# SRB measures of singular hyperbolic attractors

<EMAIL_ADDRESS>
###### Abstract.

It is known that hyperbolic maps admitting singularities have at most
countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. These maps include
the Belykh attractor, the geometric Lorenz attractor, and more general Lorenz-
type systems. In this paper, we establish easily verifiable sufficient
conditions guaranteeing that the number of ergodic SRB measures is at most
finite, and provide examples and nonexamples showing that the conditions are
necessary in general.

###### Key words and phrases:

Singular hyperbolic maps, SRB measures, strange attractors, hyperbolic sets,
physical measures.

###### 1991 Mathematics Subject Classification:

Primary: 37C05, 37C40, 37D05, 37D20, 37D45; Secondary: 37C10, 37C70, 37D35.

Dominic Veconi∗

Department of Mathematics

The Pennsylvania State University

University Park, PA 16802, USA

(Communicated by the associate editor name)

##  1\. Introduction

One primary question in smooth ergodic theory is the existence of “physical
measures” for a smooth dynamical system. Given a compact Riemannian manifold
$M$ and a smooth map $f:U\to M$, $U\subseteq M$ open, a _physical measure_ is
one in which the Birkhoff averages of continuous functions are constant on a
set of positive measure. In other words, a probability measure $\mu$ is a
_physical measure_ if

$m\bigg{\\{}x\in
U:\lim_{n\to+\infty}\frac{1}{n}\sum_{k=0}^{n-1}\left(\varphi\circ
f^{k}\right)(x)=\int_{U}\varphi\,d\mu\quad\forall\varphi\in
C^{0}(U)\bigg{\\}}>0,$

where $m$ is the Riemannian volume. Among the most significant physical
measures are the _Sinai-Ruelle-Bowen (SRB) measures_. These are invariant
measures for hyperbolic dynamical systems that have conditional measures on
unstable leaves that are absolutely continuous with respect to the Riemannian
leaf volume. For uniformly hyperbolic dynamical systems (such as transitive
Anosov diffeomorphisms and attractors of Axiom A systems), there is a unique
SRB measure [9], and the existence of SRB measures has been established for
several classes of nonuniformly hyperbolic dynamical systems [10, 14, 18] and
partially hyperbolic dynamical systems [2, 8]. It was further shown in [15]
that if $M$ is a compact Riemannian 2-manifold and $f:M\to M$ is a hyperbolic
diffeomorphism admitting an SRB measure, then this SRB measure is unique.

Many dynamical systems in engineering and natural sciences exhibit “chaotic”
behavior: their trajectories appear disordered and they are highly sensitive
to initial data. The simplest mathematical examples of such systems are
uniformly hyperbolic and uniformly expanding, and so hyperbolic dynamical
systems have been at the forefront of smooth ergodic theory since at least the
1960s. However, most stochastic dynamical systems arising from physical and
natural phenomena are not uniformly hyperbolic. In these instances, uniqueness
results for uniformly hyperbolic dynamical systems and surface maps (such as
those described in [15]) no longer apply. Examples of such dynamical systems
include the Lorenz attractor model of atmospheric convection, the associated
geometric Lorenz attractor (described in more detail below), and the Belykh
attractor of phase synchronization theory [5, 13, 17]. The latter two are maps
of the unit square that admit highly complex limit sets, so that the resulting
maps are not invariant under Lebesgue volume and may not _a priori_ admit a
unique SRB measure.

Although our results concern discrete singular hyperbolic maps, historically
many results about singular hyperbolic attractors come from investigations
into hyperbolic flows, the most famous being the flow generated by the Lorenz
equations. In [11], J. Kaplan and J. Yorke used a Poincaré return map to study
the dynamical behavior of the Lorenz attractor, such as the parameters for
which periodic points are dense. This Poincaré map was later reformulated as
the _geometric Lorenz attractor_ , which is a simplified discrete model of the
Poincaré map of the original Lorenz flow. The more general family of discrete
_Lorenz-type maps_ was introduced in [1]. In the years that followed, the
Lorenz system and related hyperbolic flows have led to active research in
singular hyperbolic attractors (see e.g. [1, 5, 13, 16, 17], and others).
There is also a large body of work on singular hyperbolic and sectional-
hyperbolic flows more generally. In [17], it is shown that singular hyperbolic
flows admit finitely many ergodic physical measures; more recently, it was
shown in [3] that flows of Hölder-$C^{1}$ vector fields admitting a sectional-
hyperbolic attracting set admit finitely many ergodic SRB measures. The proof
in [3] also relies on Poincaré return maps, and so these results extend to
discrete singular hyperbolic maps arising as Poincaré maps of hyperbolic
flows. For a detailed discussion of the ergodic properties of hyperbolic flows
and their attractors, see [4].

In this paper, we consider the class of discrete singular hyperbolic dynamical
systems. These are hyperbolic maps $f:K\setminus N\to K$, where $K\subset M$
is a precompact open subset of a Riemannian manifold $M$, and $N\subset K$ is
a closed subset of singularities on which $f$ fails to be continuous and/or
differentiable. The map $f$ is uniformly hyperbolic on the non-invariant set
$K\setminus N$, but behaves more similarly to the non-uniformly hyperbolic
setting on an invariant set that consists of trajectories passing nearby the
the singular set $N$ with a prescribed rate. Our setting includes systems that
are derived from Poincaré maps of hyperbolic flows, such as the geometric
Lorenz attractor, but also includes singular hyperbolic dynamical systems that
do not arise from flows, such as the Lozi map [12]. In [13], it was shown that
the attractors admitted by singular hyperbolic maps support at most countably
many ergodic SRB measures. In [16], conditions were given under which a
singular hyperbolic attractor admits at most finitely many ergodic SRB
measures. We provide an alternative proof of this result, with somewhat
different conditions that are easy to verify. Namely, if the singular set is a
disjoint union of finitely many embedded submanifolds that transversally
intersect unstable leaves, and if the image of neighborhoods of the singular
set remain separated from the singular set under the dynamics for sufficiently
long but finite time (conditions (SH3), (SH6), and (SH7)), then there are at
most finitely SRB measures.

Although most examples of singular hyperbolic attractors in the literature
satisfy conditions (SH3) - (SH7) (see the examples in [1, 5, 7, 12, 13]),
there exist singular hyperbolic attractors that do not satisfy these
conditions and admit infinitely many ergodic components. For this reason,
these conditions are necessary for our statement to be true in full
generality. For example, one can construct a family of Lorenz-type attractors
whose singular sets have infinitely many components, and these examples admit
countably many SRB measures, but these maps are not topologically transitive
(see Section 4.3).

Once the existence of ergodic SRB measures has been established, a natural
question to ask is when the SRB measure is unique. In [15], it is shown that
in the case of a $C^{1+\alpha}$ diffeomorphism $f$ of a compact surface, if
$f$ is topologically transitive, then $f$ admits at most one SRB measure. A
similar result may be proven for singular hyperbolic attractors, provided a
certain regularity condition on the stable foliation (Theorem 3.5, also [13]).
The regularity condition needed is _local continuity_ , which roughly means
that the smooth functions $E^{s}(x)\to M$ defining the local stable leaf
$W_{\mathrm{loc}}^{s}(x)$ vary continuously with $x\in K\setminus N$, where
$E^{s}(x)\subset T_{x}M$ is the stable subspace at $x$. In this paper, we show
that a singular hyperbolic map for which the stable foliation is locally
continuous admits a unique SRB measure if and only if the map is topologically
transitive.

This paper is structured as follows. Section 2 is devoted to preliminary
constructions and definitions needed to discuss singular hyperbolic dynamical
systems. Our main result is stated and proven in Section 3. Section 4 is spent
discussing examples of dynamical systems satisfying the hypotheses of our main
result, as well as examples of systems that fail these hypotheses and that
admit infinitey many SRB measures.

##  2\. Preliminaries

We begin by defining singular hyperbolic attractors, and discuss some of their
major properties. We consider a Riemannian manifold $M$, an open, bounded,
connected subset $K\subset M$ with compact closure, and a closed subset
$N\subset K$. We further consider a map $f:K\setminus N\to K$ satisfying:

  1. (SH1)

$f$ is a $C^{2}$ diffeomorphism from $K\setminus N$ to $f(K\setminus N)$.

We further define $N^{+}:=N\cup\partial K$ as the _discontinuity set_ for $f$
(on which the function $f$ is discontinuous), and further define

$N^{-}=\left\\{y\in K:\textrm{There are }z\in N^{+}\textrm{ and }z_{n}\in
K\setminus N^{+}\textrm{ s.t. }z_{n}\to z\textrm{ and }f(z_{n})\to
y\right\\}.$

The set $N^{-}$ is referred to as the _discontinuity set_ for $f^{-1}$. We
further assume the map $f$ satisfies:

  1. (SH2)

There exist $C_{i}>0$ and $\alpha_{i}\geq 0$, with $i=1,2$, such that

$\displaystyle\left\lVert d^{2}f_{x}\right\rVert$ $\displaystyle\leq
C_{1}\rho(x,N^{+})^{-\alpha_{1}}\quad\textrm{for }x\in K\setminus N,$
$\displaystyle\left\lVert d^{2}f_{x}^{-1}\right\rVert$ $\displaystyle\leq
C_{2}\rho(x,N^{-})^{-\alpha_{2}}\quad\textrm{for }x\in f(K\setminus N)$

where $\rho$ is the Riemannian distance in $M$.

Define the set $K^{+}$ by

$\displaystyle K^{+}=\mathop{\bigcap}_{n=0}^{\infty}\left(K\setminus
f^{-n}(N^{+})\right)=\left\\{x\in K:f^{n}(x)\not\in N^{+}\textrm{ for all
}n\geq 0\right\\},$

so that $K^{+}$ is the largest forward-invariant set on which $f$ is
continuous. Further, define

$D=\mathop{\bigcap}_{n=0}^{\infty}f^{n}(K^{+})\quad\textrm{and}\quad\Lambda=\overline{D}.$

We say $\Lambda$ is the _attractor_ for $f$.

######  Proposition 1.

[13] We have
$D=\Lambda\setminus\mathop{\bigcup}_{n\in\mathbb{Z}}f^{n}(N^{+})$.
Furthermore, $f$ and $f^{-1}$ are well-defined on $D$, and $f(D)=D$ and
$f^{-1}(D)=D$.

Given $z\in M$, $\alpha>0$, and a subspace $P\subset T_{z}M$, we denote the
cone at $z$ around $P$ with angle $\alpha$ by

$C(z,\alpha,P)=\left\\{v\in T_{z}M:\angle(v,P):=\inf_{w\in
P}\angle(v,w)\leq\alpha\right\\}.$

######  Definition 2.1.

The map $f:K\setminus N\to K$ is _singular hyperbolic_ if there is $C>0$,
$\lambda>1$, a function $\alpha:D\to\mathbb{R}$, and two distributions
$P^{s},P^{u}$ on $K\setminus N^{+}$ of dimensions $\dim P^{s}=p$, $\dim
P^{u}=q=n-p$ (with $n=\dim M$), such that the cones
$C^{s}(z)=C(z,\alpha(z),P^{s}_{z})$ and $C^{u}(z)=C(z,\alpha(z),P^{u}_{z})$,
for $z\in K\setminus N$, satisfy the following conditions:

  1. (a)

The angle between $C^{s}(z)$ and $C^{u}(z)$ is uniformly bounded below over
$K\setminus N^{+}$, and in particular, $C^{s}(z)\cap C^{u}(z)=0$;

  2. (b)

$df_{z}(C^{u}(z))\subset C^{u}(f(z))$ for $z\in K\setminus N^{+}$, and
$df^{-1}_{z}(C^{s}(z))\subset C^{s}(f^{-1}(z))$ for $z\in f(K\setminus
N^{+})$;

  3. (c)

for any $n>0$, we have:

$\displaystyle|df_{z}^{n}v|$ $\displaystyle\geq
C\lambda^{n}|v|\quad\textrm{for }z\in K^{+},v\in C^{u}(z);$
$\displaystyle|df_{z}^{-n}|$ $\displaystyle\geq
C\lambda^{n}|v|\quad\textrm{for }z\in f^{n}(K^{+}),v\in C^{s}(z).$

In this instance, the set $\Lambda$ defined above is called a _singular
hyperbolic attractor_.

Define the following subsets of $T_{z}M$ for $z\in D$:

$E^{s}_{z}=\mathop{\bigcap}_{n=0}^{\infty}df^{-n}_{f^{n}(z)}C^{s}(f^{n}(z))\quad\textrm{and}\quad
E^{u}_{z}=\mathop{\bigcap}_{n=0}^{\infty}df^{n}_{f^{-n}(z))}C^{u}(f^{-n}(z)).$

######  Proposition 2.

[13] The sets $E^{s}_{z}$ and $E^{u}_{z}$ are subspaces of $T_{z}M$, called
the _stable_ and _unstable subspaces_ at $z$ respectively. They satisfy the
following properties:

  1. (a)

the dimensions of these subspaces are the same as the respective subspaces
$P^{s}_{z}$ and $P^{u}_{z}$ around which the cones $C^{s}(z)$ and $C^{u}(z)$
are centered. That is, $\dim E^{s}_{z}=\dim P^{s}_{z}=p$ and $\dim
E^{u}_{z}=\dim P^{u}_{z}=q=n-p$;

  2. (b)

$T_{z}M=E^{s}_{z}\oplus E^{u}_{z}$;

  3. (c)

the angle between $E^{s}_{z}$ and $E^{u}_{z}$ is bounded below uniformly over
$D$;

  4. (d)

for any $n\geq 0$ and $z\in D$, we have

$\displaystyle|df^{n}_{z}v|$ $\displaystyle\leq
C\lambda^{-n}|v|\quad\textrm{for }v\in E^{s}(z),$
$\displaystyle|df^{-n}_{z}v|$ $\displaystyle\leq
C\lambda^{-n}|v|\quad\textrm{for }v\in E^{u}(z).$

The distributions $E^{s}$ and $E^{u}$ on $D$ thus form uniformly hyperbolic
structure with singularities. In particular, they are the tangent spaces of
stable and unstable foliations on $D$. To rigorously characterize the leaves
of these foliations, we need to define the subsets on which stable and
unstable manifolds may be defined.

For arbitrary $\varepsilon>0$ and $l\in\mathbb{N}$, we denote:

$\displaystyle\widehat{D}_{\varepsilon,l}^{+}$ $\displaystyle=\left\\{z\in
K^{+}:\rho\left(f^{n}(z),N^{+}\right)\geq l^{-1}e^{-\varepsilon n},\>n\geq
0\right\\};$ $\displaystyle D^{-}_{\varepsilon,l}$
$\displaystyle=\left\\{z\in\Lambda:\rho\left(f^{-n}(z),N^{-}\right)\geq
l^{-1}e^{-\varepsilon n},n\geq 0\right\\};$ $\displaystyle
D^{+}_{\varepsilon,l}$
$\displaystyle=\widehat{D}_{\varepsilon,l}^{+}\cap\Lambda;$ $\displaystyle
D^{0}_{\varepsilon,l}$ $\displaystyle=D^{-}_{\varepsilon,l}\cap
D^{+}_{\varepsilon,l};$ $\displaystyle D_{\varepsilon}^{\pm}$
$\displaystyle=\mathop{\bigcup}_{l\geq 1}D_{\varepsilon,l}^{\pm};$
$\displaystyle D_{\varepsilon}^{0}$ $\displaystyle=\mathop{\bigcup}_{l\geq
1}D_{\varepsilon,l}^{0}.$

We note that $\widehat{D}^{+}_{\varepsilon,l}$, $D^{\pm}_{\varepsilon,l}$, and
$D^{0}_{\varepsilon,l}$ are closed, and hence compact. Also observe that
$D^{0}_{\varepsilon}=D_{\varepsilon}^{+}\cap D_{\varepsilon}^{-}\subset D$ for
$\varepsilon>0$, and $D_{\varepsilon}^{0}$ is invariant under both $f$ and
$f^{-1}$. Further, $D_{\varepsilon}^{+}$ and $D_{\varepsilon}^{-}$ are
invariant under $f$ and under $f^{-1}$ respectively.

For the proof of the following proposition, see the discussion in Sections 1.5
and 2.1 of [13].

######  Proposition 3.

There exists $\varepsilon>0$ and such that:

  1. (a)

for $z\in D^{+}_{\varepsilon}$, there is an embedded (possibly disconnected)
submanifold $W_{\mathrm{loc}}^{s}(z)$ of dimension $p=\dim E^{s}_{z}$ for
which $T_{z}W^{s}_{\mathrm{loc}}(z)=E^{s}_{z}$;

  2. (b)

for $z\in D^{-}_{\varepsilon}$, there is an embedded (possibly disconnected)
submanifold $W_{\mathrm{loc}}^{u}(z)$ of dimension $q=n-p=\dim E^{u}_{z}$ for
which $T_{z}W_{\mathrm{loc}}^{u}(z)=E^{u}_{z}$.

Furthermore, define $B^{s}_{z}(y,r)$ to be the ball in
$W^{s}_{\mathrm{loc}}(z)$ of radius $r$ centered at $y\in
W_{\mathrm{loc}}^{s}(z)$, where the distance is the induced distance
$\rho^{s}$ on $W_{\mathrm{loc}}^{s}(z)$. Define $B^{u}_{z}(y,r)$ and
$\rho^{u}$ similarly. Then there is an $\alpha$ with $\lambda^{-1}<\alpha<1$
such that for $r>0$, there is a constant $C=C(r)$ such that:

  1. (c)

for $z\in D^{+}_{\varepsilon}$, $y\in W_{\mathrm{loc}}^{s}(z)$, $w\in
B^{s}_{z}(y,r)$, and $n\geq 0$, we have

$\rho^{s}(f^{n}(y),f^{n}(w))\leq C\alpha^{n}\rho^{s}(y,w);$

  2. (d)

for $z\in D^{-}_{\varepsilon}$, $y\in W_{\mathrm{loc}}^{u}(z)$, $w\in
B^{u}_{z}(y,r)$, and $n\leq 0$, we have

$\rho^{u}(f^{n}(y),f^{n}(w))\leq C\alpha^{n}\rho^{u}(y,w).$

Additionally, for $z\in D_{\varepsilon,l}^{-}$, let $B(z,\delta)$ denote the
ball of $\rho$-radius $\delta$ centered at $z$. Then there are
$\delta_{i}=\delta_{i}(z)>0$, $i=1,2,3$, with
$\delta_{1}>\delta_{2}>\delta_{3}$, so that for $w\in B(z,\delta_{3})$, the
intersection $B^{s}_{z}(z,\delta_{1})\cap W_{\mathrm{loc}}^{u}(w)$ is nonempty
and contains exactly one point, denoted $[w,z]$; and furthermore,
$B^{u}_{w}([w,z],\delta_{2})\subset W_{\mathrm{loc}}^{u}(w)$.

We denote

$W^{u}(x)=\mathop{\bigcup}_{n\geq
0}f^{n}\left(W_{\mathrm{loc}}^{u}(f^{-n}(x))\right)\quad\textrm{for }x\in K$
(1)

and

$W^{s}(x)=\mathop{\bigcup}_{n\geq
0}f^{-n}\left(W_{\mathrm{loc}}^{s}(f^{n}(x))\cap\Lambda\right)\quad\textrm{for
}x\in\Lambda.$ (2)

Given $\delta>0$ and $x\in K$, let $B_{T}^{u}(\delta,x)\subset E^{u}_{x}$
denote the open ball of radius $\delta$ in $E^{u}_{x}$. For $\delta$ less than
the injectivity radius of $M$ at $x$, suppose the connected component of
$\left(\exp_{x}\big{|}_{B_{T}^{u}(\delta,x)}\right)^{-1}(W^{u}(x))\subset
T_{x}M$ containing $0$ is the graph of some smooth function
$\psi:B_{T}^{u}(\delta,x)\to E^{s}_{x}$. If such a $\psi$ exists for a
particular $x\in M$ and $\delta>0$, we denote

$W_{\delta}^{u}(x)=\exp_{x}\left(\left\\{(u,\psi(u)):u\in
B_{T}^{u}(\delta,x)\right\\}\right).$

Such a number $\delta>0$ and such a function $\psi$ exist for each particular
$x\in K$ (in particular they form $W^{u}_{\mathrm{loc}}(x)$), but $\delta$ may
depend on $x$ (and in particular may not have a uniform lower bound). We
define $W^{s}_{\delta}(x)$ similarly.

Given local submanifolds $W_{\mathrm{loc}}^{s}(z_{1})$ and
$W_{\mathrm{loc}}^{s}(z_{2})$, we define the _holonomy map_
$\pi:W_{\mathrm{loc}}^{s}(z_{1})\to W_{\mathrm{loc}}^{s}(z_{2})$ to be
$\pi(w)=[w,z_{2}]=W_{\mathrm{loc}}^{u}(w)\cap W_{\mathrm{loc}}^{s}(z_{2})$.
Let $\nu_{z}^{s}=\nu|_{W_{\mathrm{loc}}^{s}(z)}$ and
$\nu_{z}^{u}=\nu|_{W_{\mathrm{loc}}^{u}(z)}$ denote the induced Riemannian
volumes on $W_{\mathrm{loc}}^{s}(z)$ and $W_{\mathrm{loc}}^{u}(z)$
respectively for $z\in D^{\pm}_{\varepsilon}$.

######  Proposition 4.

The local foliation $W_{\mathrm{loc}}^{s}(z)$ for $z\in D^{0}_{\varepsilon,l}$
is absolutely continuous, in the sense that for any $z_{1},z_{2}\in
D^{0}_{\varepsilon,l}$, the pushforward measure $\pi_{*}\nu_{z_{1}}^{s}$ on
$W_{\mathrm{loc}}^{s}(z_{2})$ is absolutely continuous with respect to
$\nu_{z_{2}}^{s}$.

###### Proof.

This follows from Proposition 10 of [13]. ∎

Generally, maps satisfying (SH1) and (SH2) are dissipative, and so do not
preserve Riemannian volume. Therefore, our interest is in the following class
of measures:

######  Definition 2.2.

A probability measure $\mu$ on $K$ is an _SRB (Sinai-Ruelle-Bowen) measure_ if
$\mu$ is $f$-invariant and if the conditional measures on the unstable leaves
are absolutely continuous with respect to the Riemannian leaf volume.

In [13], the existence of SRB measures for singular hyperbolic attractors is
proven under certain regulatory conditions. We will describe their
construction of SRB measures. Let $J^{u}(z)=\det\left(df|_{E^{u}_{z}}\right)$
denote the unstable Jacobian of $f$ at a point $z\in D$. For $y\in W^{u}(z)$
and $n\geq 1$, set

$\kappa_{n}(z,y)=\prod_{j=0}^{n-1}\frac{J^{u}\left(f^{-j}(z)\right)}{J^{u}\left(f^{-j}(y)\right)}.$

The functions $\kappa_{n}$ converge pointwise to a function $\kappa$ (see
[13], Proposition 6(1)). Fix $z\in D^{-}_{\varepsilon}$ and a sufficiently
small $r>0$, and set

$U_{0}:=B^{u}(z,r):=B^{u}_{z}(z,r),\quad\widetilde{U}_{n}:=f(U_{n-1}),\quad
U_{n}:=\widetilde{U}_{n}\setminus N^{+}.$

Further set

$\widetilde{C}_{0}=1\quad\textrm{and}\quad\widetilde{C}_{n}=\widetilde{C}_{n}(z)=\left(\prod_{k=0}^{n-1}J^{u}\left(f^{k}(z)\right)\right)^{-1}.$

For $n\geq 0$, define the measures
$\widetilde{\nu}_{n}=\widetilde{\nu}_{n}(z)$ on $U_{n}$ by

$d\widetilde{\nu}_{n}(y)=\widetilde{C}_{n}(z)\kappa\left(f^{n}(z),y\right)d\nu^{u}_{z}(y),$

and let $\nu_{n}$ be a measure on $\Lambda$ defined by
$\nu_{n}(A)=\widetilde{\nu}_{n}(A\cap U_{n})$ for any Borel
$A\subseteq\Lambda$. Under moderate assumptions, we have that
$\nu_{n}=f^{n}_{*}\nu_{0}$ (see [13], Proposition 8). Consider the sequence of
measures

$\mu_{n}=\frac{1}{n}\sum_{k=0}^{n-1}\nu_{k}.$

These measures admit a subsequence that converges in the weak topology to an
$f$-invariant SRB measure (_a priori_ depending on the reference point $z\in
D^{-}_{\varepsilon}$) on $\Lambda$, proving existence of SRB measures.

##  3\. Main result and proof

In this section, we give conditions under which a singular hyperbolic
attractor admits at most finitely many ergodic SRB measures, as well as
conditions under which the SRB measure is unique.

###  3.1. Main result

We begin by reviewing the assumptions we make on our map $f:K\setminus N\to
K$. Assumption (SH1) is the basic setting, and assumption (SH2) concerns the
regularity of the map. We now complete the assumptions of our setting. Below,
assumption (SH3) concerns the structure of the singular set $N$; assumptions
(SH4) - (SH6) concern the smoothness and hyperbolicity of $f$; assumption
(SH7) is a further regulatory assumption; and assumption (SH8) is a condition
on the regularity of the stable foliation.

  1. (SH3)

The singular set $N$ is the disjoint union of finitely many embedded
submanifolds $N_{i}$ with boundary, of dimension equal to the codimension of
the unstable foliation $W^{u}$.

  2. (SH4)

$f$ is continuous and differentiable in $K\setminus N^{+}$.

  3. (SH5)

$f$ possesses two families of stable and unstable cones $C^{s}(z)$,
$C^{u}(z)$, for $z\in K\setminus N^{+}$.

  4. (SH6)

The assignment $z\mapsto C^{u}(z)$ has a continuous extension in each
$\overline{K}_{i}\subset K$ (where $K_{i}$ are the connected components of
$K\setminus N^{+}$), and there exists $\alpha>0$ such that for $z\in
N\setminus\partial K$ and $v\in C^{u}(z)$, $w\in T_{z}N$, we have
$\angle(v,w)\geq\alpha$.

  5. (SH7)

$f^{j}(N^{-})\cap N^{+}=\emptyset$ for $0\leq j<k$, where $\lambda^{k}>2$ and

$\lambda=\inf_{x\in K\setminus N^{+}}\left\lVert df_{x}\right\rVert>1.$

Two general definitions are required before we state assumption (SH8). Let $M$
be a Riemannian manifold, $X\subset M$ a Borel subset, and $\mu$ a measure on
$M$ with $\mu(X)>0$. A partition $\xi$ on $X$ is a _smoothly measurable
foliation_ if for $\mu$-almost every $x\in X$, the element $\xi_{x}$ of $\xi$
containing $x$ has the form $\xi_{x}=W(x)\cap X$, where $W(x)$ is an immersed
$C^{1}$ submanifold in $M$ passing through $X$. Observe, in particular, that
the foliations $W^{s}$ and $W^{u}$ on $K\setminus N$ as above define smoothly
measurable foliations on the attractor $\Lambda$.

Given $x\in M$ and $r>0$, let $B_{r}(x)$ denote the ball of radius $r$ in $M$.
Consider $X\subset M$ a Borel subset, $\xi$ a smoothly measurable foliation on
$X$, and $x\in X$ for which $\xi_{x}=W(x)\cap X$ for some $C^{1}$ submanifold
$W(x)$. There is a radius $r=r(x)$ for which the submanifold $W(x)\cap
B_{r}(x)$ is the image of a $C^{1}$ function $\psi_{x}:U_{x}\to M$, where
$U_{x}\subset T_{x}W(x)$ is a neighborhood of $0$ and $\varphi_{x}(0)=x$,
$d(\varphi_{x})_{0}(T_{0}T_{x}W(x))=T_{x}W(x)$. We will say $\xi$ is _locally
continuous_ if for $\mu$-almost every $x\in X$, the assignments
$y\mapsto\varphi_{y}$ and $(y,u)\mapsto d(\varphi_{y})_{u}$ are continuous
over $y\in X\cap B_{r(x)}(x)$ (note $\varphi_{y}$ is defined $\mu$-almost
everywhere in $X\cap B_{r(x)}(x)$), where $u\in U_{y}\subset T_{y}W(y)$.

  1. (SH8)

The stable foliation $W^{s}$ is locally continuous.

We now state our main result.

######  Theorem 3.1.

Let $\Lambda$ be a singular hyperbolic attractor of a map $f:K\setminus N\to
K$ satisfying conditions (SH1) - (SH7).

  1. (a)

There exist at most finitely many ergodic SRB measures of the map
$f:\Lambda\to\Lambda$.

  2. (b)

If $f$ satisfies condition (SH8), then there exist a collection of pairwise
disjoint open sets $U_{1},\ldots,U_{n}$, open in $\Lambda$, for which
$\overline{\mathop{\bigcup}_{i=1}^{n}U_{i}}=\Lambda$, and each of which is
supported by exactly one ergodic SRB measure. In particular, if $f$ satisfies
condition (SH8), then $f|_{\Lambda}:\Lambda\to\Lambda$ is topologically
transitive if and only if $f$ admits exactly one ergodic SRB measure.

######  Remark 1.

The existence of an ergodic SRB measure can be proven under more general
assumptions. In particular, in Theorem 1 of [13], a regularity condition is
given under which a singular hyperbolic attractor admits at least one ergodic
SRB measure, and in Theorem 14 of [13], it is shown that conditions (SH3) -
(SH7) imply this regularity condition.

The remainder of this section is devoted to proving this result. In Section
3.2, we prove existence of SRB measures and show that a singular hyperbolic
attractor is charged by at most finitely many ergodic SRB measures. In Section
3.3, we show that $\Lambda$ admits a kind of measurable partition by open
sets, each element of which is given full measure by exactly one SRB measure,
and thus the SRB measure is unique if (SH8) is satisfied and $f|_{\Lambda}$ is
topologically transitive.

###  3.2. Existence of finitely many SRB measures

The existence of SRB measures for singular hyperbolic attractors follows from
the following result, which follows from Theorem 1 and Theorem 14 in [13].

######  Proposition 5.

Suppose $f:K\setminus N\to K$ is a singular hyperbolic map satisfying
conditions (SH1) - (SH7). Then there exists an SRB measure for $f$.

We now show that there are only finitely many ergodic SRB measures. We have
defined $W^{u}_{\delta}(x)$ to be the image under $\exp_{x}:T_{x}M\to M$ of
the graph of a function $\psi:B^{u}_{T}(\delta,x)\to E^{s}_{x}$, where
$B^{u}_{T}(\delta,x)\subset E^{u}_{x}$ is the open ball of radius $\delta$
centered at $0\in E^{u}_{x}$, provided that such a function $\psi$ and such a
number $\delta>0$ exist. For each $x\in M$, such a $\psi$ and $\delta$ do
exist. However, we may also fix $\delta>0$ and define the set $B^{-}_{\delta}$
to be the set of all $x\in D$ for which there is some $\varepsilon>0$, some
$l\in\mathbb{N}$, and some $y\in D^{-}_{\varepsilon,l}$ so that
$W^{u}_{\delta}(y)$ exists and contains $x$. Note that $x\not\in
B^{-}_{\delta}$ if, for example, $W^{u}(x)$ is not the image under $\exp_{x}$
of a smooth graph in a $\delta$-neighborhood of $0\in T_{x}M$.

######  Proposition 6.

For sufficiently small $\varepsilon>0$, we have that
$D_{\varepsilon}^{0}\neq\emptyset$.

###### Proof.

This follows from Theorem 14 and Proposition 3 of [13]. ∎

Using this proposition, assume $\varepsilon>0$ in $D^{\pm}_{\varepsilon}$ is
chosen so that $D^{0}_{\varepsilon}\neq\emptyset$. This implies, in
particular, that $D^{-}_{\varepsilon}$ has full measure with respect to any
invariant measure.

Observe that if $\delta_{1}<\delta_{2}$, then $B_{\delta_{2}}^{-}\subseteq
B_{\delta_{1}}^{-}$. Indeed, if $x\in B_{\delta_{2}}^{-}$, then $x\in
W^{u}_{\delta_{2}}(y)$ for some $y\in D_{\varepsilon,l}^{-}$. Since $f$ is
regular, $D^{-}_{\varepsilon}$ has full measure, so $D^{-}_{\varepsilon}\cap
W^{u}_{\delta_{2}}(y)$ has full conditional measure. So we can choose
$y^{\prime}\in W^{u}_{\delta_{2}}(y)$ with distance $\delta_{1}$ from $x$
along $W^{u}_{\delta_{2}}(y)$, giving us $x\in
W^{u}_{\delta_{1}}(y^{\prime})$, so $x\in B^{-}_{\delta_{1}}$.

In particular, if $\mu$ is an ergodic SRB measure that charges
$B^{-}_{\delta_{1}}$, then since $B^{-}_{\delta_{2}}\subset
B^{-}_{\delta_{1}}$, either $B_{\delta_{2}}^{-}$ is charged by $\mu$, or has
$\mu$-measure 0. In the latter case, if there is an ergodic SRB measure
$\mu_{1}$ that charges $B^{-}_{\delta_{2}}$, then both $\mu_{1}$ and $\mu$ are
ergodic SRB measures charging $B^{-}_{\delta_{1}}$. To summarize, if
$\delta_{2}>\delta_{1}$, then $B^{-}_{\delta_{1}}$ is charged by at least as
many, if not more, SRB measures as $B^{-}_{\delta_{2}}$.

Our proof of Theorem 3.1 (a) has two major components. The first is to show
that there is a $\delta_{0}>0$ for which $B_{\delta_{0}}^{-}$ is charged by
every ergodic SRB measure, and so $B_{\delta}^{-}$ and $B_{\delta_{0}}^{-}$
are charged by the same measures for every $0<\delta<\delta_{0}$. The second
is to show that every set $B_{\delta}^{-}$, and in particular
$B_{\delta_{0}}^{-}$, is charged by only finitely many ergodic SRB measures.

######  Lemma 3.2.

Suppose the singular hyperbolic map $f:K\setminus N\to K$ satisfies (SH1) -
(SH7). There is a $\delta_{0}>0$ such that for every ergodic SRB measure $\mu$
on $\Lambda$, $\mu(B_{\delta_{0}}^{-})>0$.

###### Proof.

Assumption (SH3) states that $N$ is composed of finitely many closed
submanifolds with boundary. Call these submanifolds $N_{i}$. Observe that if
$U$ is a neighborhood of $N$, then $f(U)$ is a neighborhood of $N^{-}$.
Because $f^{j}(N^{-})\cap N=\emptyset$ for $1\leq j<k$ with $\lambda^{k}>2$ by
(SH7), and because $N^{-}$ and its images are closed (as is $N$), there is a
radius $Q>0$ so that the open neighborhoods $B_{Q}(N_{i})$ of each $N_{i}$ of
radius $Q$ are pairwise disjoint and whose first $k$ images do not intersect
$N$. Let $\delta_{0}<Q$.

Fix an ergodic SRB measure $\mu$. Our strategy will be to construct a
“rectangle” $R\subset\Lambda$, formed by the local hyperbolic product
structure of $\Lambda$, with $\mu(R)>0$. Applying the map $f$ to $R$, the
unstable leaves composing $R$ will eventually grow sufficiently large so that
a certain iterate of $R$ lies inside $B^{-}_{\delta_{0}}$. Since $\mu$ is
$f$-invariant and $\mu(R)>0$, this will show that $\mu(B^{-}_{\delta_{0}})>0$
for any ergodic SRB measure $\mu$.

Proposition 6 implies $\mu(D^{-}_{\varepsilon})>0$. Therefore there is a
generic point $x\in D^{-}_{\varepsilon}$, which implies there is an $r>0$ and
an $l\geq 1$ for which $\mu(D^{-}_{\varepsilon,l}\cap B_{r}(x))>0$. By virtue
of the hyperbolic local product structure of $D$ (see Proposition 3), there is
an $\alpha>0$ and a $\beta>0$ for which $W^{s}_{\beta}(x)\subset B_{r}(x)$,
and the local unstable leaves $W^{u}_{\alpha}(y)\subset B_{r}(x)$ are well-
defined for every $y\in D^{-}_{\varepsilon,l}\cap W^{s}_{\beta}(x)$.
Furthermore, on a subset $A\subset W^{s}_{\beta}(x)$ of full conditional
measure, the leaves $W^{u}_{\alpha}(y)$ have positive conditional measure for
every $y\in A$. Therefore, the set

$R_{1}=\mathop{\bigcup}_{y\in A}W^{u}_{\alpha}(y)$

has positive $\mu$-measure and is contained in $B_{r}(x)$.

Suppose $\alpha\geq\delta_{0}$. Let $y_{0}\in A$. Then $W^{u}_{\alpha}(y_{0})$
is the union of finitely many $W^{u}_{\delta_{0}}(y_{i})$, with $y_{i}\in
W^{u}_{\alpha}(y_{0})\cap D^{-}_{\varepsilon}$. So by definition of
$B^{-}_{\delta_{0}}$, for $z\in W^{u}_{\alpha}(y_{0})$, we may take $y=y_{i}$
in the definition of $B^{-}_{\delta_{0}}$ for one of the $y_{i}$’s, so $z\in
B^{-}_{\delta_{0}}$. So $W^{u}_{\alpha}(y_{0})\subset B^{-}_{\delta_{0}}$ for
every $y_{0}\in A$. In particular, $R_{1}\subset B^{-}_{\delta_{0}}$, so since
$R_{1}$ has positive measure, $\mu(B^{-}_{\delta_{0}})>0$.

Now suppose $\alpha<\delta_{0}$, and let $x_{1}=x$. By compactness of $K$,
there is a time $j_{1}\geq 1$ at which $f^{j_{1}}(W^{u}_{\alpha}(x_{1}))$
intersects $N$ (and, by assumption, it does so transversally). For $1\leq
j\leq j_{1}$, the image $f^{j}(W^{u}_{\alpha}(x_{1}))$ is a local unstable
leaf of size $\alpha_{j}\geq\lambda^{j}\alpha$. If
$\lambda^{j}\alpha\geq\delta_{0}$ for some $j\leq j_{1}$, then using the same
arguments as in the previous paragraph, $f^{j}(W^{u}_{\alpha}(x_{1}))\subset
B^{-}_{\delta_{0}}$, for almost every $y_{0}\in f^{j}(W^{s}_{\beta}(x_{1}))$,
and more generally, $f^{j}(R)\subseteq B^{-}_{\delta_{0}}$, yielding the
desired result.

On the other hand, if $\alpha_{j}<\delta_{0}$ for $1\leq j\leq j_{1}$, then
the leaf $f^{j_{1}}(W^{u}_{\alpha}(x_{1}))$ contains a ball in
$W_{\mathrm{loc}}^{u}(f^{j_{1}}(x_{1}))$ of diameter
$\alpha^{\prime}\geq\frac{1}{2}\lambda^{j_{1}}\alpha$ that does not intersect
$N$. Because $\mu$ is an SRB measure, the conditional measure on this ball is
absolutely continuous with respect to the Riemannian measure—in particular,
the set of generic points in this leaf has full $\mu$-conditional measure. So
choose a generic point $x_{2}$ in this ball. As with $x_{1}$, we can use the
hyperbolic product structure induced by $f$ to create a rectangle $R_{2}$ of
positive $\mu$-measure defined by

$R_{2}=\mathop{\bigcup}_{y\in A_{2}}W^{u}_{\alpha^{\prime}}(y),$

where $A_{2}\subset W^{s}_{\beta_{2}}(x_{2})$ is a set of full measure for
some $\beta_{2}>0$. Relabel $\alpha_{j_{1}}=\alpha^{\prime}$ to be the
diameter of the local unstable leaf containing $x_{2}$ and not intersecting
$N$. This leaf (and in fact all of $R_{2}$) lies inside
$B_{Q}(N)=\mathop{\bigcup}_{i}B_{Q}(N_{i})$, and by our construction of
$B_{Q}(N)$, the first $k$ images of this leaf under $f$ will not intersect
$N$. Therefore, either eventually one of the images of this leaf is of
diameter $>\delta_{0}$, or this leaf intersects $N$. In the former case, as
before, we have a rectangle of positive $\mu$-measure lying inside of
$B^{-}_{\delta_{0}}$. In the latter case, the leaf is of diameter
$\geq\lambda^{k}\alpha^{\prime}\geq\frac{1}{2}\lambda^{k+j_{1}}\alpha$. Again,
there is a ball in this leaf of diameter
$\geq\frac{1}{2}\lambda^{k}\alpha^{\prime}\geq\frac{1}{4}\lambda^{k+j_{1}}\alpha$
that does not intersect $N$. As before, we may find a generic point $x_{3}$ in
this ball, and continue iterating this local leaf and resulting positive-
measure rectangle.

Proceeding in this way, we construct a sequence of local unstable leaves
$W^{u}_{\alpha_{j}}(x_{m})$, $j=j_{1}+\cdots+j_{m-1}+l$, where each leaf image
intersects $N$ at time $j_{1}+\cdots+j_{m}$, and thus admits an open ball of
sufficient size and not intersecting $N$. By construction, $j_{i+1}-j_{i}\geq
k$ for each $i$, so each leaf has

$\alpha_{j}=\alpha_{j_{1}+\ldots+j_{m-1}+l}\geq\frac{\lambda^{j}}{2^{m-1}}\alpha\geq\left(\frac{\lambda^{k}}{2}\right)^{m}\frac{\lambda^{j_{1}}}{2}\alpha.$

As $m$ increases, we eventually get
$\left(\lambda^{k}/2\right)^{m}\lambda^{j_{1}}\alpha/2\geq Q>\delta_{0}$ by
(SH7). Once this occurs, we have a rectangle of positive measure contained in
$B^{-}_{\delta_{0}}$. ∎

The next lemma shows that $B_{\delta_{0}}^{-}$ is charged by finitely many
ergodic SRB measures. Since $B_{\delta_{0}}^{-}$ is charged by every ergodic
SRB measure, this will prove Theorem 3.1.

######  Lemma 3.3.

For sufficiently small $\delta>0$, the set $B_{\delta}^{-}$ admits at most
finitely many ergodic SRB measures. In particular, there is a subset
$\Lambda^{0}\subset B_{\delta}^{-}$ that has full measure with respect to any
invariant measure, and a finite partition of $\Lambda^{0}$ each of whose
elements is charged by exactly one ergodic SRB measure.

###### Proof.

The proof is an adaptation of a Hopf argument. Define the subsets
$\Lambda^{+}\subset K$ and $\Lambda^{-}\subset\Lambda$ respectively to be the
set of points where, for every $\varphi\in C^{0}$, the limits

$\varphi_{+}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\varphi\left(f^{k}(x)\right)\quad\textrm{and}\quad\varphi_{-}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\varphi\left(f^{-k}(x)\right)$

exist. By the Birkhoff ergodic theorem, both $\Lambda^{+}$ and $\Lambda^{-}$
have full measure with respect to any invariant measure. Observe that if
$x\in\Lambda^{-}$ and $y\in W^{u}_{\alpha}(x)$ for $\alpha>0$, then
$\varphi_{-}(y)=\varphi_{-}(x)$, so $y\in\Lambda^{-}$, and so
$W^{u}_{\delta}(x)\subseteq\Lambda^{-}$ for every $x\in\Lambda^{-}$.
Similarly, $W^{s}_{\alpha}(x)\subseteq\Lambda^{+}$ for $x\in\Lambda^{+}$,
$\alpha>0$.

Recall that a point $x\in K$ lies in $B_{\delta}^{-}$ if and only if there is
a $y=y(x)\in D^{-}_{\varepsilon,l}$ for some $\varepsilon,l$ for which $x\in
W^{u}_{\delta}(y)$. So let $\Lambda^{0}$ be the set of points $x\in
B_{\delta}^{-}$ for which there is a subset $A\subseteq W^{u}_{\delta}(y)$ of
full conditional measure (with respect to Lebesgue) such that
$A\subseteq\Lambda^{+}$ and $\varphi_{+}|_{A}$ is constant for all $\varphi\in
C^{0}(K)$.

We make the following claims:

  * •

the set $\Lambda^{0}$ has full measure in $B^{-}_{\delta}$ with respect to any
invariant measure, and

  * •

the set $\Lambda^{0}$ is closed.

Granting these claims for now, using the notation in the above paragraph, for
$x\in\Lambda^{0}$, let $\varphi_{+}(W^{u}_{\textrm{loc}}(x))=\varphi_{+}(z)$
for $z\in A\subseteq W^{u}_{\delta}(y)$, where $y=y(x)$ is as in the
definition of $B_{\delta}^{-}$. We will say that $x,z\in\Lambda^{0}$ are
_equivalent_ , and write $x\sim z$, if
$\varphi_{+}(W^{u}_{\mathrm{loc}}(x))=\varphi_{+}(W^{u}_{\mathrm{loc}}(z))$
for every continuous $\varphi:K\to\mathbb{R}$.

Note that the stable foliation $W^{s}$ is absolutely continuous (see (2) and
Proposition 4 for the definition of $W^{s}$ and absolute continuity of the
foliation). So if $x\in\Lambda^{0}$ and $z\in W^{s}(x)\cap\Lambda^{0}$, then
$\varphi_{+}(x)=\varphi_{+}(z)$, and there is a set $A^{\prime}$ of full
measure in $W^{u}_{\delta}(y(z))$ on which $\varphi_{+}$ is constant and equal
to $\varphi_{+}(W^{u}_{\mathrm{loc}}(x))$. So $x\sim z$ whenever $z\in
W^{s}(x)\cap\Lambda^{0}$.

Suppose $\Lambda^{0}_{1}$ is an equivalence class, and let
$x\in\Lambda_{1}^{0}$. We claim there is an $\varepsilon>0$ so that if
$y\in\Lambda^{0}$ lies in the $\varepsilon$-ball centered at $x$, then $x\sim
y$. It will follow that each equivalence class is open in $\Lambda^{0}$, and
therefore also closed in $\Lambda^{0}$ since the equivalence classes form a
partition of $\Lambda^{0}$ and $\Lambda^{0}$ itself is closed.

To prove this claim, by virtue of Proposition 7 in [13], there is an
$\varepsilon>0$ for which $B_{\varepsilon}(x)$ has _local hyperbolic product
structure:_ for $z\in W^{u}_{\varepsilon}(x)$ and $y\in
W^{s}_{\varepsilon}(x)$, the intersection $W^{u}_{\varepsilon}(y)\cap
W^{s}_{\varepsilon}(z)$ contains exactly one point, which we denote $[y,z]$.
Let $y\in B_{\varepsilon}(x)\cap\Lambda^{0}$, let $B^{u}_{\varepsilon}(x)$
denote the ball in the local unstable manifold $W^{u}_{\mathrm{loc}}(x)$
centered at $x$ of size $\varepsilon$, and let
$\theta:B^{u}_{\varepsilon}(x)\to W^{u}_{\varepsilon}(y)$ denote the holonomy
map $\theta(z)=[y,z]$.

To show $x\sim y$, let $A\subseteq B_{\varepsilon}^{u}(x)$ be the set of
points $z$ for which $\varphi_{+}(z)=\varphi_{+}(x)$ for every continuous
function $\varphi$. By definition of the set $\Lambda^{0}$, the set $A$ has
full measure in $B_{\varepsilon}^{u}(x)$. By absolute continuity of the stable
foliation, $\theta(A)$ has full measure in
$\theta(B^{u}_{\varepsilon}(x))\subset W^{u}_{\varepsilon}(y)$. Since
$\varphi_{+}$ is constant on stable leaves,
$\varphi_{+}\circ\theta=\varphi_{+}$ for every continuous $\varphi$. Therefore
$\varphi_{+}(z_{1})=\varphi_{+}(x)$ for almost every
$z_{1}\in\theta(B^{u}_{\varepsilon}(x))$. Again, by definition of
$\Lambda^{0}$, $\varphi_{+}(z_{1})=\varphi_{+}(y)$ for every continuous
$\varphi$ and almost every $z_{1}\in\theta(B_{\varepsilon}^{u}(x))$. Therefore
$\varphi_{+}(x)=\varphi_{+}(y)$, and so $x\sim y$.

It follows from these arguments that each equivalence class is open in
$\Lambda^{0}$, and hence is also closed in $\Lambda^{0}$. By closedness of
$\Lambda^{0}$ in $K$, there is an $\varepsilon>0$ such that each pair of
equivalence classes is separated by a distance of at least $\varepsilon$. By
compactness of $K$, it follows that there may only be finitely many such
equivalence classes. Because an ergodic SRB measure must charge exactly one of
these equivalence classes, there may only be finitely many SRB measures.

It remains only to prove our previous two claims. Let $\widehat{\Lambda}$
denote those points $x\in\Lambda^{-}$ such that
$\varphi_{-}(x)=\varphi_{+}(x)$ for every continuous function $\varphi$. By
the Birkhoff ergodic theorem, $\widehat{\Lambda}$ has full measure with
respect to any invariant measure. Further, let $\widehat{\Lambda}^{0}$ denote
the set of points $x\in\widehat{\Lambda}$ such that there is a set $A\subset
W^{u}_{\gamma}(x)$ of full conditional measure such that
$A\subseteq\Lambda^{+}$ and $\varphi_{-}(z)=\varphi_{+}(z)$ for every
continuous function $\varphi$ and all points $z\in A$. Here,
$W^{u}_{\gamma}(x)$ is the connected component of $W^{u}(x)$ intersecting
$\widehat{\Lambda}$ and containing $x$ (and $W^{s}_{\gamma}(x)$ is defined
similarly). Since $\varphi_{-}(z)$ takes the same value for all $z\in
W^{u}_{\varepsilon}(x)$, $\varphi_{-}|_{A}=\varphi_{+}|_{A}$ is constant.

For $x\in\widehat{\Lambda}^{0}$, the set $\widehat{\Lambda}^{0}$ contains the
union over $y\in W^{s}_{\gamma}(x)\cap B^{-}_{\delta}$ of manifolds
$W^{u}_{\varepsilon}(y)$ that contain a subset of full conditional measure (as
this subset lies in $\widehat{\Lambda}$, which has full measure). Because
$W^{s}_{\gamma}(x)$ has full conditional measure in $\widehat{\Lambda}$, it
follows that this union also has full measure, from which it follows that
$\widehat{\Lambda}^{0}$ has full measure.

If a point $x$ has a negative semitrajectory that enters
$\widehat{\Lambda}^{0}$ infinitely often, then one can show $x\in\Lambda^{0}$.
Since $\widehat{\Lambda}^{0}$ has full measure, the set of points whose
negative semitrajectories enter $\widehat{\Lambda}^{0}$ also has full measure,
so $\Lambda^{0}$ has full measure. This proves the first of the two previous
claims.

To show that $\Lambda^{0}$ is closed, suppose $x$ is the limit point of a
sequence $(x_{n})$ in $\Lambda^{0}$. Since the stable foliation is absolutely
continuous and $W^{u}_{\varepsilon}(x_{n})$ converges to
$W^{u}_{\varepsilon}(x)$ for $\varepsilon>0$ small, we can find a set
$A\subset W^{u}_{\delta}(y)$ of full conditional measure, where $y=y(x)$ is as
in the definition of $B_{\delta}^{-}$, such on which $\varphi_{+}$ is constant
for all continuous functions $\varphi$. Therefore $x\in\Lambda^{0}$. ∎

###### Proof of Theorem 3.1(a).

Note $B^{-}:=\mathop{\bigcup}_{\delta>0}B^{-}_{\delta}$ is invariant under
$f$, and as we showed in Lemma 3.2, $\mu(B^{-})>0$ for every ergodic SRB
measure $\mu$. Therefore $\mu(B^{-})=1$ for every ergodic SRB measure $\mu$,
and since $\mu(D)=1$ as well, every ergodic SRB measure gives full volume to
$D\cap B^{-}$. If there were infinitely many ergodic SRB measures, then by
Lemma 3.2, there would be a $\delta>0$ for which $B^{-}_{\delta}$ is charged
by infinitely many SRB measures. But this contradicts Lemma 3.3. ∎

###  3.3. Uniqueness and topological transitivity

Generally speaking, a map satisfying (SH1) - (SH7) may admit more than one
ergodic SRB measure (see examples in Section 4). However, under moderate
assumptions on the regularity of the stable foliation $W^{s}$, one can show
that the components of topological transitivity of
$f|_{\Lambda}:\Lambda\to\Lambda$ are in correspondence with the number of
distinct ergodic SRB measures. We formalize this idea in this section.

Given a metric space $X$, we will call a Borel measure $\mu$ on $X$ _locally
positive_ if $\mu(U)>0$ for every nonempty open subset $U\subset X$. A
collection $\\{U_{i}\\}_{i\in I}$ of open subsets $U_{i}\subset X$, together
with a collection of Borel measures $\\{\mu_{j}\\}_{j\in J}$ on $X$, with
$J\subset I$, is an _open partition by measures_ if:

  1. (P1)

the open sets $\\{U_{i}\\}_{i\in I}$ are pairwise disjoint;

  2. (P2)

$\overline{\mathop{\bigcup}_{i\in I}U_{i}}=X$;

  3. (P3)

$\mu_{j}\left(X\setminus U_{j}\right)=0$ for all $j\in J$; and

  4. (P4)

$\mu_{j}|_{U_{j}}$ is locally positive for all $j$.

By local positivity, we may assume $I\setminus J$ has a single element, which
we denote $0\in I$. If the open sets $\\{U_{i}\\}_{i\in I}$ and the measures
$\\{\mu_{j}\\}_{j\in J}$ are chosen so that $I=J$, we say that the open
partition by measures is _complete_. If $f:X\to X$ is continuous, and the
measures $\mu_{j}$ are ergodic probability measures, we call an open partition
by the ergodic measures $\mu_{j}$ an _open ergodic partition_ if in addition
to (P1) - (P4) above, we also have

  1. (P5)

$\mu(U_{0})=0$ for any ergodic measure $\mu$.

######  Lemma 3.4.

If $X$ is a complete metric space, and $f:X\to X$ is a continuous and open map
admitting an open ergodic partition by the open sets $\\{U_{i}\\}_{i\in I}$
and the ergodic $f$-invariant Borel measures $\\{\mu_{j}\\}_{j\in
I\setminus\\{0\\}}$, then $\overline{U}_{i}$ is $f$-invariant for each $i$. If
the open ergodic partition is complete, then $f|_{\overline{U}_{i}}$ is
topologically transitive for each $i\in I$.

###### Proof.

By (P1) - (P4) and the fact that each measure $\mu_{j}$ is $f$-invariant,
$U_{i}$ is invariant for every $i$, as is $\overline{U}_{i}$. Now suppose the
open ergodic partition is complete. If $V,V^{\prime}\subset\overline{U}_{i}$
are open, then $\mathop{\bigcup}_{n\in\mathbb{Z}}f(V)$ is open and
$f$-invariant. Thus if $f^{n}(V)\cap V^{\prime}=\emptyset$ for every $n\geq
0$, ergodicity of $\mu_{i}$ implies that either $\mu_{i}(V^{\prime})=0$ or
$\mu_{i}(V)=0$. By local positivity, either $V^{\prime}=\emptyset$ or
$V=\emptyset$. ∎

Theorem 3.1(b) is now an immediate consequence of the following lemma.

######  Lemma 3.5.

If $f:K\subset N\to K$ is a singular hyperbolic map with attractor
$\Lambda\subset K$ satisfying condition (SH8) in addition to (SH1) - (SH7),
then $\Lambda$ admits a finite and complete open ergodic partition, and the
measures defining this partition are SRB measures. If in addition
$f|_{\Lambda}:\Lambda\to\Lambda$ is topologically transitive, then the open
ergodic partition consists of a single open set and a single measure, and thus
in particular, $f$ admits a unique ergodic SRB measure.

###### Proof.

The fact that $\Lambda$ admits a complete open ergodic partition by SRB
measures follows from Theorems 4 and 6 of [13]. Finiteness of this partition
follows from Theorem 3.1(a). If $f|_{\Lambda}$ is topologically transitive, by
Lemma 3.4, there is at most one ergodic SRB measure; existence of this measure
follows from Proposition 5. ∎

##  4\. Examples

Maps described in Theorem 3.1 do exist, and as the following non-example will
demonstrate, the hypotheses described in this result are necessary assumptions
in general. Examples of singular hyperbolic attractors can be found in, for
example, Lorenz-type attractors; but this class of singular hyperbolic maps
includes cases where the singular set has countably many components, and admit
countably many SRB measures.

###  4.1. Lorenz-type attractors

To begin, we describe the class of singular hyperbolic attractors generated by
Lorenz-type maps of the unit square. The definition of these maps is as
follows. Let $I=(-1,1)$, $K=I^{2}$, and
$-1=a_{0}<a_{1}<\cdots<a_{m}<a_{m+1}=1$. Define the rectangles
$P_{i}=I\times(a_{i},a_{i+1})$ for $0\leq i\leq m$, and
$N=I\times\\{a_{0},\ldots,a_{m+1}\\}$. Let $f:K\setminus N\to K$ be an
injective map given by

$f(x,y)=\big{(}\varphi(x,y),\,\psi(x,y)\big{)},\quad x,y\in I,$

where the functions $\varphi,\psi:K\to\mathbb{R}$ satisfy the following
conditions:

  1. (L1)

$\varphi$ and $\psi$ are continuous in $\overline{P}_{i}$ for each $i$, and:

$\displaystyle\lim_{y\to a_{i}^{+}}\varphi(x,y)=\varphi_{i}^{+},$
$\displaystyle\quad\lim_{y\to a_{i}^{+}}\psi(x,y)=\psi_{i}^{+},$
$\displaystyle\lim_{y\to a_{i}^{-}}\varphi(x,y)=\varphi_{i}^{-},$
$\displaystyle\quad\lim_{y\to a_{i}^{-}}\psi(x,y)=\psi_{i}^{-},$

where $\varphi_{i}^{\pm}$, $\psi_{i}^{\pm}$ do not depend on $x$ for each $i$;

  2. (L2)

$\psi$ and $\varphi$ have two continuous derivatives in $P_{i}$. Furthermore,
there are positive constants $B_{i}^{1}$, $B_{i}^{2}$, $C_{i}^{1}$, and
$C_{i}^{2}$; constants
$0\leq\nu_{i}^{1},\nu_{i}^{2},\nu_{i}^{3},\nu_{i}^{4}\leq 1$; a sufficiently
small constant $\gamma>0$; and continuous functions $A_{i}^{1}(x,y)$,
$A_{i}^{2}(x,y)$, $D_{i}^{1}(x,y)$, and $D_{i}^{2}(x,y)$ that tend to zero
uniformly over $x$ as $y\to a_{i}$ or $y\to a_{i+1}$; so that for $(x,y)\in
P_{i}$,

$\left.\begin{array}[]{l}d\varphi(x,y)=B_{i}^{1}(y-a_{i})^{-\nu_{i}^{1}}\left(1+A_{i}^{1}(x,y)\right)\\\
d\psi(x,y)=C_{i}^{1}(y-a_{i})^{-\nu_{i}^{2}}\left(1+D_{i}^{1}(x,y)\right)\end{array}\right\\}\quad\textrm{if
}y-a_{i}\leq\gamma;$
$\left.\begin{array}[]{l}d\varphi(x,y)=B_{i}^{2}(a_{i+1}-y)^{-\nu_{i}^{3}}\left(1+A_{i}^{2}(x,y)\right)\\\
d\psi(x,y)=C_{i}^{2}(a_{i+1}-y)^{-\nu_{i}^{4}}\left(1+D_{i}^{2}(x,y)\right)\end{array}\right\\}\quad\textrm{if
}a_{i+1}-y\leq\gamma;$

and additionally,
$\left\lVert\varphi_{xx}\right\rVert,\left\lVert\psi_{xx}\right\rVert,\left\lVert\varphi_{xy}\right\rVert,\left\lVert\psi_{xy}\right\rVert\leq\mathrm{const.}$;

  3. (L3)

the following inequalities hold:

$\displaystyle\left\lVert f_{x}\right\rVert,\left\lVert
g_{y}^{-1}\right\rVert$ $\displaystyle<1;$ $\displaystyle 1-\left\lVert
g_{y}^{-1}\right\rVert\left\lVert f_{x}\right\rVert$
$\displaystyle>2\sqrt{\left\lVert g_{y}^{-1}\right\rVert\left\lVert
g_{x}\right\rVert\left\lVert g_{y}^{-1}f_{y}\right\rVert};$
$\displaystyle\left\lVert g_{y}^{-1}\right\rVert\left\lVert g_{x}\right\rVert$
$\displaystyle<\left(1-\left\lVert f_{x}\right\rVert\right)\left(1-\left\lVert
g_{y}^{-1}\right\rVert\right);$

where $\left\lVert\cdot\right\rVert=\max_{i}\sup_{(x,y)\in P_{i}}|\cdot|$.

This class of maps includes the _geometric Lorenz attractor_ , for which we
have $m=1$, $a_{1}=0$, and

$\displaystyle\varphi(x,y)$
$\displaystyle=\left(-B|y|^{\nu_{0}}+Bx\,\mathrm{sgn}(y)|y|^{\nu}+1\right)\mathrm{sgn}(y),$
$\displaystyle\psi(x,y)$
$\displaystyle=\left((1+A)|y|^{\nu_{0}}-A\right)\mathrm{sgn}(y),$

where $0<A<1$, $0<B<\frac{1}{2}$, $1/(1+A)<\nu_{0}<1$, and $\nu>1$.

######  Theorem 4.1.

Let $f:I^{2}\setminus N\to I^{2}$ be a Lorenz-type map for which one of the
following properties hold:

  1. (a)

$\nu_{i}^{j}=0$, for $i=1,\ldots,m$ and $j=1,2,3,4$;

  2. (b)

$\rho\left(f^{n}(\varphi_{i}^{\pm},\psi_{i}^{\pm}),N\right)\geq
C_{i}e^{-\gamma n}$ (where $C_{i}$ are constants independent of $n$ and
$\gamma>0$ is sufficiently small).

Then $f$ admits a singular hyperbolic attractor $\Lambda$, which is supported
by at most finitely many ergodic SRB measures.

######  Remark 2.

This result is also proven in [5], and is also a consequence of the arguments
in both [3] and [16]. We present an additional proof of this result using
Theorem 3.1 directly.

###### Proof.

Properties (SH1) and (SH4)-(SH7) are shown in [13], Theorem 17. The singular
set $N$ is the disjoint union of finitely many horizontal lines
$I\times\\{a_{i}\\}$, $i=1,\ldots,m$, so (SH3) is satisfied. The statement now
follows from Theorem 3.1. ∎

######  Remark 3.

Condition (SH4) is easy to verify for the geometric Lorenz attractor, as the
map $\varphi:I^{2}\setminus\left(I\times\\{0\\}\right)\to\mathbb{R}$ extends
to $\pm 1$ as $y\to 0$ from above or below. In particular, $N^{-}\cap
K=\emptyset$, since the continuations of $\varphi$ to $N$ map $N$ to the
boundary of $K$, so (SH4) is in fact trivial. Moreover, this is true with any
Lorenz-type attractor for which $\varphi_{i}^{\pm}=\pm 1$ or $\mp 1$.

More generally, (SH4) holds if (b) is satisfied in the statement of Theorem
4.1, provided $\gamma>0$ is sufficiently small.

###  4.2. The Belykh attractor

We consider a map $f:K\setminus N\to K$, where $K=[-1,1]^{2}$, and

$N=\\{(x,kx)\in K:-1<x<1\\}$

where $|k|<1$. (More generally one can consider $N=\\{(x,h(x)):-1<x<1\\}$ for
a continuous function $h$.) We then choose constants
$\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}$ so that

$0<\lambda_{1},\mu_{1}<\frac{1}{2}\quad\textrm{and}\quad
1<\lambda_{2},\mu_{2}<\frac{2}{1+|k|},$

and define the map $T:K\setminus N\to\mathbb{R}^{2}$ by

$T(x,y)=\begin{cases}\left(\lambda_{1}(x-1)+1,\>\lambda_{2}(y-1)+1\right)&\textrm{if
}y>kx;\\\ \left(\mu_{1}(x+1)-1,\>\mu_{2}(y+1)-1\right)&\textrm{if
}y<kx.\end{cases}$

This map was first introduced in [7] as a model of phase synchronization
theory.

######  Theorem 4.2.

Define $T:K\setminus N\to\mathbb{R}^{2}$ as above.

  1. (a)

The map $T$ is a map from $K\setminus N$ into $K$, and satisfies conditions
(SH1) and (SH3)-(SH6).

  2. (b)

For any choice of $\lambda_{2}>1$, and for all but countably many $\mu_{2}>1$
(the countably many exceptions depending on $\lambda_{2}$), there is a
$\delta>0$ so that $T$ satisfies (SH7) when $|k|<\delta$, and thus admits
finitely many ergodic SRB measures for such $k$.

###### Proof.

The first of the above assertions is proven in [13]. To prove the second,
first note that [13] shows that $T$ satisfying (SH3)-(SH6) admits countably
many SRB measures. In general, (SH7) may fail; however, we will show that
given $\lambda_{2}>0$, this can happen only for countably many choices of
$\mu_{2}>0$. To see this, note that when $k=0$, (SH7) fails if the horizontal
lines forming $N^{-}$ lie inside $f^{-n}(N)$ for some $n>0$, which only
happens for countably many choices of $\mu_{2}$. Given a pair $\lambda_{2}$
and $\mu_{2}$ so that $T$ satisfies (SH7) with $k=0$, the line segments
forming $N$ and $f^{j}(N^{-})$ do not intersect for $0\leq j<k$, where
$\max(\lambda_{2},\mu_{2})^{k}>2$. Increasing $|k|$ will rotate these line
segments; by continuity, if the change in $|k|$ is sufficiently small, these
line segments will remain disjoint. So, for these choices of $\lambda_{2}$,
$\mu_{2}$, and $k$, $T$ will admit finitely many SRB measures by Theorem 3.1.
∎

###  4.3. Necessity of assumptions

The singular set $N$ may in principle have countably many components. If this
is the case, then the attractor may admit infinitely many ergodic SRB
measures, as the following example illustrates.

Let $P_{k}=\left(-1,1\right)\times\left(2^{-k}-1,\,2^{-(k-1)}-1\right)$ for
$k\geq 0$. Then $K=I^{2}=\overline{\mathop{\bigcup}_{k}P_{k}}$, and
$N^{1}:=K\setminus\mathop{\bigcup}_{k}P_{k}$ is the countable union of line
segments $(-1,1)\times\\{2^{-k}-1\\}=:N_{k}^{1}$.

Let $f:I^{2}\setminus\big{(}(-1,1)\times\\{0\\}\big{)}\to I_{2}$ be the
geometric Lorenz attractor, and let
$f_{k}:P_{k}\setminus\big{(}(-1,1)\times\left\\{\frac{2^{-k-1}+2^{-k}}{2}-1\right\\}$
be given by $f_{k}=h_{k}^{-1}\circ f\circ h_{k}$, where $h_{k}:P_{k}\to I^{2}$
is the conjugacy map given by

$h_{k}(x,y)=\left(x,2^{k+2}(y+1)-3\right).$

Now denote

$\displaystyle N$
$\displaystyle=I^{2}\setminus\left(P_{k}\setminus\left((-1,1)\times\left\\{\frac{2^{-k-1}+2^{-k}}{2}-1\right\\}\right)\right)$
$\displaystyle=(-1,1)\times\mathop{\bigcup}_{k\geq
0}\left(\left\\{2^{-k}-1,\frac{2^{-k}+2^{-k-1}}{2}-1\right\\}\right),$

and let $g:I^{2}\setminus N\to I^{2}$ be given by

$g(x,y)=f_{k}(x,y)\quad\textrm{for }(x,y)\in
P_{k}\setminus\left((-1,1)\times\left\\{\frac{2^{-k-1}+2^{-k}}{2}-1\right\\}\right).$

Effectively, we have embedded the geometric Lorenz attractor into each
disjoint rectangle $P_{k}$. The map $g$ admits a singular hyperbolic
attractor, and the singular set $N$ is the disjoint union of countably many
submanifolds. Since each orbit of $g$ is entirely contained in one of the
rectangles $P_{k}$, each $P_{k}$ supports a distinct ergodic SRB measure. So
the requirement that there are only finitely many components of the singular
set $N$ is a necessary assumption for our result to hold.

## Acknowledgments

I would like to thank Penn State University and the Anatole Katok Center for
Dynamical Systems and Geometry where this work was done. I also thank my
advisor, Y. Pesin, for introducing me to this problem and for valuable input
over the course of my investigation into singular hyperbolic attractors. I
also thank S. Luzzatto for many helpful remarks concerning the connections
between ergodicity and topological transitivity of singular hyperbolic
attractors.

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Received xxxx 20xx; revised xxxx 20xx.