


another system of transversal sections Γ(E) ⊂ Σ(E′), i.e., Σj ⊂ Σ′j for any j, such

that both connected components of πs(Σ′j − Σj) > ε.

Take any B ⊂ Γ(E). Let us assume first that:

B ∩ (∪σ∈SW s(σ)) = ∅. (4)

Lemma 3.5 implies that there is a point x ∈ B such that ω(x) ∩ S 6= ∅. Let εx > 0such that B(x, εx) ⊂ B. Fix t > 0 such that λt

u(2εx) > ε and that xt ∈ Vδ. Considerthe corresponding transition map: G = Gt

x : D(x, t) → Nxt. Recall that B(x, εx) is

a box in Nx. According to Proposition 2, there are three options; however, item 3cannot hold because we are assuming (4). If B(x, εx) ⊂ D(x, t), then Proposition 1implies that

|G(B(x, εx))|u > λtu(2εx) > ε;

and hence πs ◦ G is onto on the interval πs(B(xt, ε)). In such a case, the band

B = B(x, εx) is the one that satisfies item 2 of the statement of the Lemma. Now,

if D(x, t) is properly contained in B(x, εx), then there is a sub-band B ⊂ B(x, εx)for which the function

πs ◦ G :B−→πs(B(xt, ε))

is onto, and so it holds item 1 of the Lemma.If condition (4) does not hold, it implies that the set R introduced in Remark 2

is not empty; and hence, there are y ∈ B and ty > 0 such that (πs(y), ty) is the firstelement of R. For x ∈ B, let τy(x) > 0 be the number such that Φτy(x)(x) ∈ Γ(E),then

int{Φs(x)|x ∈ B, s ∈ (0, τy(x))} ∩ Q = ∅. (5)

Otherwise, we find a pair (y′, ty′) < (y, ty), which is a contradiction. Hence, considerthe corresponding transition map to the bigger section G : D(y, t) ⊂ B −→ Σ′

j ,for which G(y) = Φty

(y) ∈ Qj . Notice that y and Φty(y) belong to Vδ; hence

Proposition 2 yields the following three options.If item 1 holds, then it implies that B ⊂ D(y, t) and G(B) ⊂ Σ′

j. Then ty >

T0, since y ∈ Γ(E) and Φty(y) ∈ Γ(E). In this case, Proposition 1 implies that

|G(B)|u > K|B|u; which is the statement of the second item of the present Lemma.Observe that (5) implies that item 3 of Proposition 2 cannot hold. Therefore, the

only possibility left is item 2; that is, there is a sub-band B0 ⊂ B, y ∈ B0, whereB0 ⊂ D(y, t) and for which G(B0) covers the corresponding Σ′

j , i.e., πs(G(B0)) =

πs(Σ′j). Notice that in this case, for any point x ∈ G(B0)∩Γ(E), we have B(x, ε) ⊂

Σj , since both connected components of πs(Σ′j − Σj) have length greater than ε,

for any j.Consider x that ω(x) ∩ S 6= ∅, given in Lemma 3.5. Then, there is t > 0,

x = Φ−t(x), and a sub-band

x ∈ B := G−1(B(x, ε) ∩ G(B0)) ⊂ B0 ⊂ B

such that the transition map Gtx : B −→ B(x, ε) is as we claim on item 1 of the

present Lemma, and we are done.

3.2. Proof of main lemma: Fix K > 2. Consider ε > 0 and Γ(E) the systemof transversal sections given in Lemma 3.6. Take any band B of Γ(E). Recallthat there is T0 > 0 such that for any x ∈ Γ(E), the time to return to Γ(E) ist(x) ≥ T0. Let µ = exp(T0λu) = exp(K) > 2. Hence, Corollary 2 implies that||DyGt

x|Euy|| ≥ C′µ, for any y ∈ D(x, t) and any t > t(x).


On the other hand, we have to be careful to avoid splittings of the box by thestable manifold of singularities. In order to do that, we will focus on both possiblesituations after Lemma 3.6.

Case (1). There is a sub-band B0 ⊂ B, a point x ∈ B0 that ω(x) ∩ S 6= ∅ andt > 0 such that G1 := Gt

x : B0 → B(Φt(x), ε) is continuous and πs ◦G|B covers theinterval πs(B(Φt(x), ε)). Denote by x1 := Φt(x). Let t1 > 0 be the first time thepositive orbit of x1 returns to Γ(E). It holds that t1 < +∞, since ω(x) ∩ S 6= ∅.Fix j such that Φt1(x1) ∈ Σj . Consider:

G2 = Gt1x1

: D(x1, t1) ⊂ B(x1, ε) → Σj .

Observe that {x1, Φt1(x1)} ⊂ Vδ; so we can use Proposition 2 in the followingmanner.

If item 2 holds, then there is B1 ⊂ B(x1, ε) ∩ D(x1, t1), x1 ∈ B1, such that

πs(G2(B1)) = πs(Σj). Setting B = G−11 (B1) ⊂ B0, we have that Gt+t1

x = G2 ◦G1 :

B → Σj covers Σj , and we are done.If item 3 of Proposition 2 holds, then there is y ∈ B(x1, ε)∩D(x1, t1) and ty > 0

such that Φty(y) ∈ Qj′ , for some j′ ∈ {1, · · · , k}. Consider the pair (y, ty) the first

element of the associated set R (see Remark 2). Two different situations appearhere, depending on the section Σj′ where y returns.

Assume j = j′. Denote by G2 = Gt1x1

: D(x1, t1) → Σj . Notice that the imageof G2 intersects Qj . Let B1 be the box defined as the connected component ofB(x1, ε) − W s

loc(y) containing x1. Notice B1 ⊂ D(x1, t1), and hence, Proposition2 implies that G2(B1) covers all Σj . In fact, no other point in B1 hits Q before

(y, ty). Therefore, the box B = G−11 (B1) ⊂ B0 and the transition Gt+t1

x = G2 ◦ G1

covers the side of Σj that contains Φt1(x1).Nevertheless, if j 6= j′ then a different argument is needed. Denote again by B1

the box defined as the connected component of B(x1, ε) − W sloc(y) containing x1.

However, consider the transition associated to y:

G3 = Gty

y : D(y, ty) ∩ B(x, ε) → Σj′ .

It is important to recall that y ∈ G1(B0). Then, there is s > 0 such that y0 =Φ−s(y) ∈ B0, and hence, Gt

x ≡ Gsy0

in D(y0, s) ∩ D(x, t). On the other hand,x1 /∈ D(y, ty), because the first return to Γ(E) of x1 is Φt1(x1) ∈ Σj 6= Σj′ .

Therefore, there is a box B1 ⊂ D(y, ty), y ∈ B1, such that G3(B1) covers oneside of Σj′ . In fact, no point of D(y, ty) can hit Q before (y, ty), hence Proposition 2

implies that G2(B1) covers one side of Σj′ . In this case, the band B = (Gsy0

)−1(B1)

and the transition G = Gs+ty

y0= G3 ◦ G1 satisfy the Lemma.

If item 1 of Proposition 2 holds then B(x1, ε) ⊂ D(x1, t1). Therefore,

G2(B(x1, ε)) ∩ Qj 6= ∅.

This is true because:

|G2(B(x1, ε))|u ≥ λln Kln λuu ε ≥ Kε ≥

min(E)

2.

Here we are in the same situation as that where item 3 of Proposition 2 holds, sowe are done for Case (1).

Case (2). There is a point y ∈ B and t > 0 such that Φt(y) ∈ Qj , for somej ∈ {1, . . . , k}, and the corresponding transition G0 : B → Σj is continuous in B

and |G0(B)|u > K|B|u. If G0(B) do not cover one side of Σj(E) then at least


one connected component of G0(B) − W sloc(Φt(y)) has its unstable length greater

than ν|B|u, where ν = K/2 > 1. Then, there is a sub-band B0 ⊂ B such that

G0(B0) is the mentioned connected component. Now choose B1 ⊂ Σj(E) such that

πs(B1) = πs(G0(B0)). Of course |B1|u > ν|B|u. We can repeat this argument forseveral steps while in each step the band Bn does not fit into Case (1). If n0 ∈ N

is such that νn0 ≥ max(E), then we can conclude that there is a transition

G = Gn0◦ · · · ◦ G0 :Bn0

−→Σj′ (E),

for some j′ ∈ {1, . . . , k}, and such that:

|G(Bn0)|u > ν|Bn0−1|u = ν2|Bn0−2|u = νn0 |B|u > max(E).

Therefore, G(Bn0) covers one side of Σj′ (E), as we require. Nevertheless, if in some

step the band Bn fits into Case (1), then the first part of the proof gives the MainLemma.

4. Proof of main theorem. Let Λ be a singular-hyperbolic transitive attractor.Consider Γ(E) the system of transversal sections obtained in the Main Lemma, andlet A be the Markov collection obtained in Corollary 1.

It is necessary to split the set Λ into two disjoint sets, according to the α-limit:Set Λ = ΛS ∪ΛH , where ΛH = {x ∈ Λ|α(x)∩S = ∅} and ΛS = Λ−ΛH . Of course,ΛS 6= ∅; otherwise Λ would be uniform hyperbolic.

Lemma 4.1. Consider a band B ∈ A and let x ∈ GB(B) ∩ ΛS. There is B ∈ Asuch that x ∈ GB(B) and G−1

B(x) ∈ GA(A) for some A ∈ A.

Proof. Denote by t1 > 0 the number such that Φ−t1(x) = G−1B (x) =: x1. Of course,

x1 ∈ B. Consider the sequence of all returns of x1 to Γ(E), for the past; that is,an increasing sequence of real numbers {tk > 0|k ≥ 2} such that xk := Φ−tk

(x1) ∈Γ(E). In order to fix notation, assume x1 ∈ Σ+

0 , and B is a band in Σ+0 .

Fix some 0 < γ < min(E), and recall the constant C > 0 in Proposition 1. Thereis k∗ ≥ 2 that if k ≥ k∗ then:

C exp(−tkλu)(min(E) − γ) > max(E).

In fact, this inequality holds because the sequence {−tk} is strictly increasing.Moreover, there is k0 > k∗ such that d(xk0

, Q) < γ, since α(x) contains somesingular points. Assume without lose of generality that xk0

∈ Σ+1 . It may happen

that Σ1 = Σ0.

Consider then the transition T = Gtk0xk0

: D ⊂ Σ+1 → Σ0, with respect to Σ+

1 andΣ0. According to Proposition 2, we have three cases:

Case 1. If the maximal B ⊂ D such that T (B) ⊂ Σ0 is precisely Σ+1 , then Proposi-

tion 1 implies that:

|T (D)|u ≥ C exp(−λutk0)|D|u = C exp(−λutk0

)ε+0 .

However,

C exp(−λutk0)min(E) > C exp(−λutk0

)(min(E) − γ) > max(E);

and hence, |T (D)|u > max(E). Therefore T (D) covers Σ+0 (where x1 lives). Hence

D = Σ+1 ∈ A, GD ≡ T , and we are done.


Case 2. If there is a maximal band B ⊂ D such that T (B) covers Σ+0 , then this is

exactly what we wanted to prove.

Case 3. If there is y ∈ ∂D such that y ∈ W s(σ0), we can denote the two connectedcomponents of ∂D by l and l′, in a way that y ∈ l′. Notice then that T (l′) ⊂ Q0

and that T (l) ⊂ ∂Σ+0 −Q0, whenever l ∩ ∂Σ+

1 = ∅. Hence T (D) covers Σ+0 ; that is,

D ∈ A, and we are done. However, if l ∩ ∂Σ+1 6= ∅, then |D|u > min(E) − γ, since

d(xk0, Q1) < γ and xk0

∈ D. Hence, Proposition 1 implies that:

|T (D)|u > C exp(−λutk0)(min(E) − γ) > max(E);

and hence, T (D) covers Σ+0 .

Proof of Main Theorem: First we deal with points in ΛS. Recall the parameters inthe definition of Γ(E):

Γ(E) = {Σj(ε−j , ε+

j ) = Σ−j ∪ Qj ∪ Σ+

j | j ∈ {1, . . . , k}}.

Let µ±j = sup{d(x, Qj)|x ∈ ΛS ∩ Σj(ε

−j , ε+

j )} ≥ 0. Observe µ+j > 0 if and only

if Λ ∩ Σ+j 6= ∅, and the same for µ−

j . By definition µ±j ≤ ε±j . Denote by µ :=

13min{µ±

j , ηδ|µ±j > 0}. The fact that µ < ηδ, the size of transversal sections defined

in §2, will be important when considering points in ΛH . On each Σj of Γ(E) considerthe subset

Σ∗j := Σj(ε

−j + µ, ε+

j − µ) − Cl(Σj(−µ, µ)),

and denote by Γ∗(E) = {Σ∗j | j ∈ {1, . . . , k}}. Notice that for any point x ∈ Γ∗(E),

the band B(x, µ) ⊂ Σj = Σj − Qj .On the other hand, for any point x ∈ Γ(E) ∩ ΛS there is a band Bx ∈ A and

a transition G : Bx → Γ(E) such that G(Bx) covers the corresponding connected

component of Γ(E). Notice that G = Gtxx for some tx > 0.

Let Υ := ∪Gtxx (Bx) ⊂ Γ(E), where the union ranges on all x ∈ Γ(E) ∩ ΛS. Of

course, Γ(E)∩ΛS ⊂ Υ. Notice also that Υ∩Γ∗(E) 6= ∅. Let x ∈ KS := Υ∩Γ∗(E).Denote Jx = W cu

µ (x)∩Γ(E). Since x ∈ Υ, there is t1 > 0 and a band B1 = BΦ−t1(x)

such that the corresponding transition G1 covers the connected component of Γ(E)that contains x. Hence Jx ⊂ G(B1), which is to say that G−1

1 (Jx) ⊂ B1. If wedenote by x1 = Φ−t1(x) then G1 = Gt1

x1. Notice that x1 ∈ Υ, by Lemma 4.1. Now

we proceed by induction.Since xn−1 ∈ Υ, there is tn > 0 and a band Bn ∈ A such that the corresponding

transition Gn covers the connected component of Γ(E) that contains xn−1. Hence,G−1

n (G−1n−1 ◦ · · · ◦ G−1

1 (Jx)) ⊂ Bn. Therefore, Jx ⊂ Dom(G−tx ) for all t > 0; then

Jx = Wuµ (x) for any x ∈ ΛS ∩ KS . If we set sn =

∑nj=1 tj , then Proposition 1

implies that:

|G−sn

x (Jx)|u < exp(λusn)µ.

This proves the statement of item 1 and 2 of Main Theorem, for εu = µ and pointsin ΛS ∩ KS .

Now let us consider points in ΛH . Let y ∈ KH = {x ∈ Λ| O−(x) ∩ Vδ = ∅}.Denote by Jy = W cu

µ (x) ∩ Ny; then it is easy to proof that Jy = Wuµ (y), since

O−(y) ∩ Vδ = ∅. Also, Proposition 1 implies that |G−ty (Jy)|u < exp(λut)µ.

If we set εu := µ and K := KS ∪ KH , then items 1 and 2 are proved.


Item 3 of the Main Theorem claims that for any t0 > 0 the set Kt0 = ∪t>t0Φ−t(K)contains an open and dense in Λ. To get this claim, let z ∈ Λ and U ⊂ M , a 3-dimensional neighborhood of z. We need to show that there is t > t0 and x ∈ Ksuch that Φ−t(x) ∈ U .

Assume z ∈ ΛS . There is tz > 0 such that z := Φ−tz(z) ∈ Γ(E). Let Un =

B1/n(z) ∩ Γ(E), for n ≥ N such that Φtz(B1/N (z)) ⊂ U . Using the projection πs,

the set Un defines a band Bn = π−1s (πs(Un)) ⊂ Γ(E). The Main Lemma implies

there is a sub-band Bn ⊂ Bn, z ∈ Bn and a transition Gn such that Gn(Bn) coversone side of one Σj − Qj , say Σ+

j . Moreover, we have that πs(Gn(Un)) = πs(Σ+j ).

Hence, there is a point x ∈ Gn(Un) ∩ KS , and Gn = GtΦ−t(x), for some t > 0. If n

is large enough, then t − tz > t0, and we are done.Now assume that z ∈ ΛH . If O(x) ∩ Γ(E) = ∅, then Lemma 2.3 implies that

O(x) ⊂ KH , and we are done. However, if there is t ∈ R such that Φt(z) ∈ Γ(E),then we can apply the previous argument to obtain a point x ∈ KS ∩ΛS and t > t0such that Φ−t(x) ∈ U , and we are done.

5. Dynamical properties of singular-hyperbolic attractors. The existenceof unstable manifolds of uniform size (Main Theorem) allow us to portrait thedynamics inside any singular-hyperbolic transitive attractor. This is the content ofTheorem A and Theorem B. In this section we shall give a proof for these theorems.

Let Λ be a transitive singular-hyperbolic attractor for some flow Φt, and denoteby xt := Φt(x) for any point x ∈ M and t ∈ R. Let U be an open neighborhood ofΛ contained in its basin of attraction.

Proof of Theorem A: The Main Theorem states the existence of a subset K ⊂ Uand two constants εu > 0 and λu < 0. In particular, for any T > 0 the setKT = ∪t>T Φ−t(K) is a residual set. Moreover, the set K = ∩T>0KT is an openand dense set. On the other hand, the set D ⊂ Λ, the set of points in Λ with denseorbit, is also residual. Hence, there is some z ∈ D ∩ K.

Let x ∈ Λ. We need to prove that it is accumulated by periodic orbits. Wecan assume that x /∈ S; in fact, any σ ∈ S is accumulated by regular orbits ofΛ. Let {ji > 0| i ∈ N}, be an increasing sequence of real numbers such that

limi→+∞ zji= x; recall z ∈ D. On the other hand, z ∈ K, hence there is another

increasing sequence {ri} such that zri∈ K. We can assume that ji ≤ ri ≤ ji+1, for

all i ∈ N, and that {zri} is a Cauchy sequence on M . Take a positive integer n0

such that d(zi, zj) < 13 min{εs, εu}, for any i, j ≥ n0.

Fix i > n0, and denote by Z := zri. Now, for any n > i we have that

Φ−tn(Φrn

(z)) = Z, where tn = rn − ri. Hence, as a consequence of the MainTheorem we know that:

Wuεu

(zrn) = W cu

εu(zrn

) ∩ Nzrn.

On the other hand, there is a local transversal section Σ, foliated by local stablemanifolds, containing zrn

and Z, and which is C1-close to Nzrn. Notice that J :=

W cuε (zrn

) ∩ Σ is an interval, and the family of holonomy maps associated to Σbehaves the same as the one for Nzrn

. Now let:

J = G−tn

zrn(W cu

εu(zrn

)) ∩ Σ.

Finally, choose n1 > n0, such that for any n > n1, it verifies that:

exp(tnλu) = exp((rn − ri)λu) < εu/4.


This is possible because the sequence tn is not bounded and λu < 0; recall i is fixed.Consider now the 2-dimensional transversal section made of local stable manifolds

of points of J : B =⋃

y∈J W sεs

(y) ⊂ Σ. If n > n1, then J ⊂ B. This allow us to

define a function f : J −→ J by y 7→ πs(G−tnzrn

(y)) where πs denotes the projection

along stable leaves. Since f is continuous then there exists y ∈ J such that f(y) = y.This means that Φτ (y) ∈ W s

loc(y), for some τ ∈ R. This implies the existence of aperiodic point P ∈ B such that y and yτ both belong to W s

loc(P ); see Lemma 4.3of [2].

Hence, for any i > n0, and n > n1, we have constructed a periodic point P in.

The sequence of periodic points P in converges to zri

, as n → ∞. On the other hand,

Φji−ri(P i

n) →n→∞

Φji(Φ−ri

(zi)) = Φji(z) →

i→∞x,

and we are done.

Proof of Theorem B: Take some point x ∈ Λ such that their α and ω-limits aredense. As in the proof of Theorem A, we can assume that x ∈ K, of the MainTheorem, and the set of numbers r > 0 such that xr ∈ K is unbounded. Then,it is enough to prove that there is some periodic point p ∈ Λ such that, for anyri > 0 such that xri

∈ K, it is accumulated by points in the homoclinic class ofp. For that, consider an accumulation point of xri

. Since points in K have largelocal unstable manifold, they intersect with the corresponding stable ones in somesmall neighborhood of the accumulation point. Hence, arguing as in the proof ofTheorem A, we get the periodic point p.

Moreover we obtain that:

Wuεu

(Φri(x)) ∩ W s

loc(p) = {q},

Wuεu

(p) ∩ W sloc(Φri

(x)) = {q}.

Since α(x) is dense, we get that there is a sequence jk ∈ R+ such that Φ−jk

(x) →xri

, as k → ∞. Using that xri∈ K, and hence that it has large unstable manifold,

it implies that there are segments γsk contained in the local stable manifold of p such

that Φ−jk−ri(γsk) accumulate in the local stable manifold of xri

. Hence, there arepoints of intersection of the stable manifold of p and the local unstable manifold ofp, say qk, accumulating on q.

Again, using that there is nk → +∞ such that Φnk(x) accumulates on xri

, itfollows that Φnk−ri

(q) accumulates on xriand so there is a sequence of points in

the homoclinic class of p converging to xri, as we wanted.

Acknowledgements. We would like to thank the referees, whose comments im-proved the presentation of our results. The first author was partially supported byCONACYT-Mexico.

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Received August 2005; 1st revision July 2006; 2nd revision April 2007.

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Dynamical properties of singular-hyperbolic attractors

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