Having spent some time discussing spectral methods and coupling techniques as tools for studying the statistical properties of dynamical systems, we turn now to a third approach, based on convex cones and the Hilbert metric. This post is based on Will Ott’s talk from March 25.
1. Basic definitions
Let be a vector space over the reals. Ultimately we will be most interested in the case when is a function space, such as or , but for now we make the definitions in the general context.
Definition 1 A subset is a convex cone (or positive cone) if
- for each ;
- is convex; and
- for all and , we have the following property: if and for every , then .
The first three conditions are very geometric and in some sense guarantee that “looks like a cone should look”. The last condition is more topological; if is a topological vector space and is a closed subset of , then this condition holds, but we stress that the condition itself is actually weaker than this and is phrased without reference to any topology on .
Example 1 Let be the space of all real-valued functions on the unit interval with bounded variation, and let . Then is a convex cone.
We see immediately from this example that the notion of convex cone is relevant to the sorts of questions we want to ask about invariant measures of a dynamical system, because this set is exactly the set of density functions that arises when we are searching for an absolutely continuous invariant measure.
This suggests that we will ultimately want to consider the action of some operator , and in particular may want to find a fixed point of this action (for a suitable operator ). One of the most powerful methods for finding a fixed point is to find a metric in which acts as a contraction, and this is accomplished by the Hilbert metric, which we now introduce.
The distance is also called the Hilbert (projective) metric.
Several remarks are now in order. First we observe that although may be infinite-dimensional, the distance is completely determined in terms of the two-dimensional subspace spanned by and , and in particular by the points shown in Figure 1 — in the figure, the lines and are the boundary of this two-dimensional cross-section of . The lines and are parallel, as are the lines and ; then we have
An alternate description of is available in terms of this more geometric description. Let be the line through and , and let be the points where this line intersects the boundary of . We see from Figure 1 that the triangles and are similar, so
Furthermore, and are similar, so
Thus can be given in terms of the cross-ratio of the points :
Note that it is possible that the line does not intersect the boundary of twice; this corresponds to the case when either or (or both) in (1), and in this case .
This situation occurs, for example, when we take and as in the example above, and consider with disjoint supports — that is, for all . In this case and so the cone distance between and is infinite.
Because of this phenomenon, is not a true metric. Moreover, we observe that is projective: for every .
An important property of the Hilbert metric is the following theorem, due to Birkhoff, which states that a linear map from one convex cone to another is a contraction whenever its image has finite diameter.
Theorem 3 Let and be convex cones, and let be a linear map such that . (This is a sort of `positivity’ condition.) Let
where we use the convention that .
We also want to relate to a more familiar norm. Say that a norm on is adapted if the following is true: whenever are such that and , we have .
Example 2 On , the norm is adapted, but the BV norm is not.
The following lemma, due to Liverani, Saussol, and Vaienti, relates the cone metric to an adapted norm.
Convex cones and the Hilbert metric are well suited to studying nonequilibrium open systems. Consider the following setting. Let be a Riemannian manifold, volume on , and a diffeomorphism. For , let . This is a nonequilibrium closed system. (Nonequilibrium because the map changes at each time step, closed because every point can be iterated arbitrarily many times.)
Now consider sets , which we interpret as a “hole” at time . The time- survivor set is
the set of points that do not fall into a hole before time . Let . We refer to the pair as a nonequilibrium open dynamical system.
We would like an analogue of decay of correlations for such systems. Let be two probability density functions on , and evolve these under . We expect that because there is a positive probability of falling into a hole.
Let be the Perron–Frobenius operator for the closed system (with respect to ). Then to the open system we can associate the operator
Definition 5 We say that exhibits conditional memory loss in the statistical sense if for all suitably chosen , we have
The idea of this definition is that before comparing the probabilities, we need to first condition on the event that the trajectory survives. Next time we will investigate this property for piecewise expanding interval maps using the Lasota–Yorke inequality, where the holes are small and vary slowly.