An elementary way to rigorously estimate convergence to equilibrium and escape rates

We show an elementary method to have (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rate for systems satisfying a Lasota Yorke inequality. The bounds are deduced by the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiment showing the approach and some concrete result.


Introduction
The evolution of a chaotic system is unpredictable and difficult to describe, but its statistical properties are sometime predictable and (reasonably) simple to be described. Many of these properties are related to its associated transfer operator, essentially via its spectrum. This is a linear operator representing the action of the dynamics on measures space. Let us consider the set SM (X) of finite signed Borel measures on the metric space X. A Borel map T : X → X naturally induces a linear operator L T : SM (X) → SM (X) called the transfer operator, defined as follows. If µ ∈ SM (X) then L T [µ] ∈ SM (X) is the measure such that L T [µ](A) = µ(T −1 (A)).
Sometimes, when no confusion arises, we simply denote L T by L.
In this paper we address two important statistical features of the dynamics: the rate of convergence to equilibrium and the escape rate in open dynamical systems.
Convergence to equilibrium (see Section 2) is a quantitative estimation of the speed in which a starting, absolutely continuous measure approaches the physical invariant measure. This is related to the spectrum of the transfer operator, since the speed is exponential in the presence of a spectral gap, and to decay of correlations. Indeed, an upper bound on the decay of correlations can be obtained from convergence to equilibrium estimation in a large class of cases, see [1]). An estimation for these rates is a key step to deduce many other consequences: central limit theorem, hitting times, recurrence rate... (see e.g. [8,6,13,21]).
The escape rate refers to open systems, where the phase space has a "hole" (see Section 3) and one wants to understand quantitatively, the speed of loss of mass of the system trough the hole. This is related to the spectral radius of a truncated transfer operator and to the presence of metastable states (see e.g. the book [2] for an introduction).
We will present a method which allows to obtain an efficient, effective and quite elementary finite time and asymptotic upper estimations for these decay rates. The strategy is applicable under two main assumptions on the transfer operator.
• the transfer operator is regularizing on a suitable space; it satisfies a Lasota Yorke inequality (see Equation 2); • the transfer operator can be approximated in a satisfactory way by a finite dimensional one. (see Equation 3) The first item in some sense describes the small scale behavior of the system. The regularizing action implies that at a small scale we see a kind of uniform behavior. The macroscopic behavior is then described by a "finite resolution" approximation of the transfer operator. This will be represented by suitable matrices, as the transfer operator is linear, and its main features will be computable from the coefficients of the matrix.
In the following we will enter in the details of how this strategy can be implemented in general, and in some particular system for which we will present some experiment rigorously implemented by interval arithmetics.
We say that the probability measure preserving transformation (X, T, µ) has convergence to equilibrium with speed Φ with respect to these norms if for any Borel probability measure ν on X This speed can be also estimated by the rate of convergence to 0 of ||L n ξ|| w for signed measures ξ ∈ S such that ξ(X) = 0 or by estimating the decay rate of correlation integrals like | f • T n g dµ − f dµ g dµ| for observables f, g in suitable function spaces.
It is important both to have a certified quantitative estimation for this convergence at a given time (numerical purposes, rigorous computation of the invariant measure as in [14], [15], [17], see also Remark 4), or an estimation for its asymptotic speed of convergence (computer assisted proofs of the speed of decay of correlations and its statistical consequences).
In the literature, the problem of computing rigorous bounds on the decay of correlation and convergence to equilibrium rate of a given system was approached by spectral stability results (see e.g. [20]). The use of these methods is limited by the complexity of the assumptions and of the a priori estimations which are needed.
In the following we show how the Lasota Yorke inequality, which can be established in many systems, coupled with a suitable approximation of the system by a finite dimensional one allows directly to deduce finite time and asymptotic upper bounds on the convergence to equilibrium of the system in a simple and elementary way.
We also remark that condition (3) is natural for approximating operators L δ defined (as it is commonly used), by π δ Lπ δ , where π δ is a projection on a finite dimensional space with suitable properties, see Section 7, where it is shown how to obtain (3) under these assumptions.
We remark that in Equation (4) , the condition is supposed on L n δ which is supposed to be simpler than L n (e.g. a discretization with a grid of size δ represented by a matrix, see Section 5) and its properties might be checked by some feasible computation.
Under the above conditions we can effectively estimate from above the convergence to equilibrium in the system in terms of the matrix Since M is positive, its largest eigenvalue is Let (a, b) be the left eigenvector of the matrix M associated to the eigenvalue ρ, normalized in a way that a + b = 1. The condition ρ < 1 below implies that the powers of M go to zero exponentially fast. Note that the quantities δBC, δn 1 D have a chance to be small when δ is small, but this is not guaranteed since the choice of n 1 depends on δ. If we consider the case of piecewise expanding maps, with L δ being the Ulam approximation of L, this is the case (see [12], Theorem 12 ).
Theorem 2. Under the previous assumptions 2, 3, 4, if ρ < 1 then for any g ∈ V , (i) the iterates of L in1 (g) are bounded by Here indicates the componentwise ≤ relation (both coordinates are less or equal).
(ii) In particular we have Proof. (i) Let us consider g 0 ∈ V and denote g i+1 = L n1 g i . By assumption 2 we have ||L n1 g i || s ≤ Aλ n1 1 ||g i || s + B||g i || w Putting together the above assumptions 3 and 4 we get Compacting these two inequalities into a vector notation, The relation can be used because the matrix is positive. This proves (i) by an immediate induction.
(ii) Let us introduce the (a, b) balanced-norm as ||g|| (a,b) = a||g|| s + b||g|| w . The first assertion gives . From this, the statement follows directly.
(iii) Writing any integer k = in 1 + j with 0 ≤ j < n 1 , by assumption 2 we have . and the conclusion follows by the second assertion (ii).

Remark 3.
We remark that our approach being based on a vector inequality has some similarity with the technique proposed in [15]. The first inequality used is the same in both approaches, the second is different. Our inequality relies on the approximation procedure and is more general.
Remark 4. We also remark that our method allows to bound the strong norm of the iterates of a zero average starting measure. Considering L n (m−µ) where m is a suitable starting measure (Lebesgue measure in many cases) and µ is the invariant one, we can understand how many iterations of m are necessary to arrive at a given small (strong) distance from the invariant one. This, added to a way to simulate iterations of L with small errors in the strong norm, allows in principle the rigorous computation of the invariant measure up to small errors in the strong norm.

Escape rates
A system with a hole is a system where there is a subset H such that when a point falls in it, its dynamics stops there. We consider H to be not part of the set where the dynamics acts. Iterating a measure by the dynamics will lead to loose some of the measure in the hole at each iteration, letting the remaining measure decaying at a certain rate.
In many of such systems a Lasota Yorke and an approximation inequality (Equations 2, 3) can be proved, hence the procedure of the previous section allows to estimate this rate, which is related to the spectral radius of the transfer operator.
Let us hence consider a starting system without hole, with transfer operator L, consider an hole in the set H and let 1 H c be the indicator function of the complement of H. The transfer operator of the system with hole is given by Similarly to the convergence to equilibrium we can say that the escape rate of the system with respect to norms || || s , || || w is faster than Ψ if for each f : We will see in the next proposition, that an iterative procedure as the one of the previous section can be implemented to estimate the escape rate. We will need assumptions similar to the ones listed before. These are natural assumptions for open systems constructed from a system satisfying a Lasota Yorke inequality. We will verify it in some example of piecewise expanding maps with a hole (see Section 5.2).
Theorem 5. With the notations of Theorem 2 let us suppose • the system with hole satisfies for some λ 1 < 1 (10) ∀n ≥ 1, ||L n H f || s ≤ Aλ n 1 ||f || s + B||f || w . • L δ (S) ⊆ S and there is an approximation inequality: ∀g ∈ S (11) ∀n ≥ 1, ||(L n δ − L n H )g|| w ≤ δ(C||g|| s + nD||g|| w ). • Moreover, let us suppose there exists n 1 for which Aλ n1 1 < 1 and λ 2 < 1 such that In particular, as before, we have The proof of the theorem is essentially the same as the one of theorem 2, we remark that the main difference between the two propositions is that now we are looking to the behavior of iterates of L H and L δ on the whole space S and not only on the space of zero average measures V .

The Ulam method
We now give an example of a finite dimensional approximation for the transfer operator which is useful in several cases: the Ulam method.
Let us briefly recall the basic notions. Let us suppose now that X is a manifold with boundary. The space X is discretized by a partition I δ (with k elements) and the transfer operator L is approximated by a finite rank operator L δ defined in the following way: let F δ be the σ−algebra associated to the partition I δ , define the projection: π δ as π δ (f ) = E(f |F δ ), then (13) L δ (f ) := π δ Lπ δ f.
In the literature it is shown from different points of views, that taking finer and finer partitions, in suitable systems, the behavior of this finite dimensional approximation converges in some sense to the behavior of the original system, including the convergence of the spectral picture, see e.g. [11,5,10,20,3,7,3].
We remark that this approximation procedure satisfies the approximation assumption (3) if for example Bounded Variation and L 1 norms are considered, see Lemma 14 . This allows the effective use of this discretization in Theorems 2 and 5 for the sudy of piecewise expanding maps.

Piecewise expanding maps and Lasota Yorke inequalities
Let us consider a class of maps which are locally expanding but they can be discontinuous at some point. We recall some results, showing that these systems satisfy the assumptions needed in our main theorems, in particular the Lasota Yorke inequality.
1 > 2 on the set where it is defined. It is now well known (see e.g. [19]) that this kind of maps have an a.c.i.m. with bounded variation density.

Lasota Yorke inequality.
Let us consider an absolutely continuous measure µ and the following norm this is related to the classical definition of bounded variation: it is straightforward to see that if µ has density f then 1 2||f || ∞ ≤ ||µ|| BV .
The following inequality can be established (see e.g. [12], [21]) for the transfer operator of piecewise expanding maps.

5.2.
Piecewise expanding maps with holes. Let us consider a piecewise expanding map with a hole, which is an interval I. Let us see that if the system without hole satisfies a Lasota Yorke inequality and is sufficiently expanding, we can deduce a Lasota Yorke inequality for the system with an hole. Let indeed consider a starting measure with bounded variation density, f . If the piecewise expanding map without hole satisfies If on a interval I, f ≥ q then consider a function φ ∈ C 1 which is =-1 on the left of the interval and =1 on the right, and increasing inside, then φ ≥ 0 , φ dx = 2 and f φ dx ≥ 2q we can do similarly if f ≤ −q, hence ||µ|| ≥ 2||f ||∞. The existence of such interval I of course cannot be ensured in general. In this case the use of Lebesgue's density theorem allows to find an interval where the same argument applies up to a small error.
then the action of the hole is to multiply Lg by 1 I c hence introducing two new jumps which we can bound by ||Lf || ∞ ≤ 1 2 ||Lf || BV , hence ||L H g|| BV ≤ 2||Lg|| BV and the system will satisfy This is enough to obtain the Lasota Yorke inequality in interesting examples of piecewise expanding maps with holes (see Section 6.3 ).
We recall that considering the Ulam discretization for these systems, the needed approximation inequality follows again, by Lemma 14, and the assumptions of Section 3 applies.
Remark 8. We remark that for our method to be applied no conditions on the size of the hole are needed.

Numerical experiments
6.1. Convergence to equilibrium: Lanford map. In this subsection we estimate the speed of convergence to equilibrium for the map which was investigated in [18]. The map T : [0, 1] → [0, 1] is given by To apply Proposition 7 we have to take into account its second iterate F := T 2 . The data below refers to the map F and to the discretization of its transfer operator, and are some input and outputs of our algorithm. Remark 9. Using the estimates developed in [12], an approximation f δ for the invariant density f can be computed with an explicit bound on the error. The graph of the map and the computed density are drawn in figure 1a and 1b respectively. In the plotted case ||f − f δ || 1 ≤ 0.016.
Please note that since the Lorenz 1-dimensional map does not have bounded derivative, the application of Proposition 7, is not immediate, but the following is true.

Theorem 11 ([14] Theorem 21).
Let T be a 1-dimensional piecewise expanding map, possibly with infinite derivative. Denote by {d i } the set of the discontinuity points of T , increasingly ordered. Fixed a parameter l > 0 let We have that T satisfies a Lasota Yorke inequality (and in particular it is possible to choose l in a way that λ 1 < 1 ).
We apply our strategy to the map F := T 4 , please remark that, once we have the Lasota-Yorke coefficients and the discretization, the approach is the same as in Subsection 6.1. The data below refers to the map F and to the discretization of its transfer operator, and are some input and outputs of our algorithm; to compute the Lasota-Yorke constants we fixed l = 300.  The matrix that corresponds to our data is such that M 0.2915 4049 7.75·10 −8 0.058 . Remark 13. Please note that the Lorenz map we studied has some features that imply a large coefficient B in the Lasota-Yorke inequality therefore leading to a slower convergence to the equilibrium in the ||. || BV norm.
Below, some of the data (input and outputs) of our algorithm. The graph of the map and a non rigorous estimation of the conditionally invariant density are drawn in figure 3a and 3b respectively.