Hyperbolic dynamics

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Boris Hasselblatt and Yakov Pesin (2008), Scholarpedia, 3(6):2208. doi:10.4249/scholarpedia.2208 revision #91367 [link to/cite this article]
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Curator: Boris Hasselblatt

The study of dynamical systems traditionally concentrates on either continuous-time evolutions, that is, flows or semiflows (these arise from differential equations), or on discrete-time evolutions, that is, maps, either invertible (such as homeomorphisms or diffeomorphisms) or not (such as continuous maps, endomorphisms or embeddings). Several standard constructions allow one to pass from one to the other, but this entry treats both evolutions.

Smooth dynamics is the study of differentiable flows or maps, and in these situations one may try to develop local information from the infinitesimal information provided by the differential. Among smooth dynamical systems, hyperbolic dynamics is characterized by the presence of expanding and contracting directions for the derivative. This is a situation where the differential alone provides strong local, semilocal or even global information about the dynamics. Indeed, under iteration the presence of these directions produces exponential relative behavior of orbits on some set, and this affords much insight into topological and measurable aspects of the orbit structure. This stretching and folding typically gives rise to complicated long-term behavior in these systems. The dynamics appears in many ways effectively random, even though these systems are completely deterministic. The theory of hyperbolic dynamical systems provides a rigorous mathematical foundation for this remarkable phenomenon known as deterministic chaos - the appearance of chaotic motions in purely deterministic dynamical systems. The cause for these motions is the instability of trajectories that is expressed in terms of the hyperbolicity conditions.

The various aspects of the complexity present in these systems makes it natural to study their topological structure as well as statistical or probabilistic aspects of the evolution, that is, to use measure-theoretic methods.

This entry provides an introduction to hyperbolic dynamics and a detailed treatment of uniformly hyperbolic dynamical systems. This hyperbolicity theory has many applications within mathematics, such as to geometry (geodesic and frame flows, theory of foliations), modern rigidity theory, dimension theory (multifractal analysis of dynamical systems) and statistical and mathematical physics. It also serves to show the important notions, results and paradigms that motivate the study of nonuniform hyperbolicity and of partial hyperbolicity, both of which have separate entries in Scholarpedia. In fact, it is the theory of nonuniformly hyperbolic dynamical systems that in applications provides the mathematical foundation of the theory of chaos.

Contents

Introduction

A simple situation illustrates some crucial aspects of hyperbolic behavior that are common to all different versions of hyperbolicity. Let \(A\) be a linear map of the plane given by the matrix \(\begin{pmatrix}\lambda&0\\0&\mu\end{pmatrix}\ ,\) where \(0<\lambda<1<\mu\ .\) The horizontal axis is the stable (i.e., contracting) direction, and it is characterized by the property that \(A^nv\to0\) as \(n\to\infty\) for every \(v\) on this axis. Every other vector \(w\) satisfies \(A^nw\to\infty\) as \(n\to\infty\ ,\) so the vertical axis cannot be characterized by this property. To single out the vertical direction reverse time to obtain \(A^{-n}w\to0\) as \(n\to\infty\) for every vertical vector \(w\) (and only for these). Passing to the map \(A^{-1}\) instead interchanges the roles of the expanding and contracting subspaces.

The same analysis applies to any linear map without eigenvalues on the unit circle. Moreover, for any linear map \(A\) consider the sum of the generalized eigenspaces for all eigenvalues with absolute value at most \(\lambda\) for some \(\lambda\in(0,1)\ .\) This subspace is characterized by \(\|A^nv\|\le C\lambda^n\|v\|\) for some \(C>0\ .\) By passing to \(A^{-1}\) one can similarly describe the subspaces corresponding to expansion by a factor \(\mu>1\ .\) A similar analysis can be carried out for the substantially more difficult case of sequences of linear maps, which arises from the study of the action of the differential of a map along an orbit.

Indeed, this analysis of the linear situation can be transferred to study local behavior of nonlinear systems where the stable and unstable subspaces are replaced by local stable and unstable manifolds. The local stable manifold is characterized by exponential contraction in forward time, and similarly, by reversing time, the local unstable manifold is characterized by exponential contraction in backward time. Unlike in the linear case, unstable manifolds may not expand in forward time. In uniformly hyperbolic dynamical systems this is usually not the case (the unstable manifold expands exponentially when moved forward), but it is a serious obstacle in nonuniformly hyperbolic dynamics.

Uniform hyperbolicity

The strongest possible version of hyperbolicity is that the asymptotic contraction and expansion rates are uniform (uniform hyperbolicity), i.e., they are bounded away from 1 independently of the point, as is made explicit in (1) below. This is the case for Anosov diffeomorphisms. There were two important motivations for the introduction and study of this notion. One of these was the focus of the Smale school on understanding diffeomorphisms up to topological conjugacy, and particularly on structural stability, which was in subsequent decades, found to be essentially equivalent to uniform hyperbolicity (see below). The other was the Boltzmann ergodic hypothesis, which prompted the search for ergodic mechanical systems, for which geodesic flows of negatively curved Riemannian manifolds were the outstanding candidates, and for which these remain the primary example.

The quintessential example of a uniformly hyperbolic dynamical system is the linear map of the plane given by the integer matrix \(A:=\begin{pmatrix}2&1\\1&1\end{pmatrix}\ .\) This is well-defined modulo 1 (if \(\vec x=\vec y\pmod1\) then \(\vec x=\vec y+\vec n\) for an integer vector \(\vec n\ ,\) so \(A\vec x=A\vec y+A\vec n\ ,\) where \(A\vec n\) is an integer vector; thus \(\vec Ax=\vec Ay\pmod1\)) and hence gives a well-defined map on the 2-torus \(\mathbb{T}^2:=\mathbb{R}^2/\mathbb{Z}^2\ .\) Moreover, this map is invertible because \(\det A=1\ ,\) so that \(A^{-1}\) also has integer entries. \(A\) has eigenvalues \(0<\lambda^{-1}<1<\lambda\ ,\) and the corresponding eigenspaces provide the expanding and contracting directions at every point of \(\mathbb{T}^2\ .\) Having irrational slope, these spaces project to dense curves on \(\mathbb{T}^2\ .\) It is also easy to check that the set \(\mathbb{Q}^2/\mathbb{Z}^2\) of rational points is precisely the set of periodic points.

This example generalizes easily to the action on the \(n\)-torus \(\mathbb{T}^n\) of any \(n\times n\)-matrix that has integer entries and unit determinant and no eigenvalue on the unit circle. These are referred to as toral automorphisms because they are algebraic automorphisms of the additive group \(\mathbb{R}^n/\mathbb{Z}^n\ .\) A further generalization is obtained when one thinks of the torus \(\mathbb{T}^n\) as an abelian additive group and moves to nilpotent groups. Smale constructed examples of transformations on the Heisenberg group that project to a compact factor with the same feature of expanding and contracting directions.

Uniformly hyperbolic diffeomorphisms

The defining feature of uniform hyperbolicity exhibited in these examples is that the tangent space at every point splits into contracting (or stable) and expanding (or unstable) subspaces:

Definition. Suppose \(M\) is a manifold, \(f\colon M\to M\) is a diffeomorphism. We say that \(f\) is uniformly hyperbolic or an Anosov diffeomorphism 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 \(\lambda\in(0,1)\) such that for every \(n\in\mathbb{N}\) one has \[\tag{1} \|Df^n(v)\|\le C\lambda^n\|v\|\]

for \(v\in E^s(x)\) and \(\|Df^{-n}(v)\|\le C\lambda^n\|v\|\) for \(v\in
E^u(x)\ .\) 

The subspaces \(E^s(x)\) and \(E^u(x)\) are called the stable and unstable subspaces at \(x\ .\)

It is an easy consequence of this definition that the stable and unstable subspaces depend continuously on the point and are invariant. Note also that \(E^s(x)\) and \(E^u(x)\) are interchanged when one passes from a map to its inverse.

The existence of these two directions at every point imposes a topological restriction, and therefore not every manifold admits an Anosov diffeomorphism or flow. For instance, the "hairy ball theorem" shows that there is no Anosov diffeomorphism on the 2-sphere. It is unknown whether the universal cover of a manifold that admits an Anosov diffeomorphism must be \(\mathbb{R}^n\) for some \(n\ .\)

The Alekseev cone criterion

Deciding whether a given system is hyperbolic may seem to be tricky, because it may not be clear how to find the expanding and contracting subspaces in order to verify the required properties. An alternate definition that sidesteps this point, that can be checked with limited accuracy and that is clearly robust under perturbation is the cone criterion. At each point \(x\)it requires the existence of two complementary closed cones (or sectors) \(C^s_\alpha(x)\) and \(C^u_\alpha(x)\) in the tangent space that are strictly invariant in the following sense: There is a \(\gamma\in(0,1)\) such that \[Df(C^u_\alpha(x))\subset C^u_{\gamma\alpha}(f(x))\] and \(Df^{-1}(C^s_\alpha(x))\subset C^s_{\gamma\alpha}(f^{-1}(x))\ .\) Here, the cone \(C^s_\alpha(x)\) is defined to be the set of vectors in the tangent space at \(x\) that make an angle less than \(\alpha\) with \(E^s\ ,\) and similarly for \(C^u_\alpha(x)\ .\)

Whenever two such cone fields are present one obtains the stable and unstable subspaces by intersecting these cones: \[E^s(x)=\bigcap_{n\in\mathbb{N}}Df^{-n}(C^s_\alpha(f^n(x))\] and \(E^u(x)=\bigcap_{n\in\mathbb{N}}Df^{n}(C^u_\alpha(f^{-n}(x))\ .\) Cones appeared in works by Alekseev, Anosov and Sinai in the late 1960s, and cone techniques were developed and used extensively by Alekseev.

Hyperbolic sets

Up to topological conjugacy every known Anosov diffeomorphism is one of the examples we just described (or a generalization of these to automorphisms of a nilmanifold). This suggests that such diffeomorphisms are rather rare. On the other hand, there are important examples of diffeomorphisms that are hyperbolic on a proper invariant subset of interest. Here, the quintessential example was extracted by Smale from a study of relaxation oscillations due to Cartwright and Littlewood by discerning a geometric picture in horseshoe shape. The Smale horseshoe is the invariant set obtained from an embedding \(f\colon R \to \mathbb{R}^2\) of the plane that sends a rectangle \(R\subset\mathbb{R}^2\) to the plane in such a way that \(R\cap f(R)\) consists of two "horizontal" rectangles \(R_0\) and \(R_1\) and the restriction of \(f\) to the components \(R^i \subset f^{-1}(R)\ ,\) \(i = 0,1\ ,\) of \(f^{-1}(R)\) is a hyperbolic affine map, contracting in the vertical direction and expanding in the horizontal direction.

In general, a hyperbolic set is defined to be a compact invariant set \(\Lambda\) of a diffeomorphism \(f\) such that the tangent space at every \(x\in\Lambda\) admits an invariant splitting that satisfies the contraction and expansion conditions described in (1).

For the analysis of the structure of hyperbolic sets it is often useful to restrict attention to locally maximal hyperbolic sets. These are hyperbolic sets that are the largest invariant set in a small neighborhood of the set. This is the case for the horseshoe - every point for which all iterates lie in the rectangle \(R\) indeed belongs to the horseshoe.

A particular class of hyperbolic sets that has engendered much interest is that of hyperbolic attractors. By this we mean hyperbolic sets that are trapped attracting sets (as defined in the entry attractors), i.e., attractors with an isolating neighborhood, or Lyapunov-stable attractors. This implies that all unstable manifolds (which are introduced below) lie in the attractor. Well-known examples are the Plykin attractor and the Smale-Williams solenoid.

Stable and unstable manifolds

For a hyperbolic dynamical system there are nonlinear counterparts to the expanding and contracting directions in the tangent space. Just as in the case of a hyperbolic fixed point, every point in a compact hyperbolic set comes with a stable manifold and an unstable manifold. These are injectively immersed Euclidean spaces, as smooth as the diffeomorphism, and tangent to the contracting and expanding subspaces, respectively. These are a useful and often indispensable technical tool in the analysis of the dynamics. While these manifolds are smooth, their dependence on the base point is in general only Hölder continuous. (A map \(g\) between metric spaces is said to be Hölder continuous (with exponent \(\alpha\)) if \(d(g(x),g(y))\le Cd(x,y)^\alpha\) for some constant \(C\) and sufficiently small \(d(x,y)\ .\) This condition is meaningful for \(\alpha\le1\ ;\) for \(\alpha=1\) it is also called Lipschitz continuity.)

Local maximality of a hyperbolic set is equivalent to the property that for every pair of sufficiently close points a suitable small piece of unstable manifold of one of them intersects a corresponding piece of the stable manifold of the other in a unique point of the hyperbolic set (local product structure).

Hyperbolic sets and Anosov diffeomorphisms provide the two principal classes of Axiom A diffeomorphisms - diffeomorphisms for which periodic points are dense in the nonwandering set, which is hyperbolic.

Flows

Definition. Suppose \(M\) is a manifold, \(\varphi^t\colon M\to M\) is a flow. We say that \(\varphi^t\) is uniformly hyperbolic or an Anosov flow if for every \(x\in M\) there is a splitting of the tangent space \(T_xM=E^s(x)\oplus E^0(x)\oplus E^u(x)\ ,\) where \(E^0=\langle\dot\varphi^t\rangle\) is the flow direction and there are constants \(C>0\) and \(\lambda\in(0,1)\) such that for every \(t>0\) one has \[\tag{2} \|D\varphi^t(v)\|\le C\lambda^t\|v\|\]

for \(v\in E^s(x)\) and \(\|\varphi^{-t}(v)\|\le C\lambda^t\|v\|\) for \(v\in
E^u(x)\ .\) 

Examples of hyperbolic flows can be obtained from diffeomorphisms by a construction that leads to so-called special flows. Given a hyperbolic diffeomorphism \(f\) on a space \(M\) the special flow over \(f\) and under a roof function \(\varphi\) is defined by unit-speed upward motion on \(\{(x,t)\mid x\in M,0\le t\le\varphi(x)\}\ .\) An orbit that reaches the "roof" at a point \((x,\varphi(x))\) continues its upward motion from the point \((f(x),0)\) at the bottom. The resulting flow inherits from the diffeomorphism the expanding and contracting directions in \(M\) and features in addition the flow direction. An important special case is given by suspension flows which are special flows under the function \(\varphi=1\ .\)

Suspension flows are never topologically mixing because at integer times the image of \(M\times(0,1/2)\) is disjoint from \(M\times(1/2,1)\ .\) On the other hand, for a generic roof function, the corresponding special flow over a topologically mixing Anosov flow is itself topologically mixing.

Hyperbolic flows also arise directly. The central example of Anosov flows is provided by a mechanical system, namely free-particle motion (i.e., the geodesic flow) on a compact surface of negative curvature. Locally, such surfaces look like a mountain-pass landscape or the inner rim of a donut. This provides opposing curvatures, in contrast to a spheroid surface, and the effect on free-particle motion is that nearby trajectories quickly diverge from each other. Accordingly, such flows are Anosov flows: At every point the phase space can be decomposed into contracting and expanding directions plus a 1-dimensional flow direction. The rates of contration and expansion are related to the curvature. More generally, the same reasoning applies to higher-dimensional spaces with negative sectional curvatures. These are spaces for which all small 2-dimensional cross sections have negative curvature as described for surfaces.

Hyperbolicity produces several characteristic features of the orbit structure that reflect the coexistence of highly complicated long-term behavior and sensitive dependence on initial conditions on one hand with overall stability of the orbit structure on the other hand.

Sensitive dependence and expansivity

By sensitive dependence on initial conditions one means the property that there is a positive separation distance such that every point \(x\) has points arbitrarily nearby whose orbits will be separated from that of \(x\) by this distance at some (positive or negative) time. Uniformly hyperbolic dynamical systems have the stronger property of expansivity: There is a universal distance by which any two orbits will be separated at some time. Put the other way around, there is a \(\delta>0\) such that for any points \(x\) and \(y\ ,\) if \(d(f^n(x),f^n(y))\le\delta\) for all \(n\) then \(x=y\ .\)

Shadowing and structural stability

The shadowing property is that a pseudo-orbit or chain is always close to an actual orbit of the system.

Pseudo-orbits. In the discrete-time case, given \(\epsilon\ge0\ ,\) a "pseudo-orbit" or "\(\epsilon\)-orbit" (or chain) is a sequence \((x_i)_{i\in\mathbb{Z}}\) of points such that \(d(x_{i+1},f(x_i))\le\epsilon\) for \(i\in\mathbb{Z}\ .\) A sequence \((x_i)_{i=1}^{n-1}\) of points is said to be a "closed \(\epsilon\)-pseudo-orbit" if \(d(x_{i+1},f(x_i))\le\epsilon\) for \(0\leq i<n\) and \(d(x_0,f(x_{n-1}))\le\epsilon\ .\) For \(\epsilon=0\) an \(\epsilon\)-orbit is just an orbit, and for small \(\epsilon\) one can think of this as a computed orbit with round-off error or as the orbit of a perturbation of the system.

A pseudo-orbit is within "round-off error" \(\delta\) for some \(\delta>0\) that depends only on the system. More precisely, we have the following

Shadowing Lemma. If \(\Lambda\) is a hyperbolic set for a diffeomorphism \(f\) and \(\delta>0\ ,\) then there exists an \(\epsilon>0\) such that for each \(\epsilon\)-pseudo-orbit \((x_i)_{i\in\mathbb{Z}}\) in \(\Lambda\) there is a point \(x\) such that \(d(x_i,f^i(x))<\delta\) for all \(i\in\mathbb{Z}\ .\) If \(\Lambda\) is locally maximal then \(x\in\Lambda\ .\)

A strengthening of the Shadowing Lemma -- the "Shadowing Theorem" -- says that a continuous family of pseudo-orbits for a perturbation of a hyperbolic diffeomorphism is shadowed by a continuous family of genuine orbits for the diffeomorphism itself. This result is closely related to structural stability. In particular, it is a tool for establishing that hyperbolic sets are structurally stable. Given a hyperbolic set \(\Lambda_f\) for a diffeomorphism \(f\) there is an \(\epsilon>0\) such that every \(\epsilon\)-perturbation \(g\)of \(f\) (in the \(C^1\)-topology) has a hyperbolic set \(\Lambda_g\) in an \(\epsilon\)-neighborhood of \(\Lambda_f\) and such that \(f\upharpoonright\Lambda_f\) and \(g\upharpoonright\Lambda_g\) are topologically conjugate. Moreover, the conjugacy can be chosen to move points very little, and is then also unique. It is always Hölder continuous but rarely smooth. For instance, given any \(\alpha\in(0,1)\) there is a symplectic automorphism \(A\) of a torus and a \(C^k\)-neighborhood \(U\) of \(A\) in the space of symplectic diffeomorphisms such that for an open dense subset of \(f\in U\) the conjugacy to \(A\) and its inverse are \(\alpha\)-Hölder only on a set of measure zero.

A consequence of structural stability is that hyperbolic systems constitute an open subset of the space of diffeomorphisms or flows. Accordingly, the list of examples given above can be augmented by including all sufficiently small perturbations of those examples.

Conversely, \(C^1\)-structural stability has been shown to be equivalent to hyperbolicity (more precisely, Axiom A with strong transversality); this is the stability theorem of Smale, Robbin, Robinson, Palis, Mañé, Liao, Hayashi.

Specification and entropy

The Anosov Closing Lemma tells us that a periodic pseudo-orbit is shadowed by a genuine periodic orbit.

Anosov Closing Lemma. If \(\Lambda\) is a hyperbolic set for a diffeomorphism \(f\ ,\) then there exists a \(K>0\) such that given any sufficiently small \(\epsilon>0\) and a closed \(\epsilon\)-pseudo-orbit \((x_i)_{i=0}^{n-1}\) in \(\Lambda\) there is an \(n\)-periodic point \(x\) of \(f\) such that \(d(x_i,f^i(x))<K\epsilon\) whenever \(0\leq i<n\ .\) If \(\Lambda\) is locally maximal then \(x\in\Lambda\ .\)

In fact, this and the Shadowing Lemma are strengthened significantly by Bowen's Specification Theorem, which says that topologically transitive hyperbolic systems have the following Specification Property.

Specification Property. Given any \(\epsilon>0\) there is a relaxation time \(M\in\mathbb{N}\) such that every \(M\)-spaced collection of orbit segments is \(\epsilon\)-shadowed by an actual orbit. More precisely, for points \(x_1,\dots,x_n\) and segment lengths \(k_1,\dots,k_n\in\mathbb{N}\) one can find times \(a_1,\dots,a_n\) such that \(a_{i+1}\le a_i+k_i+M\) and a point \(x\) such that \(d(f^{a_i+j}(x),f^j(x_i))<\epsilon\) whenever \(0\le j\le n_i\ .\) Moreover, one can choose \(x\) to be a periodic point with period no more than \(a_n+k_n+M\ .\) (If the dynamical system is also mixing, then one can also prescribe the exact transition time between the orbit segments.)

Thus, one can prescribe the evolution of a periodic orbit to the extent of specifying a finite collection of arbitrarily long orbit segments and any fixed precision: As long as one allows for enough time between the specified segments one can find a periodic orbit approximating this itinerary. Note that the time between the segments depends only on the quality of the approximation and not on the length of the specified segments. The Specification Theorem is a tool of great utility and importance in the study of both the topological structure of hyperbolic sets and the statistical properties of orbits within such sets. Indeed, many of these properties hold for any expansive system with the Specification Property.

One application of the Specification Property is that for any expansive homeomorphism there are constants \(c_1\) and \(c_2\) such that the number \(P(n)\) of periodic points up to period \(n\) satisfies \[\tag{3} c_1\cdot e^{nh}\leq P(n)\leq c_2\cdot e^{nh}\ ,\]


where \(h\) is the topological entropy. This gives rather tight bounds on the growth rate of periodic points and shows that hyperbolic dynamical systems have an abundance of periodic points. Indeed, if \(h=0\) then this result implies that there are only finitely many periodic points, and the Specification Property then tells us that the space is a single point. Consequently, every nontrivial hyperbolic set has positive topological entropy and the number of periodic points grows at a precisely known rate. (This can be refined further by using the Bowen-Margulis measure.)

For a diffeomorphism \(f\) and a real-valued function \(\varphi\) that is a coboundary, i.e., it has the form \(\varphi(x)=h(f(x))-h(x)\) for some function \(h\ ,\) one immediately has \(\sum_{i-0}^{n-1}\varphi(f^i(x))=0\) whenever \(f^n(x)=x\ .\) If \(f\) is hyperbolic then the abundance of periodic points produces a great many constraints of this type. Remarkably, these are all the constraints:

Livshitz Theorem. If \(\Lambda\) is a topologically transitive hyperbolic set for a diffeomorphism \(f\) and \(\varphi\colon\Lambda\to\mathbb{R}\) is an \(\alpha\)-Hölder continuous function such that \(\sum_{i-0}^{n-1}\varphi(f^i(x))=0\) whenever \(f^n(x)=x\ ,\) then there is a continuous function \(h\colon\Lambda\to\mathbb{R}\) such that \(\varphi=h\circ f-h\ .\) This function is unique up to an additive constant, and it is \(\alpha\)-Hölder continuous.

Spectral decomposition

While the overall orbit structure of hyperbolic dynamical systems is quite complex, hyperbolic sets permit a decomposition into somewhat elementary parts as follows. A locally maximal hyperbolic set is a finite union of disjoint closed invariant subsets, each of which is topologically transitive, that is, contains a dense orbit. While these have no interesting closed invariant subsets, there is a further decomposition. Each of these transitive components for a diffeomorphism \(f\) in turn is a finite union of closed subsets \(\Lambda_1,\dots,\Lambda_k\) such that \(f(\Lambda_i)=\Lambda_{i+1}\) when \(0\le i<k\) and \(f(\Lambda_k)=\Lambda_0\) with the property that \(f^k\) is topologically mixing on each of the \(f^k\)-invariant sets \(\Lambda_1,\dots,\Lambda_k\ .\)

A simple corollary of this spectral decomposition is that an Anosov diffeomorphism of a connected manifold is topologically mixing - a connected manifold cannot be decomposed into disjoint closed sets.

Markov partitions

Figure 1: Markov partition for \(\begin{pmatrix}2&1\\1&1\end{pmatrix}\)

An altogether different decomposition of a compact locally maximal hyperbolic set \(\Lambda\) provides a remarkable tool for a fine analysis of the dynamics on this set. This is the construction of a Markov partition, i.e., a suitable finite collection \(R_1,\dots,R_m\) of closed subsets of \(\Lambda\) whose interiors (in the relative topology of the hyperbolic set) are nonempty and pairwise disjoint and such that these sets have the Markov property that if there are points \(x_i\in R_{k_i}\) for \(i\in\{1,\dots,m\}\) such that \(f(x_i)\in R_{k_{i+1}}\) then there is a point \(z\in R_{k_1}\) such that \(f^i(z)\in R_{k_i}\) for \(i\in\{1,\dots,m\}\ .\) The sets \(R_i\) are usually referred to as rectangles because when \(\Lambda=M\) their boundaries consist of pieces of stable and unstable manifolds. In general, the boundaries consist of intersection of pieces of stable and unstable manifolds with \(\Lambda\ ,\) and in dimension higher than 2 the boundary typically has a rather complicated geometry. The Markov condition can be realized in a simple geometric way; it is only necessary to require that the image of each rectangle overlaps the other rectangles in the right way. Specifically, when projected to the unstable direction, the image of a rectangle goes entirely across every rectangle it meets, and when projected to the stable direction the preimage of a rectangle goes entirely across every rectangle it meets. Thus, we can state the Markov property in the following alternate form: if \(x\in R_i\ ,\) \(f(x)\in R_j\) and \(f^{-1}(x)\in R_k\) then \(f(W^s(x)\cap R_i)\subset W^s(f(x))\cap R_j\) and \(f^{-1}(W^u(x)\cap R_i)\subset W^u(f^{-1}(x))\cap R_k\ .\)


The rectangles in a Markov partition may have overlapping boundaries. Away from these one obtains a correspondence between points of the hyperbolic set and sequences whose entries are \(0,\dots,k\ .\) Specifically, each point \(x\) of \(\Lambda\) has an itinerary (or code) \((a_i)_{i\in\mathbb{Z}}\) defined by \(f^i(x)\in R_{a_i}\ .\) The resulting sequence \((a_i)_{i\in\mathbb{Z}}\) clearly satisfies the property that for each \(i\) there is a point in \(R_i\) that is mapped to \(R_{i+1}\ ;\) such a sequence is said to be admissible. Conversely, every admissible sequence \((a_i)_{i\in\mathbb{Z}}\) defines a unique point \(x\in\Lambda\) by taking \(\{x\}=\bigcap_{i\in\mathbb{Z}}f^{-i}(R_{a_i})\ ,\) provided this intersection contains no more than a single point (which is what is meant by "suitable" above). This situation can be described more explicitly as follows. The set of admissible sequences is a topological Markov chain or subshift of finite type, which means that it is described by an \(m\times m\) transition matrix \(A\) whose entries \(a_{ij}\) are 1 or 0 according to whether the symbol \(j\) can follow the symbol \(i\) or not. The set of these sequences is denoted by \(\Sigma_A\) (or by \(\Omega_A\)), and the map \(h\colon\Sigma_A\to\Lambda\) that sends an \(a\in\Sigma_A\) to \(\bigcap_{i\in\mathbb{Z}}f^{-i}(R_{a_i})\) is well-defined and onto (since every point has an itinerary), and, in fact, Hölder continuous if one chooses a natural metric on \(\Sigma_A\ .\)

This factor or coding map also makes it possible to transport invariant measures for the shift with nice ergodic properties to invariant measures for the map that have nice ergodic properties. Also topologically the tools of symbolic dynamics now provide much information about the orbit structure because after suitable adjustments for the boundary overlaps, where this correspondence is ambiguous, the coding provides a means of detecting the presence of orbits with specified properties such as periodicity or density.

Measure-theoretic properties

Since by equation (3) nontrivial hyperbolic sets contain infinitely many (isolated) periodic orbits, there are many invariant measures because each periodic orbit carries an atomic invariant measure (this is defined by setting the measure of any set equal to the percentage of points from this orbit that it contains). Since the space of invariant measures in compact, any weak limit of invariant measures is again an invariant measure. This yields many more invariant measures. For hyperbolic sets, all invariant measures are obtained in this way, and many invariant measures have strong ergodic properties.

Volume-preserving Anosov systems provide a natural illustration of this. Among these are small perturbations of hyperbolic toral automorphisms and of geodesic flows on negatively curved manifolds. Ergodicity of volume follows from an argument due to Hopf (which exploits the fact that a continuous invariant function is constant on stable and unstable manifolds), and ideas such as in the spectral decomposition or the use of Markov partitions establishes that this measure is indeed mixing and constitutes a Bernoulli-system. The same reasoning holds for any smooth invariant measure.

An important and subtle technical issue that underlies the Hopf argument is that the stable and unstable foliations of an Anosov system are absolutely continuous. This means that a measurable subset of the manifold that intersects almost every stable leaf (or almost every unstable leaf) in a set of leaf (Riemannian) measure zero must be a null set. There are examples of foliations that do not have this property ("Fubini's nightmare"), and establishing absolute continuity of the stable and unstable foliations was a fundamental breakthrough by Anosov.

A broad class of invariant measures with an abundance of ergodic properties - known as Gibbs or equilibrium measures - can be constructed by utilizing some methods from statistical physics adapted to the setting of dynamical systems.

The Variational Principle for topological entropy says that for any homeomorphism of a compact metric space the topological entropy is the supremum of measure-theoretic entropies, i.e., \(h(f)=\sup h_\mu(f)\ ,\) where the supremum is taken over all \(f\)-invariant Borel probability measures. If the homeomorphism is also expansive, then this supremum is attained, that is, there is a measure of maximal entropy. If, in addition, the homeomorphism has the Specification Property, then the measure of maximal entropy is unique, and this measure is known as the Bowen-Margulis measure. Careful study of this measure permits great refinements of the asymptotic orbit growth estimate provided by equation (3). In particular, periodic orbits are uniformly distributed with respect to this measure.

Topological entropy can be generalized to topological pressure \(P(\varphi)\) associated with any sufficiently regular function \(\varphi\) on the phase space (such as a Hölder-continuous one). It corresponds to the special case of \(\varphi=0\ ,\) i.e., \(h(f)=P(0)\ .\) The pressure of an invariant measure \(\mu\) is \(P(\mu)=h_\mu(f)+\int\varphi d\mu\ ,\) and there is an analogous variational principle for the pressure\[P(\varphi)=\sup P(\mu)\ ,\] where the supremum is taken over all \(f\)-invariant Borel probability measures. Again, expansivity ensures the existence of a maximizing measure and the Specification Property guarantees uniqueness of this measure, which is called an equilibrium measure. One obtains important examples of equilibrium measures by choosing \(\varphi=\log\hbox{Jac}(df\upharpoonright E^u)\) (in the case of hyperbolic attractors this is the Sinai-Ruelle-Bowen Measure) and \(s\varphi\ ,\) where \(\varphi\) is as before and \(s\) is the root of Bowen's equation \(P(s\varphi)=0\) (this value of \(s\) is the Hausdorff dimension of the hyperbolic set along unstable manifolds).


Here, the definition of a Sinai-Ruelle-Bowen measure is that this is a measure \(\nu\) on a hyperbolic attractor for which there is a set \(B\) of positive Lebesgue measure, called the the basin of attraction of \(\nu\ ,\) such that for every \(x\in B\) and every continuous function \(\varphi\colon M\to\mathbb{R}\) we have \[\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\varphi(f^k(x))=\int_M\varphi d\nu\ .\]

For a uniformly hyperbolic set this property is equivalent to absolute continuity of the conditionals induced by \(\nu\) on unstable leaves.

Nonuniform and partial hyperbolicity

The notion of uniform hyperbolicity can be generalized in several ways. One of these is to retain hyperbolicity without uniformity by assuming that there is a dichotomy of exponential contraction and expansion, but without uniformity of the rates of contraction or expansion. This leads to the theory of nonuniformly hyperbolic dynamical systems, whose assumptions are broad enough to include a wide range of applications. Another generalization is to retain uniformity without hyperbolicity by allowing a center direction in which any expansion or contraction is in a uniform way slower than the expansion and contraction in the unstable and stable subspaces. This leads to the theory of partially hyperbolic systems, which has demonstrated that a limited amount of uniform expansion and contraction is often sufficient to produce ergodicity and topological transitivity.

These two areas are the subject of separate Scholarpedia entries. A further generalization, to nonuniformly partially hyperbolic dynamical systems, is treated in Barreira, Pesin 2007.

Select historical remarks

The beginnings of hyperbolic dynamics go back to Poincaré who perceived the possibility of complex dynamics arising from homoclinic tangles. In the 1890s Hadamard followed up these ideas and studied geodesic flows on surfaces of negative curvature; in this work he noted the presence of salient sets of Cantor type, only a few years after Cantor had constructed these. He also used a representation of orbits that some authors credit as the first step towards symbolic dynamics.

The study of geodesic flows of manifolds of negative curvature continued in the work of Hedlund, Hopf and others through the 1930s with ergodicity as a major objective, motivated by the Boltzmann Ergodic Hypothesis. This thread was not picked up again until Anosov and Sinai were able in the 1960s to overcome the main technical challenge (absolute continuity) to establishing ergodicity of these types of flows, which have since been known as Anosov flows. These developments were contemporaneous with the work of Smale and his school that was motivated by issues of structural stability.

References

  • D. Anosov, Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature. Proc. Steklov Inst. Math., 90 (1969), 1-235
  • L. Barreira, Y. Pesin Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics. Handbook of Dynamical Systems 1B, 57-263, Elsevier North Holland, 2005
  • L. Barreira and Y. Pesin: Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge University Press, 2007
  • B. Hasselblatt, Y. Pesin Partially Hyperbolic Dynamical Systems. Handbook of Dynamical Systems 1B, 1-55, Elsevier North Holland, 2005
  • B. Hasselblatt, Hyperbolic dynamical systems. Handbook of Dynamical Systems 1A, 239-319, Elsevier North Holland, 2002
  • A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995
  • R. Mañe, Ergodic Theory and Differentiable Dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete 8, Springer-Verlag, 1987
  • Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity. Zürich Lectures in Advanced Mathematics, EMS, 2004
  • S. Smale, Differentiable Dynamical Systems. Bulletin of the American Mathematical Society (N.S.) 73 (1967), 747-817
  • J.-C. Yoccoz: Introduction to Hyperbolic Dynamics. Real and complex dynamical systems, 265-291. Proceedings of the NATO Advanced Study Institute held in Hillerød, June 20-July 2, 1993. Edited by Bodil Branner and Paul Hjorth. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 464. Kluwer Academic Publishers, Dordrecht, 1995

Internal references

  • John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
  • Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
  • Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.
  • Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
  • Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
  • Ernest Barreto (2008) Shadowing. Scholarpedia, 3(1):2243.
  • Steve Smale and Michael Shub (2007) Smale horseshoe. Scholarpedia, 2(11):3012.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
  • David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
  • Roy Adler, Tomasz Downarowicz, Michał Misiurewicz (2008) Topological entropy. Scholarpedia, 3(2):2200.

See also

Anosov Diffeomorphism, Dynamical Systems, Entropy, Ergodic Theory, Invariant Measures, Mapping, Nonuniform Hyperbolicity, Normal Hyperbolicity, Partial Hyperbolicity, Shadowing, Topological Dynamics

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