Partial hyperbolicity

The definition of a uniformly hyperbolic dynamical system can be generalized in a different direction from that of nonuniformly hyperbolic dynamics by imposing a uniform condition that is weaker than hyperbolicity.

Partial hyperbolicity in the broad sense

Definition. Suppose \(M\) is a manifold, \(f\colon M\to M\) is a diffeomorphism. We say that \(f\) is (uniformly) partially hyperbolic in the broad sense if for every \(x\in M\) there is a splitting of the tangent space \(T_xM=E^s(x)\oplus E^u(x)\) and there are constants \(C>0\) and \(\lambda<\mu\) such that for every \(n\in\mathbb{N}\) we obtain \[\|Df^n(v)\|\le C\lambda^n\|v\|\] for \(v\in E^s(x)\) and \(\|Df^{-n}(v)\|\le C\mu^{-n}\|v\|\) for \(v\in E^u(x)\ .\)

If one takes \(\lambda<1<\mu\) here then this becomes the definition of an Anosov diffeomorphism. Thus, the definition of hyperbolicity has been relaxed by allowing the rate gap to occur in a different place. Accordingly, Anosov systems satisfy this definition in an obvious way. An Anosov system may satisfy this definition in a less obvious way, however: Consider a linear map of the \(n\)-torus with at least one eigenvalue outside the unit circle. The set of the absolute values of the eigenvalues is finite, and if we take \([\lambda,\mu]\) in a complementary interval then the above definition applies. If the map in question also has eigenvalues on the unit circle then it satisfies the definition of partial hyperbolicity only in this nontrivial way.

Partial hyperbolicity in the narrow sense

A slightly narrower notion of partial hyperbolicity has been studied a great deal and is motivated by the question of how much of the hyperbolic paradigm survives if one replaces the complete hyperbolicity such as in an Anosov system by "some" hyperbolicity - is it possible that the presence of some uniform expansion and contraction is sufficient for the creation of an interesting degree of the ergodic and topological complexities that one sees in uniformly hyperbolic dynamical systems. Thus, this generalization requires stable and unstable directions, but also allows the presence of a center direction, in which nothing contracts as rapidly as in the stable direction and nothing expands as fast as in the unstable direction.

Definition. Suppose \(M\) is a manifold, \(f\colon M\to M\) is a diffeomorphism. We say that \(f\) is (uniformly) partially hyperbolic (in the narrow sense) or simply partially hyperbolic if for every \(x\in M\) there is a splitting of the tangent space \(T_xM=E^s(x)\oplus E^c(x)\oplus E^u(x)\) and there are constants \(C>0\) and \[\tag{1} 0<\lambda_1\le\mu_1<\lambda_2\le\mu_2<\lambda_3\le\mu_3\]

with \(\mu_1<1<\lambda_3\) such that for every \(n\in\mathbb{N}\) we obtain

\(\lambda_1^n\|v\|/C\leq\|Df^n(v)\|\le C\mu_1^n\|v\|\) for \(v\in E^s(x)\ ,\)
\(\lambda_2^n\|v\|/C\leq\|Df^n(v)\|\le C\mu_2^n\|v\|\) for \(v\in E^c(x)\ ,\)
\(\lambda_3^n\|v\|/C\leq\|Df^n(v)\|\le C\mu_3^n\|v\|\) for \(v\in E^u(x)\ .\)

Let \(E^{cs}:=E^c\oplus E^s\) and \(E^{cu}:=E^c\oplus E^u\ .\) These 5 subbundles are Hölder continuous. As in the case of uniformly hyperbolic dynamical systems, one can verify this property using cone fields. For partial hyperbolicity in the broad sense there are two cone families corresponding to the two parts of the rate spectrum. For systems that are partially hyperbolic in the narrow sense, each of the two rate gaps gives rise to 2 complementary cone families. This does not include a cone family associated with the center, which is related to the fact that the behavior of the center direction is far more delicate than that of the two other subbundles.

As in the case of uniformly hyperbolic dynamical systems, this definition extends readily to that of partial hyperbolicity on compact invariant sets, including partially hyperbolic attractors.

The same techniques that produce stable and unstable manifolds for uniformly hyperbolic systems also produce invariant contracting and expanding foliations \(W^s\) and \(W^u\) with smooth leaves to which \(E^s\) and \(E^u\) are tangent. Moreover, these have the same absolute continuity property that was used in an essential way to establish ergodicity of uniformly hyperbolic dynamical systems. Thus there is hope for applying the Hopf argument to partially hyperbolic systems.

By contrast, the center bundle may not even have a corresponding foliation, and even if there is a center foliation, it often (maybe generically) fails to have this absolute continuity property (although the leaves are moderately smooth). Together with the lack of strong expansion in the center this is the reason that hyperbolic arguments need to be carried over to this situation in a way that uses only the stable and unstable directions.

Examples

Time-1 maps of Anosov flows are partially hyperbolic; the center direction is the flow direction.

Taking the Cartesian product of an Anosov diffeomorphism with the identity or an isometry, such as a rotation, provides trivial examples of partially hyperbolic dynamical systems; the second factor is the central direction. The same holds if the second factor is any dynamical system whose maximal expansion is separated from the slowest expansion rate of the Anosov diffeomorphism and likewise for the contraction rates. A slight generalization of this idea is that of skew products. These are obtained from an Anosov diffeomorphism \(f\) on a manifold \(M\) and a family \(\{g_x\colon N\to N\mid x\in M\}\) whose rates are again uniformly inside the rate gap of \(f\) by setting \(F(x,y):=(f(x),g_x(y))\ .\) A special case is that of group extensions: Here \(N\) is a compact Lie group and \(g_x(y)=\varphi(x)y\) for \(y\in N\ ,\) where \(\varphi\colon M\to N\) is smooth. Note that the group translation \(y\mapsto\varphi(x)y\) is an isometry.

A geodesic flow on a manifold \(M\) of negative curvature is an Anosov flow, so its time-1 map is partially hyperbolic, as noted above. One can obtain from this also an example of a partially hyperbolic flow, however. Consider the frame bundle \(N\) on \(M\ ,\) which is the bundle of positively oriented orthonormal \(n\)-frames. The flow acts on these by moving the base point along the geodesic defined by the first vector of a frame and by moving all vectors of the frame along the geodesic by parallel transport. This gives positively oriented orthonormal frames at all points of the geodesic. Each fiber of the bundle \(N\to SM\) given by projecting to the first vector is a copy of \(SO(n-1)\ .\) The center bundle consists of the flow direction and this fiber direction and therefore has dimension \(1+\dim SO(n-1)\ .\)

As noted above, linear maps of the \(n\)-torus with at least one eigenvalue outside the unit circle are partially hyperbolic where the central direction corresponds to the generalized eigenspace associated with the eigenvalues on the unit circle. More generally affine diffeomorphisms on finite volume homogeneous spaces whose adjoint operator has an eigenvalue outside the unite circle are partially hyperbolic.

Products lead to rather nongeneric examples, of course, and it is well to note here that perturbations of partially hyperbolic dynamical systems are again partially hyperbolic. Thus, one can augment this collection of examples by \(C^1\) perturbations of all of them. Note that a perturbation of a partially hyperbolic set may fail to be partially hyperbolic, but this is true for partially hyperbolic attractors.

On the other hand, as in the case of Anosov systems, the presence of the invariant foliations constitutes a restriction on the topological type of the underlying manifold. In particular, there are no partially hyperbolic diffeomorphisms on the 3-sphere.

Ergodicity, transitivity and accessibility

Uniformly hyperbolic dynamical systems are topologically transitive (up to the spectral decomposition into finitely many pieces) and ergodic if they preserve volume. Moreover, both properties persist under perturbations of the system (in the case of ergodicity, one restricts to volume-preserving perturbations). The question of how much of the hyperbolic paradigm survives for partially hyperbolic dynamical systems can therefore be made concrete by asking:

Are partially hyperbolic dynamical systems topologically transitive? Ergodic? Is either property stable under perturbations?

The idea is that even if one does not have complete hyperbolicity, the expansion of stable manifolds should permeate the entire space. Several of the above examples show that this does not always work: In a product or skew product an expanding fiber does not have access to the entire space because it is confined to one slice of the product. Accordingly, the above questions are meaningful in the presence of some irreducibility assumptions or hypotheses that imply these.

The essential feature that is needed for the Hopf argument to prove ergodicity and that is also useful for establishing transitivity is the exact opposite of the confinement of stable and unstable leaves to a slice of the phase space:

Accessibility. A partially hyperbolic diffeomorphism is said to be accessible if any two points are joined by a path that consists of finitely many pieces, each of which lies in a stable or unstable manifold. Sometimes essential accessibility is enough, which means that up to a set of measure zero (with respect to a smooth invariant measure) there is only one accessibility class (of the equivalence relation defined by joining points with these "\(us\)-paths").

If the stable and unstable foliations are smooth then accessibility implies the non-integrability of the subbbundle \(E^s\oplus E^u\) since otherwise each leaf of the resulting foliation will be an accessibility class.

Theorem. If a partially hyperbolic diffeomorphism preserves a smooth measure and has the essential accessibility property then almost every point has a dense orbit.

This result is stronger than transitivity. The conclusion is exactly what one would get from ergodicity by the Birkhoff Ergodic Theorem. On the other hand, there are accessible nontransitive partially hyperbolic diffeomorphisms.

Ergodicity can also be obtained from accessibility, although this requires great refinements in the Hopf argument (to get around the lack of absolute continuity of the center bundle). An important measure-theoretic ingredient is the use of juliennes instead of balls. Juliennes are highly predistorted balls that are finely adapted to the dynamics. It is then shown that these can be used instead of balls to define Lebesgue density points. These arguments use the additional mildly restrictive assumption that the center behavior is not only dominated by the stable and unstable behavior, but that there is a specific degree of domination (which is conjectured to be unnecessary):

Definition. We say that \(f\) is center-bunched if \(\max\{\mu_1,\lambda_3^{-1}\}<\lambda_2/\mu_2\) in equation (1). Here these parameters are allowed to depend on the point (contrary to the original intent of equation (1).

This assumption holds for example whenever the center direction is one-dimensional.

Theorem. A \(C^2\) volume-preserving partially hyperbolic essentially accessible center-bunched diffeomorphism is ergodic.

Stable ergodicity

Smooth ergodic dynamical systems can be constructed in a variety of ways, and hyperbolicity is by no means essential in this respect. Hyperbolicity plays a greater role when it comes to more refined statistical or mixing properties, or when ergodicity of all \(C^1\) perturbations is desired as well. This stability of ergodicity is clear for volume-preserving Anosov systems, and the study of partially hyperbolic dynamical systems was motivated in no small part by the question of whether a reduced amount of hyperbolicity suffices for stable ergodicity.

In the situation of the previous theorem, partial hyperbolicity and center bunching are open conditions. Hence:

Theorem. A \(C^2\) volume-preserving partially hyperbolic stably essentially accessible center-bunched diffeomorphism is ergodic (and, in fact, stably a K-automorphism).

The situations of above results about stable accessibility provide applications of this theorem. A particularly clean one is that the time-1 map of a volume-preserving Anosov flow is stably ergodic unless the stable and unstable foliations are jointly integrable.

Essential accessibility is known not always to be stable, but it is conjectured that accessibility is stable under perturbation. This suggests the following:

Pugh-Shub Conjecture. A volume-preserving accessible partially hyperbolic dynamical system is stably ergodic.

Persistence of accessibility is \(C^1\)-open-dense in

the space of \(C^r\) partially hyperbolic diffeomorphisms

the subspace of volume-preserving ones

group extensions of Anosov diffeomorphisms

a neighborhood of skew products over Anosov diffeomorphisms.

Also, there is an open dense set of roof functions for which the special flow over an Anosov diffeomorphism is stably accessible. For time-1 maps of Anosov flows, accessibility is stable.

It is conjectured that persistence of accessibility is \(C^r\)-open-dense in the space of \(C^r\) partially hyperbolic diffeomorphisms and the subspace of volume-preserving ones (this is known in the case when the central distribution is one-dimensional).

Frame flows. There are also several cases in which the frame flow and its time-\(t\) maps are known to be ergodic:

Theorem. Let \(\Phi_t\) be the frame flow on an \(n\)-dimensional compact smooth Riemannian manifold with sectional curvatures between \(-\Lambda^2\) and \(-\lambda^2\ .\) Then in each of the following cases the flow is Bernoulli, and the time-one map of the frame flow is stably ergodic and stably K:

if the curvature is constant

for a set of metrics of negative curvature which is open and dense in the \(C^3\) topology

if \(n\) is odd and \(n\ne 7\)

if \(n\) is even, \(n\ne 8\ ,\) and \(\lambda/\Lambda>0.93\)

if \(n=7\) and \(\lambda/\Lambda>0.99023\dots\)

if \(n=8\) and \(\lambda/\Lambda>0.99023\dots\)

Partially hyperbolic attractors

u-measures

An invariant Borel probability measure \(\mu\) on a partially hyperbolic attractor \(\Lambda\) is said to be a \(u\)-measure if the conditional measures \(\mu^u(x)\) generated by \(\mu\) on local unstable leaves \(V^u(x)\) are absolutely continuous with respect to the Riemannian volume on \(V^u(x)\ .\)

Every ergodic component of a \(u\)-measure is again a \(u\)-measure. A \(u\)-measure depends continuously on the attractor: If \(f_n\to f\) in the \(C^2\)-topology, \(\nu_n\) is a \(u\)-measure for \(f_n\) and \(\nu_n\to\nu\) weakly, then \(\nu\) is a \(u\)-measure for \(f\ .\)

Consider a smooth measure \(\nu\) on \(U\) with the density function \(\psi\) with respect to the Riemannian volume \(m\ ,\) i.e., \(\hbox{supp }\psi\subset U\) and \(\int_U\,\psi dm=1\ .\) The sequence of measures \[\nu_n=\frac{1}{n}\sum_{i=0}^{n-1}\,f_*^i\nu\]

is the evolution of the measure \(\nu\) under the system \(f\ .\) Even if the sequence \(\nu_n\) does not converge, any limit measure \(\mu\) is supported on \(\Lambda\ ,\) and it is an \(f\)-invariant \(u\)-measure on \(\Lambda\ .\) This describes \(u\)-measures as a result of the evolution of an absolutely continuous measure in a neighborhood of the attractor.

Another approach for constructing \(u\)-measures on \(\Lambda\) is to determine \(u\)-measures as limit measures for the evolution of an absolutely continuous measure supported on a local unstable manifold: For \(x\in\Lambda\) and \(y\in V^u(x)\) consider the function \[\kappa(x,y)=\prod_{i=0}^{n-1}\frac{J(df\upharpoonright E^u(f^i(y)))}{J(df\upharpoonright E^u(f^i(x)))}\ .\]

Define the probability measure \(\tilde m_n\) on \(V_n(x)=f^n(V^u(x))\) by \[d\tilde m_n(y)=c_n\kappa(f^n(x),y)dm_{V_n(x)}\] for \(y\in V_n(x)\ ,\) where \(c_n\) is normalizing factor and \(m_{V_n(x)}\) is the Riemannian volume on \(V_n(x)\) induced by the Riemannian metric. We define the Borel measure \(m_n\) on \(\Lambda\) by \[m_n(A)=\tilde m_n(A\cap V_n(x))\ ,\]

when \(A\subset\Lambda\) is a Borel set. One can show that \(m_n(A)=m_0(f^{-n}(A))\ ,\) and again, any limit measure of the sequence of measures \(m_n\) is an \(f\)-invariant \(u\)-measure on \(\Lambda\ .\) In fact, unlike with the previous construction, every \(u\)-measure is obtained in this way.

If the unstable distribution \(E^u\) splits into the sum of two invariant subdistributions \(E^u=E_1\oplus E_2\) with \(E_1\) expanding more rapidly than \(E_2\ ,\) then one can view \(f\) as a partially hyperbolic diffeomorphism with \(E_1\) as the new unstable distribution (and \(E_2\oplus E^c\) as the new center subbundle) and construct \(u\)-measures, associated with this subbundle, as limit measures for the evolution of an absolutely continuous measure supported on a local unstable manifold. It then turns out that any \(u\)-measure associated with the distribution \(E^u\) is a \(u\)-measure associated with the distribution \(E_1\ .\) (This is obtained from the fact that the leaves of the \(W_1(y)\) depend smoothly on \(y\in W^u(x)\ .\))

u-measures and Sinai-Ruelle-Bowen measures

Unlike in the uniformly hyperbolic situation, \(u\)-measures need not be unique for partially hyperbolic attractors. There are, however, interesting situations where this is the case.

Definition. We say that a \(u\)-measure \(\nu\) on a partially hyperbolic attractor \(\Lambda\) exhibits mixed hyperbolicity if it has negative central exponents (i.e., there exists a set \(A\subset\Lambda\) of positive \(\nu\)-measure such that the Lyapunov exponent \(\chi(x,v)\) is negative for every \(x\in A\) and \(v\in E^c(x)\)).

The reason for the terminology is that this assumption mixes (uniform) partial hyperbolicity and nonuniform hyperbolicity.

Theorem. If \(\Lambda\) is a partially hyperbolic attractor and \(\nu\) is a \(u\)-measure that exhibits mixed hyperbolicity then every ergodic component of \(f\upharpoonright A\) of positive measure is open\(\pmod 0\ .\) If, in addition, the orbit of almost every \(x\in\Lambda\) is dense in \(\Lambda\ ,\) then \(\nu\) is ergodic (indeed Bernoulli).

The very last assumption follows, for example, if for every \(x\in\Lambda\) the global strongly unstable manifold \(W^u (x)\) is dense. Moreover, under this assumption there is a unique \(u\)-measure which is also a Sinai-Ruelle-Bowen measure, i.e., there is a set \(B\subset U\) (the isolating neighborhood of the attractor \(\Lambda\)) such that for every \(x\in B\) and every continuous function \(\varphi\) on \(U\) we have \[\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\varphi(f^k(x))=\int_U\varphi d\nu\ .\] An SRB measure for a partially hyperbolic attractor is a \(u\)-measure whose basin has full volume in the topological basin of attraction. This latter condition always holds for a \(u\)-measure that is unique.

Specifically, we have:

Theorem. Let \(f\) be a \(C^2\) diffeomorphism possessing a partially hyperbolic attractor \(\Lambda\ .\) Assume that

there exist \(x\in\Lambda\) and a disk \(D^u(x)\subset W^u(x)\) centered at \(x\) for which \(\chi(y, v)<0\) for a positive Lebesgue measure subset of points \(y\in D^u\) and every vector \(v\in E^c(y)\ ;\)
every leaf of the foliation \(W^u\) is dense in \(\Lambda\ .\)

Then \(f\) has a unique \(u\)-measure and it is ergodic (indeed Bernoulli) and a Sinai-Ruelle-Bowen measure. Its support coincides with \(\Lambda\ .\)

Another criterion is the following:

Theorem. If \(f\) is nonuniformly expanding along the center-unstable direction, i.e., there is a set \(A\subset M\) of positive Lebesgue measure such that \[\limsup_{n\to\infty}\frac{1}{n}\,\sum_{j=1}^n\,\log\|df^{-1}\upharpoonright E^{cu}_{f^j(x)}\|<0\] for all \(x\) in \(A\) then \(f\) has an ergodic Sinai-Ruelle-Bowen measure supported in \(\bigcap_{j=0}^\infty f^j(M)\ .\) Moreover, if the above limit is bounded away from zero then \(A\) is contained\(\pmod0\) in the union of the basins of finitely many Sinai-Ruelle-Bowen measures.

The following statement describes a version of stable ergodicity for partially hyperbolic attractors.

Theorem. Assume that there exist a \(u\)-measure \(\nu=\nu_f\) for \(f\) that exhibits mixed hyperbolicity. Assume also that for every \(x\in\Lambda_f\) the global strongly unstable manifold \(W^u(x)\) is dense in \(\Lambda_f\ .\) Then any \(C^2\) diffeomorphism \(g\) which is sufficiently close to \(f\) also has negative central exponents almost everywhere with respect to a \(u\)-measure \(\nu_g\ ;\) this measure is the only Sinai-Ruelle-Bowen measure for \(g\) and \(g\upharpoonright \Lambda_g\) is ergodic with respect to \(\nu_g\) (indeed Bernoulli, so \(f\) is stably Bernoulli).

References

Boris Hasselblatt, Yakov Pesin Partially Hyperbolic Dynamical Systems. Handbook of Dynamical Systems 1B, 1-55, Elsevier North Holland, 2005

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity. Zürich Lectures in Advanced Mathematics, EMS, 2004

C. Pugh, M. Shub, Stable Ergodicity. Bulletin of the American Mathematical Society (N.S.) 41 (2004), no. 1, 1-41

Partial hyperbolicity

Contents

Introduction

Partial hyperbolicity

Partial hyperbolicity in the broad sense

Partial hyperbolicity in the narrow sense

Examples

Ergodicity, transitivity and accessibility

Stable ergodicity

Partially hyperbolic attractors

u-measures

u-measures and Sinai-Ruelle-Bowen measures

References

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