statistical properties of dynamical systems – simulation and abstract computation

Chaos, Solitons & Fractals 45 (2012) 1–14

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos

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Statistical properties of dynamical systems – Simulation andabstract computation

Stefano Galatolo a,⇑, Mathieu Hoyrup b, Cristóbal Rojas c

a Dipartimento di Matematica Applicata, Università di Pisa, Italyb LORIA, INRIA Lorraine, Francec Departamento de Matematica, Universidad Andres Bello, Chile

a r t i c l e i n f o

Article history:Received 24 March 2011Accepted 8 September 2011

0960-0779/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.chaos.2011.09.011

⇑ Corresponding author.E-mail address: [email protected] (S. Galatolo

a b s t r a c t

We survey an area of recent development, relating dynamics to theoretical computer sci-ence. We discuss some aspects of the theoretical simulation and computation of the longterm behavior of dynamical systems. We will focus on the statistical limiting behaviorand invariant measures. We present a general method allowing the algorithmic approxi-mation at any given accuracy of invariant measures. The method can be applied in manyinteresting cases, as we shall explain. On the other hand, we exhibit some examples wherethe algorithmic approximation of invariant measures is not possible. We also explain howit is possible to compute the speed of convergence of ergodic averages (when the system isknown exactly) and how this entails the computation of arbitrarily good approximations ofpoints of the space having typical statistical behaviour (a sort of constructive version of thepointwise ergodic theorem).

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The advent of automated computation led to a series ofgreat successes and achievements in the study of manynatural and social phenomena which are modeled bydynamical systems. The use of computers and simulationsallowed to predict the behavior of many important modelsand, on the other hand, led to the discovery of importantgeneral aspects of dynamics. This motivated the hugeamount of work that was made by hundreds of scientiststo improve practical simulation and computation tech-niques. It also motivated the study of the theoretical limitsof simulation and computation techniques, and the theo-retical understanding of related problems.

In this paper we shall focus on some of these theoreticalaspects related to rigorous computation and simulation of(discrete time) dynamical systems.

The simulation and investigation of dynamical systemsstarted with what we call the naïve approach, in which the

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).

user just implements the dynamics without taking rigor-ous account of numerical errors and rounding. Then theuser simply sees what happens on the screen.

Evidently, the sensitivity to initial conditions and thetypical instability of many interesting systems (to pertur-bations on initial conditions but also on the map generat-ing the dynamics) implies that what is being observed onthe computer screen could be completely unrelated towhat was meant to be simulated. In spite of this, the näiveapproach turns out to work almost unreasonably well inmany situations and it is even today the most commonlyused approach in simulations. The theoretical reasons ofwhy this method works and what are its limits, are inour opinion yet to be understood (e.g. some aspects havebeen investigated in [36,41,29,7,8,34]).

Opposite to the naïve approach, there is the ‘‘absolutelyrigorous’’ approach, which will be the main theme of thispaper. Here the user seeks for an algorithm able to producea description up to any desired precision of the object xwhich is meant to be computed. In case such an algorithmexits, we say that the object x is computable (suitable pre-cise definitions will be given below).

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2 S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14

For example, in this approach the constant e is a com-putable number because there is an algorithm that is ableto produce a rational approximation of e at any given accu-racy (for instance by finding the right m and calculatingPm

11n!

such that the error is smaller than requested).In an ‘‘absolutely rigorous’’ simulation (computation),

both the initial point (or initial distribution) and the tran-sition map are supposed to be computable, and the ques-tion arises if interesting objects related to the dynamics(e.g. invariant sets or invariant measures) can be computedfrom the description of the system (or perhaps by usingsome additional information).

In this paper we shall survey a number of results relatedto these computational aspects, some of which are updatedwith new information. We will see that in many cases theobjects of interest can indeed be computed, but there aresome subtleties, and cases where the interesting objectscannot be computed from the description of the system,or cannot be computed at all. In other words, these resultsprovide theoretical limits to such computations. In partic-ular, we shall recall some examples where this non-com-putability phenomenon appears, in the case of invariantmeasures and Julia sets.

1.1. Computing invariant measures

An important fact motivating the study of the statisticalproperties of dynamical systems is that the pointwise longterm prediction of a chaotic system is not possible,whereas, in many cases, the estimation or forecasting ofaverages and other long term statistical properties is. Thisforecasting procedure, often corresponds in mathematicalterms to the computation of invariant measures or to theestimation of some of their properties – measures containinformation on the statistical behavior of the system (X,T)and on the potential behavior of averages of observablesalong typical trajectories of the system (see Section 3).

An invariant measure is a Borel probability measure lon X such that for each measurable set A it holds thatl(A) = l(T�1(A)). They represent equilibrium states, in thesense that probabilities of events do not change in time.

To rigorously compute an invariant measure means tofind an algorithm which is able to output a description ofthe measure (for example an approximation of the mea-sure made by a combination of delta measures placed on‘‘rational’’ points) up to any prescribed precision.

We remark that once an interesting invariant measureis computed, this information is useful for the computationof other dynamical properties: Lyapunov exponents, entro-py, escape rates, etc. For example, in dimension one, oncean ergodic invariant measure l has been approximated,the Lyapunov exponent kl can be estimated using the for-mula kl ¼

R 10 log2T 0dl, where T0 denotes the derivative of

the map generating the dynamics. In higher dimensions,similar techniques can be applied (see, e.g. [21] for moreexamples of derivation of dynamical quantities from thecomputation of the invariant measure and similar objects).

Before giving more details about the computation ofinvariant measures, we remark that, since there are onlycountably many ‘‘algorithms’’ (computer programs), what-ever we mean by ‘‘approximating a measure by an algo-

rithm’’ would imply that only countable many measurescan be computed. In general, however, a dynamical systemmay have uncountably many invariant measures (usuallyan infinite dimensional set). So, a priori most of them willnot be algorithmically describable. This is not a seriousproblem because we should put our attention on the mostmeaningful ones. An important part of the theory ofdynamical systems is indeed devoted to the understandingof physically relevant invariant measures. Informallyspeaking, these are measures which represent the asymp-totic statistical behavior of many (positive Lebesgue mea-sure) initial conditions (see Section 3 for more details).

The existence and uniqueness of physical measures is awidely studied problem (see [48]), which has been solvedfor some important classes of dynamical systems. Thesemeasures are some of the good candidates to be computed.Other measures of particular interest in certain cases, aswe shall see below, are the maximal entropy ones.

The computation of interesting invariant measures (inmore or less rigorous ways) is the main goal of a significantpart of the literature related to computation in dynamics.An important role here is played by the transfer operatorinduced by the dynamics. Indeed, the map T defining thedynamics, induces a transition map LT on the space ofprobability measures over X, LT:PM(X) ? PM(X). LT is calledthe transfer operator associated to T (definition and basicresults about this are recalled in Section 3). Invariant mea-sures are fixed points of this operator. The main part of themethods which are used to compute invariant measuresdeals with suitable finite dimensional approximations ofthis operator. In Section 3 we will review briefly some ofthese methods, and give some references. We then con-sider the problem from the abstract point of view men-tioned above, and give some general results on rigorouscomputability of the physical invariant measure. In partic-ular we will see that the transfer operator is computableup to any approximation in a general context (see Theo-rem 16). The corresponding invariant measure is comput-able, provided we are able to give a description of aspace of ‘‘regular’’ measures where the physical invariantmeasure is the only invariant one (see Theorem 17 and fol-lowing corollaries).

We will also show how the measure can be obtainedfrom a description of the system in a class of examples(piecewise expanding maps) by using the Lasota–Yorkeinequality.

Rational transformations of the Riemann Sphere are an-other interesting class where our methods can be applied.These complex dynamics are very well studied in the liter-ature, and enormous attention has been addressed to theJulia sets associated to them, which have emerged as themost studied examples of fractal sets generated by adynamical system. One of the reasons for their popularityis the beauty of the computer-generated images of suchsets. The algorithms used to draw these pictures vary;the naïve approach in this case works by iterating the cen-ter of a pixel to determine if it lies in the Julia set. There ex-ists also more sophisticated algorithms (using classicalcomplex analysis, see [42]) which work quite well formany examples, but it is well known that in some particu-lar cases computation time will grow very rapidly with

S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14 3

increase of the resolution. Moreover, there are examples,even in the family of quadratic polynomials, where no sat-isfactory pictures of the Julia set exist.

In the rigorous approach, a set is computable if, roughlyspeaking, its image can be generated by a computer withan arbitrary precision. Under this notion of computability,the question arise if Julia sets are always computable. In aseries of papers [5,4,12,13] it was shown that even thoughin many cases (hyperbolic, parabolic) the Julia set is indeedcomputable, there exists quadratic polynomials which arecomputable (again, in the sense that all the trajectoriescan be approximated by an algorithm at any desired accu-racy), and yet the Julia set is not.

So we cannot simulate the set of limits points on whichthe chaotic dynamics takes place, but, what about the sta-tistical distribution? In fact, it was shown by Brolin andLuybich that there exists a unique invariant measurewhich maximizes entropy, and that this measure is sup-ported on the Julia set. The question of whether this mea-sure can be computed has been recently solved in [6],where it is proved that the Brolin–Lyubich measure is al-ways computable. So that even if we cannot visualize theJulia set as a spatial object, we can approximate its distri-bution at any finite precision. In Section 3.3 we shall brieflyreview the mentioned results on uncomputability of Juliasets, and explain how our method can be applied to showcomputability of the Brolin–Lyubich measure.

After such general statements one could conjecture thatall computable dynamical systems should always have acomputable invariant measure. We will see that, perhapssurprisingly, this is not true. Not all computable systems(the dynamics can be computed up to any prescribedapproximation) have computable invariant measures (seeSection 3.4). The existence of such examples reveals somesubtleties in the computation of invariant measures.

Finally, to further motivate these results, we remarkthat from a technical point of view, computability of theconsidered invariant measure is a requirement in severalresults establishing connections between computation,probability, randomness and pseudo-randomness (see Sec-tion 5 and e.g. [3,23–25])

1.2. Computing the speed of convergence and pseudorandompoints

For several questions in ergodic theory the knowledgeof the speed of convergence to ergodic behavior is impor-tant to deduce other practical consequences. In the compu-tational framework, the question turns out to be theeffective estimation of the speed of convergence in theergodic theorems.1 From the numerical–practical point ofview this has been done in some classes of systems, havinga spectral gap for example. In this case a suitable approxima-tion of the transfer operator allows to compute the rate ofdecay of correlations [22,37] and from this, other rates ofconvergence can be easily deduced.

1 Find a N such that 1n

Pf � Tn differs from

Rf dl less than a given error

for each nP, in some sense.

Other classes of systems could be treated joining theabove spectral approach, with combinatorial constructions(towers, see, e.g. [38]), but the general case need a differentapproach.

In [2] it was shown that much more in general, if thesystem can be described effectively, then the rate of conver-gence in the pointwise ergodic theorem can be effectively esti-mated (not only the asymptotical speed, but explicitestimates on the error).

We give in Section 4 a very short proof of a statement ofthis kind (see Theorem 39) for ergodic systems, and showsome consequences. Among these, a constructive versionof pointwise ergodic theorem. If the system is computable(in some wide sense that will be described), then it is pos-sible to compute points having typical statistical behavior.Such points could hence be called pseudorandom points ofthe system (see Section 5).

Since the computer can only handle computable initialconditions, any simulation can start only from thesepoints. Pseudorandom initial conditions are hence in prin-ciple the good points where to start a simulation.

We remark that it is widely believed that naïve com-puter simulations very often produce correct statisticalbehavior. The evidence is mostly heuristic. Most argu-ments are based on the various ‘‘shadowing’’ results (see,e.g. [33, chapter 18]). In this kind of approach (differentfrom ours), it is possible to prove that in a suitable systemevery pseudo-trajectory (as the ones which are obtained insimulations with some computation error) is close to a realtrajectory of the system. However, even if we know thatwhat we see in a simulation is near to some real trajectory,we do not know if this real trajectory is typical in somesense. Another limitation of this approach is that shadow-ing results hold only in systems having some stronghyperbolicity, while many physically interesting systemsare not like this. In our approach we consider real trajecto-ries instead of ‘‘pseudo’’ ones and we ask if there existscomputable points which behaves as a typical point ofthe dynamics.

2. Computability on metric spaces

To have formal results and precise assumptions on thecomputability (up to any given error) of continuous ob-jects, we first need to introduce some concepts. In particu-lar, we shall introduce recursive versions of open andcompact sets, and characterize the functions which arewell suited to operate with those sets (computable func-tions). In this section, we try to explain this theory as sim-ple and self contained as possible. Other details anddifferent approaches to the subject can be found in theintroductory texts [9,47].

2.1. Computability

The starting point of recursion theory was the introduc-tion of a mathematical definition making precise the intu-itive notions of algorithmic or effective procedure onsymbolic objects. Several different formalizations havebeen independently proposed (by Church, Kleene, Turing,


Post, Markov. . . ) in the 30s. They all turned out to beequivalent: they compute the same functions from N toN. The class of functions thus defined is now called theclass of recursive functions. As an algorithm is allowed torun forever on an input, these functions may be partial,i.e. not defined everywhere. The domain of a recursivefunction is the set of inputs on which the algorithm even-tually halts. A recursive function whose domain is N is saidto be total.

We now recall an important concept from recursiontheory. A set E # N is said to be recursively enumerable(r.e.) if there is a (partial or total) recursive functionu : N! N enumerating E, that is E ¼ fuðnÞ : n 2 Ng. IfE – ;, u can be effectively converted into a total recursivefunction w which enumerates the same set E.

2.2. Algorithms and uniform algorithms

Strictly speaking, recursive functions only work on nat-ural numbers. However, this can be extended to the objects(thought as ‘‘finite’’ objects) of any countable collection,once a numbering of its elements has been chosen. We willuse the word algorithm instead of recursive function whenthe inputs or outputs are interpreted as finite objects.The operative power of algorithms on the objects of sucha numbered set obviously depends on what can be effec-tively recovered from their numbers.

More precisely, let X and Y be numbered sets such thatthe numbering of X is injective (it is then a bijection be-tween N and X). Then any recursive function u : N! N in-duces an algorithm A : X ! Y . The particular case X ¼ N

will be much used.For instance, the set Q of rational numbers can be injec-

tively numbered Q ¼ fq0; q1; . . .g in an effective way: thenumber i of a rational a/b can be computed from a and b,and vice versa. We fix such a numbering: from now onthe rational number with number i will be denoted by qi.

Now, let us consider computability notions on the set R

of real numbers.

Definition 2. Let x be a real number. We say that:

� x is lower semi-computable if the set fi 2 N : qi < xg isr.e.� x is upper semi-computable if the set fi 2 N : qi > xg is

r.e.� x is computable if it is lower and upper semi-

computable.

It is worth noticing that a real number is computable ifand only if there exists an algorithmic enumeration of a se-quence of rational numbers converging exponentially fastto x. That is:

Proposition 3. A real number is computable if there is analgorithm A : N! Q such that jAðnÞ � xj 6 2�n for all n.

We remark that the notion of computable real numberwas already introduced (in a different but equivalentway) by Turing [44].

Uniformity. Algorithms can be used to define comput-ability notions on many classes of mathematical objects.The precise definitions will be particular to each class ofobjects, but they will always follow the following scheme:

An object O is computable if there is an algorithm

A : X ! Y;

which computes O in some way.

Each computability notion comes with a uniform ver-sion. Let ðOiÞi2N be a sequence of computable objects:

Oi is computable uniformly in i if there is an algorithm

A : N� X ! Y

such that for all i; Ai :¼ Aði; �Þ : X ! Y computes Oi.

For instance, the elements of a sequence of real numbersðxiÞi2N are uniformly computable if there is a algorithmA : N�N! Q such that jAði;nÞ � xij 6 2�n for all i, n.

In other words a set of objects is computable uniformlywith respect to some index if they can be computed with a‘‘single’’ algorithm starting from the value of the index.

In each particular case, the computability notion maytake a particular name: computable, recursive, effective,r.e., etc. so the term ‘‘computable’’ used above shall bereplaced.

2.3. Computable metric spaces

A computable metric space is a metric space with anadditional structure allowing to interpret input and outputof algorithms as points of the metric space. This is done inthe following way: there is a dense subset (called idealpoints) such that each point of the set is identified with anatural number. The choice of this set is compatible withthe metric, in the sense that the distance between twosuch points is computable up to any precision by an algo-rithm getting the names of the points as input. Using thesesimple assumptions many constructions on metric spacescan be implemented by algorithms.

Definition 4. A computable metric space (CMS) is a tripleX ¼ ðX; d; SÞ, where

(i) (X,d) is a separable metric space.(ii) S ¼ fsigi2N is a dense, numbered, subset of X called

the set of ideal points.(iii) The distances between ideal points d(si,sj) are all

computable, uniformly in i, j (there is an algorithmA : N3 ! Q such that jAði; j;nÞ � dðsi; sjÞj < 2�n).

Point (iii) says that the information that can be recov-ered from the numbers of the numbered set S is their mu-tual distances.

The algorithms involved in the following definitions (upto Definition 9) are assumed to be total.

Definition 5. We say that in a metric space (X,d), asequence of points ðxnÞn2N converges recursively to a pointx if there is an algorithm D : Q! N such that d(xn,x) 6 �for all n P D(�).


Definition 6. A point x 2 X is said to be computable ifthere is an algorithm A : N! S such that ðAðnÞÞn2N con-verges recursively to x.

We define the set of ideal balls to be B :¼fBðsi; qjÞ : si 2 S;0 < qj 2 Qg where B(x,r) = {y 2 X : d(x,y) <r} is an open ball. We fix a numbering B ¼ fB0;B1; . . .gwhich makes the number of a ball effectively computablefrom its center and radius and vice versa. B is a countablebasis of the topology.

Definition 7 (Effective open sets). We say that an open setU is effective if there is an algorithm A : N! B such thatU ¼

SnAðnÞ.

Observe that an algorithm which diverges on each inputn enumerates the empty set, which is then an effectiveopen set. Sequences of uniformly effective open sets arenaturally defined. Moreover, if ðUiÞi2N is a sequence of uni-formly effective open sets, then

SiUi is an effective open

set.

Definition 8 (Effective Gd-set). An effective Gd-set is anintersection of a sequence of uniformly effective open sets.

Obviously, an uniform intersection of effective Gd-sets isalso an effective Gd-set.

Let X; SX ¼ fsX1 ; s

X2 ; . . .g; dX

� �and Y ; SY ¼ fsY

1 ; sY2 ; . . .g; dY

� �be computable metric spaces. Let also BX

i and BYi be enu-

merations of the ideal balls in X and Y. A computable func-tion X ? Y is a function whose behavior can be computedby an algorithm up to any precision. For this it is sufficientthat the pre-image of each ideal ball can be effectively enu-merated by an algorithm.

Definition 9 (Computable functions). A function T : X ? Yis computable if T�1ðBY

i Þ is an effective open set, uniformlyin i. That is, there is an algorithm A : N�N! BX such thatT�1ðBY

i Þ ¼S

nAði;nÞ for all i.A function T : X ? Y is computable on D # X if there

are uniformly effective open sets Ui such thatT�1ðBY

i Þ \ D ¼ Ui \ D.

Remark 10. Intuitively, a function T is computable (onsome domain D) if there is a computer program whichcomputes T(x) (for x 2 D) in the following sense: on input� > 0, the program, along its run, asks the user for approx-imations of x, and eventually halts and outputs an idealpoint s 2 Y satisfying d(T(x),s) < �. This idea can be formal-ized, using for example the notion of oracle computation.The resulting notion coincides with the one given in theprevious definitions.

Recursive compactness is an assumption which will beneeded in the following. Roughly, a compact set is recur-sively compact if the fact that it is covered by a finite col-lection of ideal balls can be tested algorithmically (forequivalence with the �-net approach and other propertiesof recursively compact set see [26]).

Definition 11. A set K # X is recursively compact if it iscompact and there is a recursive function u : N� ! N suchthat u(i1, . . . , ip) halts if and only if ðBi1 ; . . . ;Bip Þ is a coveringof K.

3. Computability of invariant measures

3.1. Invariant measures and statistical properties

Let X be a metric space, T : X ? X a Borel measurablemap and l a T-invariant Borel probability measure. A setA is called T -invariant if T�1(A) = A (mod 0). The system(X,T,l) is said to be ergodic if each T-invariant set has totalor null measure. In such systems the famous Birkhoff ergo-dic theorem says that time averages computed along l-typical orbits coincides with space average with respectto l. More precisely, for any f 2 L1(X,l) it holds

limn!1

SfnðxÞn¼Z

f dl; ð1Þ

for l almost each x, where Sfn ¼ f þ f � T þ � � � þ f � Tn�1.

This shows that in an ergodic system, the statisticalbehavior of observables, under typical realizations of thesystem is given by the average of the observable madewith the invariant measure.

We say that a point x belongs to the basin of attraction ofan invariant measurel if (1) holds at x for each bounded con-tinuous f. In case X is a manifold (possibly with boundary), aphysical measure is an invariant measure whose basin ofattraction has positive Lebesgue measure (for more detailsand a general survey see [48]). Computation of such mea-sures will be the main subject of the first part of this section.

3.1.1. The transfer operatorA function T between metric spaces naturally induces a

function LT between probability measure spaces. This func-tion LT is linear and is called transfer operator (associatedto T). Measures which are invariant for T are fixed pointsof LT.

Let us consider a computable metric space X and a mea-surable function T : X ? X. Let us also consider the spacePM(X) of Borel probability measures on X.

Let us define the linear function LT : PM(X) ? PM(X) byduality in the following way: if l 2 PM(X) then LT(l) is suchthatZ

f dLTðlÞ ¼Z

f � T dl

for each bounded continuous f.The computation of invariant measures (and many

other dynamical quantities) very often is done by comput-ing the fixed points (and some other spectral information)of this operator in a suitable function space. The most ap-plied and studied strategy is to find a suitable finite dimen-sional approximation of LT (restricted to a suitable functionspace) so reducing the problem to the computation of thecorresponding relevant eigenvectors of a finite matrix.

An example of this is done by discretizing the space Xby a partition Ai and replacing the system by a (finite state)Markov Chain with transition probabilities

Pij ¼mðAi \ AjÞ

mðAiÞ;

where m is the Lebesgue measure on X (see, e.g.[20,21,37]). Then, taking finer and finer partitions it is pos-


sible to obtain in some cases that the finite dimensionalmodel will converge to the real one (and its natural invari-ant measure to the physical measure of the original sys-tem). In some cases there is an estimation for this speedof convergence (see, e.g. [21] for a discussion), but a rigor-ous bound on the error (and then a real rigorous computa-tion) is known only in a few cases (piecewise expanding orexpanding maps, see [37,28]).

Similar approaches consists of applying a kind of Fae-do–Galerkin approximation to the transfer operator byconsidering a complete Hilbert base of the function spaceand truncating the operator to the action on the first ele-ments (see [46]).

Another approach is to consider a perturbation of thesystem by a small noise. The resulting transfer operatorhas a kernel. This operator canthen be approximated by afinite dimensional one (again by the Faedo–Galerkin meth-od for instance) and the relevant eigenvectors can be thuscalculated (see, e.g. [17,16]). This method is useful in caseswhere it is possible to prove that the physical measure ofthe original system can be obtained as the limit whenthe size of the noise tends to zero (this happens, for exam-ple, for uniformly hyperbolic systems).

Variations on the method of partitions are given in[18,19], while in [40] a different method, fastly converging,based on periodic points is exploited for piecewise analyticMarkov maps. Another strategy to face the problem ofcomputation of invariant measures consist in followingthe way the measure l can be constructed and check thateach step can be realized in an effective way. In some inter-esting examples we can obtain the physical measure aslimit of iterates of the Lebesgue measurel ¼ limn!1Ln

TðmÞ (recall that m is the Lebesgue measure).To prove computability of l the main point is to explicitlyestimate the speed of convergence to the limit. This some-times can be done using techniques related to decay of cor-relations [24].

Concluding, if the goal is to rigorously compute aninvariant measure, most of the results in modern litera-ture are partial. Indeed, besides of being applied to re-stricted classes of systems, they usually need additionalinformation in order to compute the measure. For exam-ple, the calculation of the finite dimensional approxima-tions is often not done Turing-rigorously, or the rate ofconvergence of the approximations is computed up tosome constants (depending on the system) which arenot estimated.

In the remaining part of this section we present somegeneral results, mainly from [26,6], and explain how theycan be applied to rigorously compute invariant measures.These results have the advantage of being simple and quitegeneral with all the needed assumptions made explicit. Onthe other hand, they are not well suited for a complexity(in time or space) analysis, so it is not clear if they can beimplemented and used in practice.

The rigorous framework in which they are proved, how-ever, allows to see them as a study about the theoreticallimits of (Turing-)rigorous computation of invariant mea-sures and, in fact, also negative results can be obtained.In particular, we present examples of computable systemshaving no computable invariant measures.

3.2. Computability of measures

In this section we explain precisely what we mean bycomputing a measure, namely to have an algorithm ableto approximate the measure by ‘‘simple measures’’ up toany desired accuracy.

Let us consider the space PM(X) of Borel probabilitymeasures over X. Let C0(X) be the set of real-valuedbounded continuous functions on X. We recall the notionof weak convergence of measures:

Definition 12. ln is said to be weakly convergent to l ifRf dln !

Rf dl for each f 2 C0(X).

Let us introduce the Wasserstein–Kantorovich distancebetween measures. Let l1 and l2 be two probability mea-sures on X and consider:

W1ðl1;l2Þ ¼ supf21�LipðXÞ

Zf dl1 �

Zf dl2

��;

where 1-Lip (X) is the space of functions on X having Lips-chitz constant less than one. The distance W1 has the fol-lowing useful properties:

Proposition 13 (see [1] Prop 7.1.5).

1. W1 is a distance and if X is bounded, separable and com-plete, then PM(X) with this distance is a separable andcomplete metric space.

2. If X is bounded, a sequence is convergent in the W1 metricif and only if it is convergent for the weak topology.

3. If X is compact then PM(X) is compact with this metric.

Item (1) has an effective version: PM(X) inherits thecomputable metric structure of X. Indeed, given the set SX

of ideal points of X we can naturally define a set of idealpoints SPM(X) in PM(X) by considering finite rational convexcombinations of the Dirac measures ds supported on idealpoints s 2 SX. This is a dense subset of PM(X). The proof ofthe following proposition can be found in [31].

Proposition 14. If X is bounded, then (PM(X),W1,SPM(X)) is acomputable metric space.

A measure l is then computable if there is a sequenceln 2 SPM(X) converging recursively fast to l in the W1 metric(and hence in the weak topology).

3.3. Computable invariant ‘‘regular’’ measures

Here we describe a general method to compute invari-ant measures, that can be applied in several situations.We start by illustrating the method in its simplest form,namely to show that.

Proposition 15. If a computable system T : X ? X is uniquelyergodic, then its invariant measure is computable.

Proof. The method is based on the following very simpleand well-known fact: the singleton set {x0} is recursivelycompact if and only if its unique element x0 is a computable


point. The idea is then to show that the set {l} � PM(X),where l is the unique invariant measure, is recursivelycompact. This is done in the following way: first we observethat the set of probability measures PM(X) is recursivelycompact (since X is). Second we observe that the set PMT(X)c

of probability measures which are NOT invariant under T isrecursively open (here it is essential that T is an everywherecomputable map). Finally we use the general fact that arecursively compact set minus a recursively open set isalways recursively compact. In our case: PM(X) � PMT(X)c ={l} is recursively compact, as was to be shown. h

The obstructions to extend this idea to more generalsituations come from two main facts: (i) when the mapT is not everywhere computable and then the set PMT(X)of invariant measures is not (recursively) compact any-more and (ii) when the map T is not uniquely ergodic,the set PMT(X) is not a singleton, and it therefore doesnot need to contain computable points (even if recur-sively compact).

To overcome these difficulties, we essentially need tofind a different recursively compact condition whichwould allow us to separate a distinguished invariant mea-sure from the rest, while making use of T only on its do-main of computability.

In the following, we present a general result which setsthe requirements in order to accomplish this task. It will beconvenient to spell the principle in a more equation solv-ing way, namely as an abstract theorem allowing the com-putation of isolated fixed points (see [26] Corollary 2.4.3)of maps computable on a suitable subset.

This result will be then applied to the transfer operator,and invariant measures will be found as fixed points thatcan be somehow distinguished by means of analytic ordynamical properties (as being physical or absolutely con-tinuous, for example).

It will be therefore necessary to consider a space wherethe transfer operator is computable (this space hopefullywill contain the distinguished invariant measure).

We remark that if T is not continuous then LT is not nec-essarily continuous (this can be realized by applying LT tosome delta measure placed near a discontinuity point)and hence not computable. Still, we have that LT is contin-uous (and its modulus of continuity is computable) at allmeasures l which are ‘‘far enough’’ from the discontinuityset D. This is technically expressed by the conditionl(D) = 0.

2 If the invariant measure is unique in V the isolating ball is notnecessary.

Proposition 16. Let X be a computable metric space and T :X ? X be a function which is computable on XnD. Then LT iscomputable on the set of measures

PMDðXÞ :¼ l 2 PMðXÞ : lðDÞ ¼ 0f g: ð2Þ

From a practical point of view, this proposition providessufficient conditions to rigorously approximate the trans-fer operator by an algorithm.

The above tools allow us to ensure the computability ofLT on a large class of measures and obtain in a way similarto Proposition 15:

Theorem 17 [26], Theorem 3.2. Let X be a computablemetric space and T a function which is computable on XnD.Suppose there is a recursively compact set of probabilitymeasures V � PM(X) such that for every l 2 V, l(D) = 0 holds.Then every invariant measure isolated in V is computable.Moreover the theorem is uniform: there is an algorithm whichtakes as inputs finite descriptions of T, V and an ideal ball inM(X) which isolates2 an invariant measure l, and outputs afinite description of l.

Remark 18. Notice that there is no computability condi-tion on the set D. We also remark that unless T is uniquelyergodic, invariant measures are never isolated in PM(X), sothe task is to find a (recursively compact) condition thatonly a single invariant measure would satisfy.

Clearly, Proposition 15 is now a trivial corollary of The-orem 17. As already said, the main difficulty in the applica-tion of Theorem 17 is the requirement that the invariantmeasure we are trying to compute be isolated in V. In gen-eral the space of invariant measures of a given dynamicalsystem could be very large (an infinite dimensional convexsubset of PM(X)) but there is often some kind of particularregularity that can be exploited to characterize a particularinvariant measure, isolating it from the rest. For example,let us consider the following seminorm:

klka ¼ supx2X;r>0

lðBðx; rÞÞra :

If a and K are computable and X is recursively compactthen

Va;K ¼ l 2 PMðXÞ : klka 6 K� �

ð3Þ

is recursively compact [26]. This implies

Proposition 19. Let X be recursively compact and T becomputable on XnD, with dimH(D) <1. Then any invariantmeasure isolated in Va,K with a > dimH(D) is computable. Onceagain, this is uniform in T, a, K.

We recall that in the above proposition dimH denotesthe Hausdorff dimension of a set. The above general prop-osition allow us to obtain as a corollary the computabilityof many systems having absolutely continuous invariantmeasures. For example, let us consider maps on theinterval.

Proposition 20. Let X = [0,1], T be computable on XnD, withdimH(D) < 1 and suppose (X,T) has a unique absolutelycontinuous invariant measure l with bounded density, thenl is computable (starting from T and a bound for the L1 normof the invariant density).

Similar results hold for maps on manifolds (again see[26]).

3.3.1. Computing the measure from a description of the systemin the class of piecewise expanding maps

As it is well known, interesting examples of systemshaving a unique absolutely continuous invariant measure


(with bounded density as required) are topologically tran-sitive piecewise expanding maps on the interval or expand-ing maps on manifolds.

We show how to find a bound for the invariant density onpiecewise expanding maps. This implies that the invariantmeasure can be calculated starting from a description of T.

Definition 21. A nonsingular function T :([0,1],m) ? ([0,1],m) is said to be piecewise expanding if3

1. There is a finite set of points d1 = 0, d2, . . . ,dn = 1 suchthat Tjðdi ;diþ1Þ is C2 and can be extended to a C2 map on[di,di+1].

2. inf (T0) > 1 on the set where it is defined.3. T is topologically mixing.4

It is now well known that such maps have a uniqueergodic invariant measure with bounded variation density.

Such a density is also the unique fixed point of the (Per-ron Frobenius) operator5 L : L1[0,1] ? L1[0,1] defined by

½Lf ðxÞ ¼X

y2T�1x

f ðyÞT 0ðyÞ

:

We now show how to find a bound for such a density,starting from the description of the system, and then com-pute the associated invariant measure.

The following proposition was proved in the celebratedpaper [35] (Theorem 1 and its proof) and it is now calledLasota–Yorke inequality. We give a precise statementwhere the constants are explicited.

Proposition 22. Let T be a piecewise expanding map. Let f beof bounded variation in [0,1] and let denote its variation byVar(f). Let d1, . . . , dn be the discontinuity points of T.

If k ¼ infx2½0;1�fd1 ;...;dngT0ðxÞ. Then

VarðLf Þ 6 2kVarðf Þ þ Bkfk1;

where

B ¼supx2½0;1�fd1 ;...;dng

T 00 ðxÞðT 0 ðxÞÞ2

�� infx2½0;1�fd1 ;...;dng

1T 0 ðxÞ

�� þ 2min di � diþ1ð Þ :

The following is an elementary fact about the behaviorof real sequences.

Lemma 23. If a real sequence an is such that an+1 6 lan + k forsome l < 1, k > 0, then

supðanÞ 6max a0;k

1� l

:

3 For the sake of simplicity we will consider the simplest setting in whichwe can work and give precise estimations. Such a class was generalized inseveral ways. We then warn the reader that in the current literature, bypiecewise expanding maps it is meant something slightly more generalthan the definition we give here.

4 A system is said to be topologically mixing if, given sets A and B, thereexists an integer N, such that, for all n > N, one has fn(A) \ B – ;.

5 Note that this operator corresponds to the above cited transfer operatoracting on densities instead of measures.

Proposition 24. If f is the density of the physical measure ofa piecewise expanding map T as above and k > 2. Then

Varðf Þ 6 B1� 2k

;

where B is defined as above.

Proof (Sketch). The topological mixing assumptionimplies that the map has only one invariant physical mea-sures (see [45]). Let us use the above results iterating theconstant density corresponding to the Lebesgue measure.Proposition 22 and Lemma 23 give that the variation ofthe iterates is bounded by B

1�2k. Suppose that the limit mea-sure has density f. By compactness of uniformly boundedvariation functions in L1, the above properties give a boundon the variation of f (see [35] proof of Theorem 1). h

The following is a trivial consequence of the fact that

kfk1 6 Varðf Þ þZ

fdl:

Corollary 25. In the above situation kfk1 6 B1�2kþ 1.

The bound on the density of the invariant measure, to-gether with Corollary 20 gives the following uniform resulton the computation of invariant measures of such maps.

Theorem 26. Suppose a piecewise expanding map T and itsfirst and second derivatives are computable on[0,1] � {d1, . . . , dn}. Suppose that also its extension to theclosed intervals [di,di+1] are computable. Then, the physicalmeasure can be computed starting from a description of thesystem (the points di and the programs computing the mapand its derivatives).

Proof (Sketch). Since T and T0 are computable on eachinterval [di,di+1] we can compute a number k such that1 < k6 inf (T0) (see [26] Proposition 3) If we consider theiterate TN instead of T the associated invariant density willbe the same as the one of T, and if kN > 2, then TN will satisfyall the assumptions needed on Proposition 24. In the sameway it is possible to compute an upper bound for the related

B1�2k. Then we have a bound on the density and we can applyCorollary 20 to compute the invariant measure. h

3.3.2. Unbounded densities and non uniformly hyperbolicmaps

The above results ensure computability of some abso-lutely continuous invariant measure with bounded density.If we are interested in situations where the density is un-bounded, we can consider a new norm, ‘‘killing’’ singularities.

Let us hence consider a computable function f : X ! R

and

klkf ;a ¼ supx2X;r>0

f ðxÞlðBðx; rÞÞra :

If a and K are computable and X is recursively compactthen


Va;K ¼ l 2 PMðXÞ : klkf ;a 6 Kn o

ð4Þ

is recursively compact, and Proposition 19 also hold for theseminorm k�kf,a. If f is such that f(x) = 0 whenlimr!0

lðBðx;rÞÞra ¼ 1 this can let the norm be finite when the

density diverges.As an example, where this can be applied, let us con-

sider the Manneville Pomeau type maps on the unit inter-val. These are maps of the type x ? x + xz(mod1).

When 1 < z < 2 the dynamics has a unique absolutelycontinuous invariant measure lz. This measure has a den-sity ez(x) which diverges at the origin. Moreover,ez(x) x�z + 1 and is bounded elsewhere (see, e.g. [32] Sec-tion 10 and [45] Section 3). If we consider the norm k�kf,1

with f(x) = x2 we have that klzkf,1 is finite for each such z.It follows that for each such z the measure lz iscomputable.

3.3.3. Julia sets and the Brolin–Lyubich measureHere we explain how our method can be applied to

compute the measure of maximal entropy (or Brolin–Lyub-ich measure) associated to rational maps of the RiemannSphere.

It is interesting to say that this measure is supported onthe Julia set (associated to the rational map) which mayhappen to be uncomputable!

For sake of completeness we now attempt to brieflyintroduce computability notions for invariant compactsets, and the main results on Julia sets.

Let K be a compact subset of the plane. Informallyspeaking, in order to draw the set K on a computer screenwe need a program which is able to decide, given someprecision �, if pixel p has to be colored or not. By represent-ing pixel p by a ball B(p,�) where ðp; �Þ 2 Q

2, the questionone would like to answer is: does B(p,e) intersects K? Thefollowing definition captures exactly this idea:

Definition 27. A compact set K � R2 is said to be comput-able if there is an algorithm A such that, upon input (p,�):

� halts and outputs ‘‘yes’’ if K \ B(p,�) – ;,� halts and outputs ‘‘no’’ if K \ Bðp; �Þ ¼ ;,� run forever otherwise.

We remark that if a compact set is computable underthe definition above, then it is recursively compact in thesense of Definition 11. The converse is however false.

The question of whether Julia sets are computable un-der this definition has been completely solved in a seriesof papers by Binder, Braverman and Yampolsky. See[4,5,12–14]. Here we review some of their results. For sim-plicity, we give the definition of the Julia set only for qua-dratic polynomials. Consider a quadratic polynomial

PcðzÞ ¼ z2 þ c : C! C:

Obviously, there exists a number M such that if jzj > M,then the iterates of z under P will uniformly scape to 1.The filled Julia set is the compact set defined by:

Kc ¼ z 2 C : supnjPn

c ðzÞj <1� �

:

That is, the set of points whose orbit remains bounded. Forthe filled Julia set one has the following result:

Theorem 28 [14]. The filled Julia set Kc of a computablequadratic polynomials Pc is always computable.

The Julia set can now be defined by:

Jc ¼ @Kc;

where @(A) denotes the boundary of A. The Julia set is therepeller of the dynamical system generated by Pc. For allbut finitely many points, the limit of the nth preimagesP�n

c ðzÞ coincides with the Julia set Jc. The dynamics of Pc

on the set Jc is chaotic, again rendering numerical simula-tion of individual orbits impractical. Yet Julia sets areamong the most drawn mathematical objects, and count-less programs have been written for visualizing them.

In spite of this, the following result was shown in [12].

Theorem 29. There exist computable quadratic polynomialsPc(z) = z2 + c such that the Julia set Jc is not computable.

This phenomenon of non-computability is rather subtleand rare. For a detailed exposition, the reader is referredto the monograph [13].

Thus, we cannot accurately simulate the set of limitpoints of the preimages (Pc)�n(z), but what about their sta-tistical distribution? The question makes sense, as for allz –1 and every continuous test function w, the followingholds:

12n

Xw2ðPcÞ�nðzÞ

wðwÞ !n!1

Zwdk;

where k is the Brolin–Lyubich probability measure [15,39]supported on the Julia set Jc. We can thus ask whether theBrolin–Lyubich measure is computable. Even if Jc = Supp(k)is not a computable set, the answer does not a priori haveto be negative. In fact, the following result holds:

Theorem 30 [6]. The Brolin–Lyubich measure of computablequadratic polynomial is always computable.

This result essentially follows from Theorem 17 as well.The difficulty is once again to find a recursively compactcondition which characterizes Brolin–Lyubich measure.This measure happens to be singular with respect to Lebes-gue, so that the absolute continuity argument does not ap-ply. The key property allowing the application ofTheorem 17 in this case is balance: a probability measurel on C is said to be balanced with respect to a rationalfunction R (of degree at least 2), if for every set A � C onwhich R is injective, we have:

lðRðAÞÞ ¼ d � lðAÞ;

where d denotes the degree of R.Computation of the Brolin–Lyubich measure is then re-

duced to show that for probability measures, being bal-anced is a recursively compact condition. The proof ofthis fact is not very difficult, but it does require to over-come some problems like the fact that the functionl ? l(A) is not computable. For more details the inter-ested reader is referred to [6].


3.4. Computable systems without computable invariantmeasures

We have seen several techniques to establish the com-putability of many important invariant measures. Thisraises naturally the following question: does a computablesystem necessarily have a computable invariant measure?What about ergodic physical measures?

The following is an easy example o a simple system forwhich the last answer is negative, showing that the generalquestion of computing invariant measures has somesubtlety.

Let us consider a system on the unit interval given asfollows. Let s 2 (0,1) be a lower semi-computable realnumber which is not computable. There is a computablesequence of rational numbers si such that supisi = s. Foreach i consider Ti(x) = max(x,si) and

TðxÞ ¼XiP1

2�iTiðxÞ:

The functions Ti are uniformly computable so T is alsocomputable.

The system ([0,1],T) is hence a computable dynamicalsystem (see Fig. 1). T is non-decreasing, and T(x) > x ifand only if x < s.

This system has a physical ergodic invariant measurewhich is ds, the Dirac measure placed on s. The measureis physical because s attracts all the interval at its left.Since s is not computable then ds is not computable. Weremark that coherently with the previous theorems ds isnot isolated.

It is easy to prove, by a simple dichotomy argument,that a computable function from [0,1] to itself must havea computable fixed point. Hence it is not possible to con-struct a system over the interval having no computable

Fig. 1. The map T.

invariant measure at all (we always have the d over thefixed point). With some more work one can construct suchan example on the circle. Indeed the following can beestablished.

Proposition 31. There is a computable, continuous map T onthe circle having no computable invariant probabilitymeasure.

For the description of the system and for applications toreverse mathematics see [26] (Proposition 12).

4. Computing the speed of ergodic convergence

As recalled before, the Birkhoff ergodic theorem tellsthat, if the system is ergodic, there is a full measure setof points for which the averages of the values of the obser-vable f along its trajectory (time averages) coincides withthe spatial average of the observable f. Similar results canbe obtained for the convergence in the L2 norm, and others.Many, more refined results are linked to the speed of con-vergence of this limit. And the question naturally arise, ifthere is a possibility to compute this speed of convergencein a sense similar to Definition 5.

In the paper [2] some abstract results imply that in acomputable ergodic dynamical system, the speed of con-vergence of such averages can be algorithmically esti-mated. On the other hand it is also shown that there arenon ergodic systems where this kind of estimations arenot possible. In [27] a very short proof of this result forergodic systems was given. We present the precise result(Theorem 39) with its proof in this section.

4.1. Convergence of random variables

We first make precise what is meant by ‘‘compute thespeed of convergence’’ in a pointwise a.e. convergentsequence.

Definition 32. A random variable on (X,l) is a measur-able function f : X ! R.

Definition 33. Random variables fn effectively convergein probability to f if for each � > 0, l{x : jfn(x) � f(x)j < �}converges effectively to 1, uniformly in �. That is, there isa computable function n(�,d) such that for all n P n(�,d),l{jfn � fjP �} < d.

Definition 34. Random variables fn effectively convergealmost surely to f if f 0n ¼ supkPnjfk � f j effectively convergein probability to 0.

Definition 35. A computable probability space is a pair(X,l) where X is a computable metric space and l a com-putable Borel probability measure on X.

Let Y be a computable metric space. A function f :(X,l) ? Y is almost everywhere computable (a.e. comput-able for short) if it is computable on an effective Gd-set ofmeasure one, denoted by domf and called the domain ofcomputability of f.


A morphism of computable probability spaces f :(X,l) ? (Y,m) is a morphism of probability spaces whichis a.e. computable.

Remark 36. A sequence of functions fn is uniformly a.e.computable if the functions are uniformly computable ontheir respective domains, which are uniformly effective Gd-sets. Observe that in this case, intersecting all the domainsprovides an effective Gd-set on which all fn are computable.In the following we will apply this principle to the iteratesfn = Tn of an a.e. computable function T : X ? X, which areuniformly a.e. computable.

Remark 37. The space L1(X,l) (resp. L2(X,l)) can be madea computable metric space, choosing some dense set ofbounded computable functions as ideal elements. We saythat an integrable function f : X ! R is L1(X,l)-computableif its equivalence class is a computable element of the com-putable metric space L1(X,l). Of course, if f = g l-a.e., then fis L1(X,l)-computable if and only if g is. Basic operationson L1(X,l), such as addition, multiplication by a scalar,min, max, etc. are computable. Moreover, if T : X ? X pre-serves l and T is a.e. computable, then f ? f � T (from L1

to L1) is computable (see [30]).Let us call (X,l,T) a computable ergodic system if (X,l)

is a computable probability space where T is a measurepreserving morphism and (X,l,T) is ergodic. Let kfk denotethe L1 norm or the L2 norm.

Proposition 38. Let (X,l, T) be a computable ergodic system.Let f be a computable element of L1(X,l) (resp. L2(X,l)).

The L1 convergence (resp. L2 convergence) of the Birkhoffaverages An = (f + f � T +� � �+ f�Tn�1)/n is effective. That is:there is an algorithm n : Q! N such that for eachm P nð�Þ; kAm �

Rfdlk 6 �. Moreover the algorithm

depends effectively on T, l, f.

Proof. Replacing f with f �R

fdl, we can assume thatRfdl ¼ 0. The sequence kAnk is computable (see Remark

37) and converges to 0 by the ergodic theorems.Given p 2 N, we write m 2 N as m = np + k with 0 6 k < p.

Then

Anpþk ¼1

npþ k

Xn�1

i¼0

pAp � Tpi þ kAk � Tpn

!;

kAnpþkk 61

npþ kðnpkApk þ kkAkkÞ

6 kApk þkAkk

n

6 kApk þkfkn:

Let � > 0. We can compute some p = p(�) such thatkApk < �/2. Then we can compute some nð�ÞP 2

� kfk. Thefunction m(�) :¼ n(�)p(�) is computable and for allm P m(�), kAmk 6 �. h

4.2. Effective almost sure convergence

Now we use the above result to find a computable esti-mation for the a.s. speed of convergence.

Theorem 39. Let (X,l, T) be a computable ergodic system. If fis L1(X,l)-computable, then the a.s. convergence is effective.Moreover, the rate of convergence can be computed as abovestarting from T, l, f.

This will be proved by the following.

Proposition 40. If f is L1(X,l) -computable as above, andkfk1 is bounded, then the almost-sure convergence is effective(uniformly in f and a bound on kfk1 and on T, l).

To prove this we will use the well known Maximal ergo-dic theorem:

Lemma 41 (Maximal ergodic theorem). For f 2 L1(X,l) andd > 0,

l supnjAf

nj > d

� � 6

1dkfk1:

The idea is simple: compute some p such that kAfpk1 is

small, apply the maximal ergodic theorem to g :¼ Afp, and

then there is n0, that can be computed, such that Afn is close

to Agn for n P n0.

Proof of Proposition 40. Let �, d > 0. Compute p such thatkAf

pk 6 d�=2. Applying the maximal ergodic theorem tog :¼ Af

p gives:

l supnjAg

nj > d=2� �

6 �: ð5Þ

Now, Agn is not far from Af

n: expanding Agn, one can check

that

Agn ¼ Af

n þu � Tn � u

np;

where u = (p � 1)f + (p � 2)f � T +� � �+ f � Tp�2. kuk16

pðp�1Þ2 kfk1 so if n P n0 P 4(p � 1)kfk1/d, then

kAgn � Af

nk1 6 d=2. As a result, if jAfnðxÞj > d for some

n P n0, then jAgnðxÞj > d=2. From (5), we then derive

l supnPn0

jAfnj > d

( ) !6 �:

As n0 can be computed from d and �, we get theresult. h

Remark 42. This result applies uniformly to a uniformsequence of computable L1(X,l) observables fn.

We now extend this to L1(X,l)-computable functions,using the density of L1(X,l) in L1(X,l).

Proof of Theorem 39. Let �, d > 0. For M 2 N, let usconsider f 0M 2 L1ðX;lÞ defined as

f 0MðxÞ ¼minðf ;MÞ if f ðxÞP 0maxðf ;�MÞ if f ðxÞ 6 0:

�


Compute M such that kf � f 0Mk1 6 d�. Applying Proposi-

tion 40 to f 0M gives some n0 such that l fsupnPn0j

�A

f 0Mn j >

dgÞ < �. Applying Lemma 41 to f 00M ¼ f � f 0M gives

l fsupnjAf 00Mn j > dg

� �< �. As a result, l fsupnPn0

jAfnj >

�2dgÞ < 2�. h

Remark 43. We remark that a bounded a.e. computablefunction, as defined in Definition 35 is a computable ele-ment of L1(X,l) (see [30]). Conversely, if f is a computableelement of L1(X,l) then there is a sequence of uniformlycomputable functions fn that effectively converge l-a.e.to f.

5. Computing pseudorandom points, constructiveergodic theorem

In this section we show how the previous results oncomputation of invariant measures and speed of conver-gence allow to compute points which are statistically typ-ical for the dynamics. Let X again be a computable metricspace and l a computable probability measure on it. Sup-pose X is complete.

Points satisfying the above recalled pointwise ergodictheorem for an observable f, will be called typical for f.

Points which are typical for each f which is continuouswith compact support are called typical for the measure l(and for the dynamics).

The set of computable points in X (see Definition 6)being countable, is a very small (invariant) set, comparedto the whole space. For this reason, a computable pointcould rarely be expected to be typical for the dynamics,as defined before. More precisely, the Birkhoff ergodic the-orem and other theorems which hold for a full measure set,cannot help to decide if there exist a computable pointwhich is typical for the dynamics.

A number of theoretical questions arise naturally fromall these facts. Due to the importance of forecasting andsimulation of a dynamical system’s behavior, these ques-tions also have some practical motivation.

Problem 44. Since simulations can only start with com-putable initial conditions, given some typical statisticalbehavior of a dynamical system, is there some computableinitial condition realizing this behavior? How to choosesuch points?

Such points could be called pseudorandom points, and aresult showing its existence and their computability fromthe description of the system could be seen as a construc-tiveversion of the pointwise ergodic theorem.

Meaningful simulations, showing typical behaviors ofthe dynamics can be performed only if computable, pseu-dorandom initial conditions exist. Computable points arethe only points we can use when we perform a simulationor some explicit computation on a computer.

Based on a sort of effective Borel–Cantelli lemma, in[24] the above problem was solved affirmatively for a classof systems which satisfies certain technical assumptionswhich includes systems whose decay of correlation is fas-

ter than Clog2(time). Using the results on the estimationof the rate of convergence given in the previous section,it is possible to remove these technical assumptions onthe speed of decay of correlations. It is also worth to re-mark that the result is uniform also in T and l (the pseudo-random points will be calculated from a description of thesystem).

The following result ([24], Theorem 2 or [27]) showsthat from a sequence fn which converges effectively a.s.to f and from its speed of convergence, it is possible tocompute points xi for which fn(xi) ? f(xi).

Theorem 45. Let X be a complete metric space. Let fn, f beuniformly a.e. computable random variables. If fn effectivelyconverges almost surely to f then the set {x : fn(x) ? f(x)}contains a sequence of uniformly computable points which isdense in Supp(l). Moreover, the effective sequence foundabove depends algorithmically on fn and on the functionn(d,�) giving the rate of convergence

Since by the results of the previous section n(d,�) can becalculated starting from T, l and f, we can directly applythe above Theorem to compute typical points for thedynamics. Indeed, the following holds (see [27] for thedetails).

Theorem 46. If (X,l,T) is a computable ergodic system and fis L1(X,l) and a.e. computable, then there is a uniformsequence xi of computable points which is dense on thesupport of l such that for each i

limn!1

1n

Xf ðTnðxiÞÞ ¼

Zf dl:

Moreover this sequence can be computed starting from adescription of T, l and f.

The uniformity in f of the above result together with theexistence of a r.e. collection of computable observableswhich is dense in the space of compactly supported contin-uous functions, imply the following (see again [27] for thedetails).

Theorem 47. If (X,l, T) is a computable ergodic system thenthere is a uniform sequence xn of computable points which isdense on the support of l such that for each n, xn is typical forl. Moreover this sequence can be computed starting from adescription of T and l.

In particular, for the classes of systems where the inter-esting invariant measure can be computed by the descrip-tion of the system (see Section 3.3.1, Theorem 26), weobtain that in turn the pseudorandom points can be com-puted starting from a description of the system alone.

Corollary 48. Each piecewise expanding map, with comput-able derivatives, as in the assumptions of Theorem 26 has asequence of pseudorandom points which is dense in thesupport of the measure. Moreover this sequence can becomputed starting from the description of the system.

These two statements can be seen as constructive/effec-tive versions of the ergodic theorem.


6. Conclusions and directions

In this article we have reviewed some recent resultsabout the rigorous computation of invariant measures,invariant sets and typical points. Here, the sentence rigor-ous computation means ‘‘computable (up to any requiredprecision) by a Turing Machine’’. Thus, this can be seenas a theoretical study of which infinite objects in dynamicscan be arbitrarily well approximated by a modern com-puter (in an absolute sense), and which cannot. In this line,we presented some general principles and techniques thatallow the computation of the relevant objects in severaldifferent situations. On the other hand, we also presentedsome examples in which the computation of the relevantobjects is not possible at all, setting some theoretical limitsto the abilities of computers when used to simulatedynamical systems.

The examples of the second kind, however, seem to berather rare. An important question is therefore whetherthis phenomenon of non-computability is robust or preva-lent in any sense, or if it is rather exceptional. For example,one could ask whether the non-computability is destroyedby small changes in the dynamics or whether the non-computability occurs with small or null probability.

Besides, in this article we have not considered the effi-ciency (and therefore the feasibility) of any of the algo-rithms we have developed. An important (and moredifficult) remaining task is therefore the development ofa resource bounded version of the study presented in thispaper. In the case of Julia sets, for instance, it has beenshown in [10,43,11] that hyperbolic Julia sets, as well assome Julia sets with parabolics, are computable in polyno-mial time. On the other hand, in [5] it was shown thatthere exists computable Siegel quadratic Julia sets whosetime complexity is arbitrarily high.

For the purpose of computing the invariant measureform the description of the system, in Section 3.3.1 we hadto give explicit estimations on the constants in the Lasota–Yorke inequality. This step is important also when tech-niques different from ours are used (see, e.g. [37]). Similarestimations could be done in other classes of systems, fol-lowing the way the Lasota–Yorke inequality is proved ineach class (although, sometimes this is not a completelytrivial task and requires the ‘‘effectivization’’ of some stepin the proof). A more general method to have an estimationfor the constants or other ways to get information on theregularity of the invariant measure would be useful.

Acknowledgement

We would like to thank The Abdus Salam InternationalCentre for Theoretical Physics (Trieste, IT) for support andhospitality during this research.

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