Chaos, Solitons & Fractals 45 (2012) 1–14
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals
Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos
Frontiers
Statistical properties of dynamical systems – Simulation and
abstract computation
Stefano Galatolo a,⇑, Mathieu Hoyrup b, Cristóbal Rojas c
a
Dipartimento di Matematica Applicata, Università di Pisa, Italy
LORIA, INRIA Lorraine, France
c
Departamento de Matematica, Universidad Andres Bello, Chile
b
a r t i c l e
i n f o
Article history:
Received 24 March 2011
Accepted 8 September 2011
a b s t r a c t
We survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long
term behavior of dynamical systems. We will focus on the statistical limiting behavior
and invariant measures. We present a general method allowing the algorithmic approximation at any given accuracy of invariant measures. The method can be applied in many
interesting cases, as we shall explain. On the other hand, we exhibit some examples where
the algorithmic approximation of invariant measures is not possible. We also explain how
it is possible to compute the speed of convergence of ergodic averages (when the system is
known exactly) and how this entails the computation of arbitrarily good approximations of
points of the space having typical statistical behaviour (a sort of constructive version of the
pointwise ergodic theorem).
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The advent of automated computation led to a series of
great successes and achievements in the study of many
natural and social phenomena which are modeled by
dynamical systems. The use of computers and simulations
allowed to predict the behavior of many important models
and, on the other hand, led to the discovery of important
general aspects of dynamics. This motivated the huge
amount of work that was made by hundreds of scientists
to improve practical simulation and computation techniques. It also motivated the study of the theoretical limits
of simulation and computation techniques, and the theoretical understanding of related problems.
In this paper we shall focus on some of these theoretical
aspects related to rigorous computation and simulation of
(discrete time) dynamical systems.
The simulation and investigation of dynamical systems
started with what we call the naïve approach, in which the
⇑ Corresponding author.
E-mail address: galatolo@dm.unipi.it (S. Galatolo).
0960-0779/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2011.09.011
user just implements the dynamics without taking rigorous account of numerical errors and rounding. Then the
user simply sees what happens on the screen.
Evidently, the sensitivity to initial conditions and the
typical instability of many interesting systems (to perturbations on initial conditions but also on the map generating the dynamics) implies that what is being observed on
the computer screen could be completely unrelated to
what was meant to be simulated. In spite of this, the näive
approach turns out to work almost unreasonably well in
many situations and it is even today the most commonly
used approach in simulations. The theoretical reasons of
why this method works and what are its limits, are in
our opinion yet to be understood (e.g. some aspects have
been investigated in [36,41,29,7,8,34]).
Opposite to the naïve approach, there is the ‘‘absolutely
rigorous’’ approach, which will be the main theme of this
paper. Here the user seeks for an algorithm able to produce
a description up to any desired precision of the object x
which is meant to be computed. In case such an algorithm
exits, we say that the object x is computable (suitable precise definitions will be given below).
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
For example, in this approach the constant e is a computable number because there is an algorithm that is able
to produce a rational approximation of e at any given accuracy (for instance by finding the right m and calculating
Pm 1
1 n! such that the error is smaller than requested).
In an ‘‘absolutely rigorous’’ simulation (computation),
both the initial point (or initial distribution) and the transition map are supposed to be computable, and the question arises if interesting objects related to the dynamics
(e.g. invariant sets or invariant measures) can be computed
from the description of the system (or perhaps by using
some additional information).
In this paper we shall survey a number of results related
to these computational aspects, some of which are updated
with new information. We will see that in many cases the
objects of interest can indeed be computed, but there are
some subtleties, and cases where the interesting objects
cannot be computed from the description of the system,
or cannot be computed at all. In other words, these results
provide theoretical limits to such computations. In particular, we shall recall some examples where this non-computability phenomenon appears, in the case of invariant
measures and Julia sets.
1.1. Computing invariant measures
An important fact motivating the study of the statistical
properties of dynamical systems is that the pointwise long
term prediction of a chaotic system is not possible,
whereas, in many cases, the estimation or forecasting of
averages and other long term statistical properties is. This
forecasting procedure, often corresponds in mathematical
terms to the computation of invariant measures or to the
estimation of some of their properties – measures contain
information on the statistical behavior of the system (X, T)
and on the potential behavior of averages of observables
along typical trajectories of the system (see Section 3).
An invariant measure is a Borel probability measure l
on X such that for each measurable set A it holds that
l(A) = l(T1(A)). They represent equilibrium states, in the
sense that probabilities of events do not change in time.
To rigorously compute an invariant measure means to
find an algorithm which is able to output a description of
the measure (for example an approximation of the measure made by a combination of delta measures placed on
‘‘rational’’ points) up to any prescribed precision.
We remark that once an interesting invariant measure
is computed, this information is useful for the computation
of other dynamical properties: Lyapunov exponents, entropy, escape rates, etc. For example, in dimension one, once
an ergodic invariant measure l has been approximated,
the Lyapunov exponent kl can be estimated using the forR1
mula kl ¼ 0 log2 T 0 dl, where T0 denotes the derivative of
the map generating the dynamics. In higher dimensions,
similar techniques can be applied (see, e.g. [21] for more
examples of derivation of dynamical quantities from the
computation of the invariant measure and similar objects).
Before giving more details about the computation of
invariant measures, we remark that, since there are only
countably many ‘‘algorithms’’ (computer programs), whatever we mean by ‘‘approximating a measure by an algo-
rithm’’ would imply that only countable many measures
can be computed. In general, however, a dynamical system
may have uncountably many invariant measures (usually
an infinite dimensional set). So, a priori most of them will
not be algorithmically describable. This is not a serious
problem because we should put our attention on the most
meaningful ones. An important part of the theory of
dynamical systems is indeed devoted to the understanding
of physically relevant invariant measures. Informally
speaking, these are measures which represent the asymptotic statistical behavior of many (positive Lebesgue measure) initial conditions (see Section 3 for more details).
The existence and uniqueness of physical measures is a
widely studied problem (see [48]), which has been solved
for some important classes of dynamical systems. These
measures are some of the good candidates to be computed.
Other measures of particular interest in certain cases, as
we shall see below, are the maximal entropy ones.
The computation of interesting invariant measures (in
more or less rigorous ways) is the main goal of a significant
part of the literature related to computation in dynamics.
An important role here is played by the transfer operator
induced by the dynamics. Indeed, the map T defining the
dynamics, induces a transition map LT on the space of
probability measures over X, LT:PM(X) ? PM(X). LT is called
the transfer operator associated to T (definition and basic
results about this are recalled in Section 3). Invariant measures are fixed points of this operator. The main part of the
methods which are used to compute invariant measures
deals with suitable finite dimensional approximations of
this operator. In Section 3 we will review briefly some of
these methods, and give some references. We then consider the problem from the abstract point of view mentioned above, and give some general results on rigorous
computability of the physical invariant measure. In particular we will see that the transfer operator is computable
up to any approximation in a general context (see Theorem 16). The corresponding invariant measure is computable, provided we are able to give a description of a
space of ‘‘regular’’ measures where the physical invariant
measure is the only invariant one (see Theorem 17 and following corollaries).
We will also show how the measure can be obtained
from a description of the system in a class of examples
(piecewise expanding maps) by using the Lasota–Yorke
inequality.
Rational transformations of the Riemann Sphere are another interesting class where our methods can be applied.
These complex dynamics are very well studied in the literature, and enormous attention has been addressed to the
Julia sets associated to them, which have emerged as the
most studied examples of fractal sets generated by a
dynamical system. One of the reasons for their popularity
is the beauty of the computer-generated images of such
sets. The algorithms used to draw these pictures vary;
the naïve approach in this case works by iterating the center of a pixel to determine if it lies in the Julia set. There exists also more sophisticated algorithms (using classical
complex analysis, see [42]) which work quite well for
many examples, but it is well known that in some particular cases computation time will grow very rapidly with
S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
increase of the resolution. Moreover, there are examples,
even in the family of quadratic polynomials, where no satisfactory pictures of the Julia set exist.
In the rigorous approach, a set is computable if, roughly
speaking, its image can be generated by a computer with
an arbitrary precision. Under this notion of computability,
the question arise if Julia sets are always computable. In a
series of papers [5,4,12,13] it was shown that even though
in many cases (hyperbolic, parabolic) the Julia set is indeed
computable, there exists quadratic polynomials which are
computable (again, in the sense that all the trajectories
can be approximated by an algorithm at any desired accuracy), and yet the Julia set is not.
So we cannot simulate the set of limits points on which
the chaotic dynamics takes place, but, what about the statistical distribution? In fact, it was shown by Brolin and
Luybich that there exists a unique invariant measure
which maximizes entropy, and that this measure is supported on the Julia set. The question of whether this measure can be computed has been recently solved in [6],
where it is proved that the Brolin–Lyubich measure is always computable. So that even if we cannot visualize the
Julia set as a spatial object, we can approximate its distribution at any finite precision. In Section 3.3 we shall briefly
review the mentioned results on uncomputability of Julia
sets, and explain how our method can be applied to show
computability of the Brolin–Lyubich measure.
After such general statements one could conjecture that
all computable dynamical systems should always have a
computable invariant measure. We will see that, perhaps
surprisingly, this is not true. Not all computable systems
(the dynamics can be computed up to any prescribed
approximation) have computable invariant measures (see
Section 3.4). The existence of such examples reveals some
subtleties in the computation of invariant measures.
Finally, to further motivate these results, we remark
that from a technical point of view, computability of the
considered invariant measure is a requirement in several
results establishing connections between computation,
probability, randomness and pseudo-randomness (see Section 5 and e.g. [3,23–25])
1.2. Computing the speed of convergence and pseudorandom
points
For several questions in ergodic theory the knowledge
of the speed of convergence to ergodic behavior is important to deduce other practical consequences. In the computational framework, the question turns out to be the
effective estimation of the speed of convergence in the
ergodic theorems.1 From the numerical–practical point of
view this has been done in some classes of systems, having
a spectral gap for example. In this case a suitable approximation of the transfer operator allows to compute the rate of
decay of correlations [22,37] and from this, other rates of
convergence can be easily deduced.
R
P
Find a N such that 1n f T n differs from f dl less than a given error
for each nP, in some sense.
1
3
Other classes of systems could be treated joining the
above spectral approach, with combinatorial constructions
(towers, see, e.g. [38]), but the general case need a different
approach.
In [2] it was shown that much more in general, if the
system can be described effectively, then the rate of convergence in the pointwise ergodic theorem can be effectively estimated (not only the asymptotical speed, but explicit
estimates on the error).
We give in Section 4 a very short proof of a statement of
this kind (see Theorem 39) for ergodic systems, and show
some consequences. Among these, a constructive version
of pointwise ergodic theorem. If the system is computable
(in some wide sense that will be described), then it is possible to compute points having typical statistical behavior.
Such points could hence be called pseudorandom points of
the system (see Section 5).
Since the computer can only handle computable initial
conditions, any simulation can start only from these
points. Pseudorandom initial conditions are hence in principle the good points where to start a simulation.
We remark that it is widely believed that naïve computer simulations very often produce correct statistical
behavior. The evidence is mostly heuristic. Most arguments are based on the various ‘‘shadowing’’ results (see,
e.g. [33, chapter 18]). In this kind of approach (different
from ours), it is possible to prove that in a suitable system
every pseudo-trajectory (as the ones which are obtained in
simulations with some computation error) is close to a real
trajectory of the system. However, even if we know that
what we see in a simulation is near to some real trajectory,
we do not know if this real trajectory is typical in some
sense. Another limitation of this approach is that shadowing results hold only in systems having some strong
hyperbolicity, while many physically interesting systems
are not like this. In our approach we consider real trajectories instead of ‘‘pseudo’’ ones and we ask if there exists
computable points which behaves as a typical point of
the dynamics.
2. Computability on metric spaces
To have formal results and precise assumptions on the
computability (up to any given error) of continuous objects, we first need to introduce some concepts. In particular, we shall introduce recursive versions of open and
compact sets, and characterize the functions which are
well suited to operate with those sets (computable functions). In this section, we try to explain this theory as simple and self contained as possible. Other details and
different approaches to the subject can be found in the
introductory texts [9,47].
2.1. Computability
The starting point of recursion theory was the introduction of a mathematical definition making precise the intuitive notions of algorithmic or effective procedure on
symbolic objects. Several different formalizations have
been independently proposed (by Church, Kleene, Turing,
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
Post, Markov. . . ) in the 30s. They all turned out to be
equivalent: they compute the same functions from N to
N. The class of functions thus defined is now called the
class of recursive functions. As an algorithm is allowed to
run forever on an input, these functions may be partial,
i.e. not defined everywhere. The domain of a recursive
function is the set of inputs on which the algorithm eventually halts. A recursive function whose domain is N is said
to be total.
We now recall an important concept from recursion
theory. A set E # N is said to be recursively enumerable
(r.e.) if there is a (partial or total) recursive function
u : N ! N enumerating E, that is E ¼ fuðnÞ : n 2 Ng. If
E – ;, u can be effectively converted into a total recursive
function w which enumerates the same set E.
Uniformity. Algorithms can be used to define computability notions on many classes of mathematical objects.
The precise definitions will be particular to each class of
objects, but they will always follow the following scheme:
2.2. Algorithms and uniform algorithms
For instance, the elements of a sequence of real numbers
ðxi Þi2N are uniformly computable if there is a algorithm
A : N N ! Q such that jAði; nÞ xi j 6 2n for all i, n.
In other words a set of objects is computable uniformly
with 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 may
take a particular name: computable, recursive, effective,
r.e., etc. so the term ‘‘computable’’ used above shall be
replaced.
Strictly speaking, recursive functions only work on natural 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 will
use the word algorithm instead of recursive function when
the inputs or outputs are interpreted as finite objects.
The operative power of algorithms on the objects of such
a numbered set obviously depends on what can be effectively recovered from their numbers.
More precisely, let X and Y be numbered sets such that
the numbering of X is injective (it is then a bijection between N and X). Then any recursive function u : N ! N induces 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 injectively numbered Q ¼ fq0 ; q1 ; . . .g in an effective way: the
number i of a rational a/b can be computed from a and b,
and vice versa. We fix such a numbering: from now on
the rational number with number i will be denoted by qi.
Now, let us consider computability notions on the set R
of real numbers.
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 version. Let ðOi Þi2N be a sequence of computable objects:
Oi is computable uniformly in i if there is an algorithm
A:NX !Y
such that for all i; Ai :¼ Aði; Þ : X ! Y computes Oi.
2.3. Computable metric spaces
A computable metric space is a metric space with an
additional structure allowing to interpret input and output
of algorithms as points of the metric space. This is done in
the following way: there is a dense subset (called ideal
points) such that each point of the set is identified with a
natural number. The choice of this set is compatible with
the metric, in the sense that the distance between two
such points is computable up to any precision by an algorithm getting the names of the points as input. Using these
simple assumptions many constructions on metric spaces
can be implemented by algorithms.
Definition 2. Let x be a real number. We say that:
Definition 4. A computable metric space (CMS) is a triple
X ¼ ðX; d; SÞ, where
x is lower semi-computable if the set fi 2 N : qi < xg is
r.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 semicomputable.
It is worth noticing that a real number is computable if
and only if there exists an algorithmic enumeration of a sequence of rational numbers converging exponentially fast
to x. That is:
Proposition 3. A real number is computable if there is an
algorithm A : N ! Q such that jAðnÞ xj 6 2n for all n.
We remark that the notion of computable real number
was already introduced (in a different but equivalent
way) by Turing [44].
(i) (X, d) is a separable metric space.
(ii) S ¼ fsi gi2N 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 algorithm
A : N3 ! Q such that jAði; j; nÞ dðsi ; sj Þj < 2n ).
Point (iii) says that the information that can be recovered from the numbers of the numbered set S is their mutual distances.
The algorithms involved in the following definitions (up
to Definition 9) are assumed to be total.
Definition 5. We say that in a metric space (X, d), a
sequence of points ðxn Þn2N converges recursively to a point
x if there is an algorithm D : Q ! N such that d(xn, x) 6
for all n P D().
S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
Definition 6. A point x 2 X is said to be computable if
there is an algorithm A : N ! S such that ðAðnÞÞn2N converges recursively to x.
We define the set of ideal balls to be B :¼
fBðsi ; qj Þ : 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 ; . . .g
which makes the number of a ball effectively computable
from its center and radius and vice versa. B is a countable
basis of the topology.
Definition 7 (Effective open sets). We say that an open set
U is effective if there is an algorithm A : N ! B such that
S
U ¼ n AðnÞ.
Observe that an algorithm which diverges on each input
n enumerates the empty set, which is then an effective
open set. Sequences of uniformly effective open sets are
naturally defined. Moreover, if ðU i Þi2N is a sequence of uniS
formly effective open sets, then iUi is an effective open
set.
Definition 8 (Effective Gd-set). An effective Gd-set is an
intersection of a sequence of uniformly effective open sets.
Obviously, an uniform intersection of effective Gd-sets is
also an effective Gd-set.
Let X; SX ¼ fsX1 ; sX2 ; . . .g; dX and Y; SY ¼ fsY1 ; sY2 ; . . .g; dY
X
Y
be computable metric spaces. Let also Bi and Bi be enumerations of the ideal balls in X and Y. A computable function X ? Y is a function whose behavior can be computed
by an algorithm up to any precision. For this it is sufficient
that the pre-image of each ideal ball can be effectively enumerated by an algorithm.
Definition 9 (Computable functions). A function T : X ? Y
is computable if T 1 ðBYi Þ is an effective open set, uniformly
in i. That is, there is an algorithm A : N N ! BX such that
S
T 1 ðBYi Þ ¼ n Aði; nÞ for all i.
A function T : X ? Y is computable on D # X if there
are uniformly effective open sets Ui such that
T 1 ðBYi Þ \ D ¼ U i \ D.
Remark 10. Intuitively, a function T is computable (on
some domain D) if there is a computer program which
computes T(x) (for x 2 D) in the following sense: on input
> 0, the program, along its run, asks the user for approximations of x, and eventually halts and outputs an ideal
point s 2 Y satisfying d(T(x), s) < . This idea can be formalized, using for example the notion of oracle computation.
The resulting notion coincides with the one given in the
previous definitions.
Recursive compactness is an assumption which will be
needed in the following. Roughly, a compact set is recursively compact if the fact that it is covered by a finite collection of ideal balls can be tested algorithmically (for
equivalence with the -net approach and other properties
of recursively compact set see [26]).
Definition 11. A set K # X is recursively compact if it is
compact and there is a recursive function u : N ! N such
that u(i1, . . . , ip) halts if and only if ðBi1 ; . . . ; Bip Þ is a covering
of K.
5
3. Computability of invariant measures
3.1. Invariant measures and statistical properties
Let X be a metric space, T : X ? X a Borel measurable
map and l a T-invariant Borel probability measure. A set
A is called T -invariant if T1(A) = A (mod 0). The system
(X, T, l) is said to be ergodic if each T-invariant set has total
or null measure. In such systems the famous Birkhoff ergodic theorem says that time averages computed along ltypical orbits coincides with space average with respect
to l. More precisely, for any f 2 L1(X, l) it holds
lim
n!1
Sfn ðxÞ
¼
n
Z
f dl;
ð1Þ
for l almost each x, where Sfn ¼ f þ f T þ þ f T n1 .
This shows that in an ergodic system, the statistical
behavior of observables, under typical realizations of the
system is given by the average of the observable made
with the invariant measure.
We say that a point x belongs to the basin of attraction of
an invariant measure l if (1) holds at x for each bounded continuous f. In case X is a manifold (possibly with boundary), a
physical measure is an invariant measure whose basin of
attraction has positive Lebesgue measure (for more details
and a general survey see [48]). Computation of such measures will be the main subject of the first part of this section.
3.1.1. The transfer operator
A function T between metric spaces naturally induces a
function LT between probability measure spaces. This function LT is linear and is called transfer operator (associated
to T). Measures which are invariant for T are fixed points
of LT.
Let us consider a computable metric space X and a measurable function T : X ? X. Let us also consider the space
PM(X) of Borel probability measures on X.
Let us define the linear function LT : PM(X) ? PM(X) by
duality in the following way: if l 2 PM(X) then LT(l) is such
that
Z
f dLT ðlÞ ¼
Z
f T dl
for each bounded continuous f.
The computation of invariant measures (and many
other dynamical quantities) very often is done by computing the fixed points (and some other spectral information)
of this operator in a suitable function space. The most applied and studied strategy is to find a suitable finite dimensional approximation of LT (restricted to a suitable function
space) so reducing the problem to the computation of the
corresponding relevant eigenvectors of a finite matrix.
An example of this is done by discretizing the space X
by a partition Ai and replacing the system by a (finite state)
Markov Chain with transition probabilities
Pij ¼
mðAi \ Aj Þ
;
mðAi Þ
where m is the Lebesgue measure on X (see, e.g.
[20,21,37]). Then, taking finer and finer partitions it is pos-
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
sible to obtain in some cases that the finite dimensional
model will converge to the real one (and its natural invariant measure to the physical measure of the original system). In some cases there is an estimation for this speed
of convergence (see, e.g. [21] for a discussion), but a rigorous bound on the error (and then a real rigorous computation) is known only in a few cases (piecewise expanding or
expanding maps, see [37,28]).
Similar approaches consists of applying a kind of Faedo–Galerkin approximation to the transfer operator by
considering a complete Hilbert base of the function space
and truncating the operator to the action on the first elements (see [46]).
Another approach is to consider a perturbation of the
system by a small noise. The resulting transfer operator
has a kernel. This operator canthen be approximated by a
finite dimensional one (again by the Faedo–Galerkin method for instance) and the relevant eigenvectors can be thus
calculated (see, e.g. [17,16]). This method is useful in cases
where it is possible to prove that the physical measure of
the original system can be obtained as the limit when
the size of the noise tends to zero (this happens, for example, 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 analytic
Markov maps. Another strategy to face the problem of
computation of invariant measures consist in following
the way the measure l can be constructed and check that
each step can be realized in an effective way. In some interesting examples we can obtain the physical measure as
limit
of
iterates
of
the
Lebesgue
measure
l ¼ limn!1 LnT ðmÞ (recall that m is the Lebesgue measure).
To prove computability of l the main point is to explicitly
estimate the speed of convergence to the limit. This sometimes can be done using techniques related to decay of correlations [24].
Concluding, if the goal is to rigorously compute an
invariant measure, most of the results in modern literature are partial. Indeed, besides of being applied to restricted classes of systems, they usually need additional
information in order to compute the measure. For example, the calculation of the finite dimensional approximations is often not done Turing-rigorously, or the rate of
convergence of the approximations is computed up to
some constants (depending on the system) which are
not estimated.
In the remaining part of this section we present some
general results, mainly from [26,6], and explain how they
can be applied to rigorously compute invariant measures.
These results have the advantage of being simple and quite
general with all the needed assumptions made explicit. On
the other hand, they are not well suited for a complexity
(in time or space) analysis, so it is not clear if they can be
implemented and used in practice.
The rigorous framework in which they are proved, however, allows to see them as a study about the theoretical
limits of (Turing-)rigorous computation of invariant measures and, in fact, also negative results can be obtained.
In particular, we present examples of computable systems
having no computable invariant measures.
3.2. Computability of measures
In this section we explain precisely what we mean by
computing a measure, namely to have an algorithm able
to approximate the measure by ‘‘simple measures’’ up to
any desired accuracy.
Let us consider the space PM(X) of Borel probability
measures over X. Let C0(X) be the set of real-valued
bounded continuous functions on X. We recall the notion
of weak convergence of measures:
Definition 12. ln is said to be weakly convergent to l if
R
R
f dln ! f dl for each f 2 C0(X).
Let us introduce the Wasserstein–Kantorovich distance
between measures. Let l1 and l2 be two probability measures on X and consider:
W 1 ðl1 ; l2 Þ ¼
sup
f 21LipðXÞ
Z
f dl1
Z
f dl2 ;
where 1-Lip (X) is the space of functions on X having Lipschitz constant less than one. The distance W1 has the following useful properties:
Proposition 13 (see [1] Prop 7.1.5).
1. W1 is a distance and if X is bounded, separable and complete, then PM(X) with this distance is a separable and
complete metric space.
2. If X is bounded, a sequence is convergent in the W1 metric
if 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 the
computable metric structure of X. Indeed, given the set SX
of ideal points of X we can naturally define a set of ideal
points SPM(X) in PM(X) by considering finite rational convex
combinations of the Dirac measures ds supported on ideal
points s 2 SX. This is a dense subset of PM(X). The proof of
the following proposition can be found in [31].
Proposition 14. If X is bounded, then (PM(X), W1, SPM(X)) is a
computable metric space.
A measure l is then computable if there is a sequence
ln 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 invariant 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 uniquely
ergodic, then its invariant measure is computable.
Proof. The method is based on the following very simple
and well-known fact: the singleton set {x0} is recursively
compact if and only if its unique element x0 is a computable
7
S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
point. The idea is then to show that the set {l} PM(X),
where l is the unique invariant measure, is recursively
compact. This is done in the following way: first we observe
that the set of probability measures PM(X) is recursively
compact (since X is). Second we observe that the set PMT(X)c
of probability measures which are NOT invariant under T is
recursively open (here it is essential that T is an everywhere
computable map). Finally we use the general fact that a
recursively compact set minus a recursively open set is
always 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 general
situations come from two main facts: (i) when the map
T is not everywhere computable and then the set PMT(X)
of invariant measures is not (recursively) compact anymore and (ii) when the map T is not uniquely ergodic,
the set PMT(X) is not a singleton, and it therefore does
not need to contain computable points (even if recursively compact).
To overcome these difficulties, we essentially need to
find a different recursively compact condition which
would allow us to separate a distinguished invariant measure from the rest, while making use of T only on its domain of computability.
In the following, we present a general result which sets
the requirements in order to accomplish this task. It will be
convenient to spell the principle in a more equation solving way, namely as an abstract theorem allowing the computation 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 that
can be somehow distinguished by means of analytic or
dynamical properties (as being physical or absolutely continuous, for example).
It will be therefore necessary to consider a space where
the transfer operator is computable (this space hopefully
will contain the distinguished invariant measure).
We remark that if T is not continuous then LT is not necessarily continuous (this can be realized by applying LT to
some delta measure placed near a discontinuity point)
and hence not computable. Still, we have that LT is continuous (and its modulus of continuity is computable) at all
measures l which are ‘‘far enough’’ from the discontinuity
set D. This is technically expressed by the condition
l(D) = 0.
Proposition 16. Let X be a computable metric space and T :
X ? X be a function which is computable on XnD. Then LT is
computable on the set of measures
PMD ðXÞ :¼ fl 2 PMðXÞ : lðDÞ ¼ 0g:
ð2Þ
From a practical point of view, this proposition provides
sufficient conditions to rigorously approximate the transfer operator by an algorithm.
The above tools allow us to ensure the computability of
LT on a large class of measures and obtain in a way similar
to Proposition 15:
Theorem 17 [26], Theorem 3.2. Let X be a computable
metric space and T a function which is computable on XnD.
Suppose there is a recursively compact set of probability
measures 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 which
takes as inputs finite descriptions of T, V and an ideal ball in
M(X) which isolates2 an invariant measure l, and outputs a
finite description of l.
Remark 18. Notice that there is no computability condition on the set D. We also remark that unless T is uniquely
ergodic, invariant measures are never isolated in PM(X), so
the task is to find a (recursively compact) condition that
only a single invariant measure would satisfy.
Clearly, Proposition 15 is now a trivial corollary of Theorem 17. As already said, the main difficulty in the application of Theorem 17 is the requirement that the invariant
measure we are trying to compute be isolated in V. In general the space of invariant measures of a given dynamical
system could be very large (an infinite dimensional convex
subset of PM(X)) but there is often some kind of particular
regularity that can be exploited to characterize a particular
invariant measure, isolating it from the rest. For example,
let us consider the following seminorm:
klka ¼ sup
x2X;r>0
lðBðx; rÞÞ
ra
:
If a and K are computable and X is recursively compact
then
V a;K ¼
l 2 PMðXÞ : klka 6 K
ð3Þ
is recursively compact [26]. This implies
Proposition 19. Let X be recursively compact and T be
computable on XnD, with dimH(D) < 1. Then any invariant
measure isolated in Va,K with a > dimH(D) is computable. Once
again, this is uniform in T, a, K.
We recall that in the above proposition dimH denotes
the Hausdorff dimension of a set. The above general proposition allow us to obtain as a corollary the computability
of many systems having absolutely continuous invariant
measures. For example, let us consider maps on the
interval.
Proposition 20. Let X = [0, 1], T be computable on XnD, with
dimH(D) < 1 and suppose (X, T) has a unique absolutely
continuous invariant measure l with bounded density, then
l is computable (starting from T and a bound for the L1 norm
of the invariant density).
Similar results hold for maps on manifolds (again see
[26]).
3.3.1. Computing the measure from a description of the system
in the class of piecewise expanding maps
As it is well known, interesting examples of systems
having a unique absolutely continuous invariant measure
2
If the invariant measure is unique in V the isolating ball is not
necessary.
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
(with bounded density as required) are topologically transitive piecewise expanding maps on the interval or expanding maps on manifolds.
We show how to find a bound for the invariant density on
piecewise expanding maps. This implies that the invariant
measure 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 such
that Tjðdi ;diþ1 Þ 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 unique
ergodic invariant measure with bounded variation density.
Such a density is also the unique fixed point of the (Perron Frobenius) operator5 L : L1[0, 1] ? L1[0, 1] defined by
X f ðyÞ
:
½Lf ðxÞ ¼
T 0 ðyÞ
1
y2T
x
We now show how to find a bound for such a density,
starting from the description of the system, and then compute the associated invariant measure.
The following proposition was proved in the celebrated
paper [35] (Theorem 1 and its proof) and it is now called
Lasota–Yorke inequality. We give a precise statement
where the constants are explicited.
Proposition 22. Let T be a piecewise expanding map. Let f be
of bounded variation in [0, 1] and let denote its variation by
Var(f). Let d1, . . . , dn be the discontinuity points of T.
If k ¼ inf x2½0;1 fd1 ;...;dn g T 0 ðxÞ. Then
VarðLf Þ 6 2kVarðf Þ þ Bkf k1 ;
where
B¼
supx2½0;1 fd1 ;...;dn g
inf x2½0;1 fd1 ;...;dn g
T 00 ðxÞ
ðT 0 ðxÞÞ2
1
T 0 ðxÞ
þ
2
:
min ðdi diþ1 Þ
The following is an elementary fact about the behavior
of real sequences.
Lemma 23. If a real sequence an is such that an+1 6 lan + k for
some l < 1, k > 0, then
k
:
supðan Þ 6 max a0 ;
1l
3
For the sake of simplicity we will consider the simplest setting in which
we can work and give precise estimations. Such a class was generalized in
several ways. We then warn the reader that in the current literature, by
piecewise expanding maps it is meant something slightly more general
than the definition we give here.
4
A system is said to be topologically mixing if, given sets A and B, there
exists 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 operator
acting on densities instead of measures.
Proposition 24. If f is the density of the physical measure of
a piecewise expanding map T as above and k > 2. Then
Varðf Þ 6
B
;
1 2k
where B is defined as above.
Proof (Sketch). The topological mixing assumption
implies that the map has only one invariant physical measures (see [45]). Let us use the above results iterating the
constant density corresponding to the Lebesgue measure.
Proposition 22 and Lemma 23 give that the variation of
B
the iterates is bounded by 12k
. Suppose that the limit measure has density f. By compactness of uniformly bounded
variation functions in L1, the above properties give a bound
on the variation of f (see [35] proof of Theorem 1). h
The following is a trivial consequence of the fact that
kf k1 6 Varðf Þ þ
Z
fdl:
B
Corollary 25. In the above situation kf k1 6 12k
þ 1.
The bound on the density of the invariant measure, together with Corollary 20 gives the following uniform result
on the computation of invariant measures of such maps.
Theorem 26. Suppose a piecewise expanding map T and its
first and second derivatives are computable on
[0, 1] {d1, . . . , dn}. Suppose that also its extension to the
closed intervals [di, di+1] are computable. Then, the physical
measure can be computed starting from a description of the
system (the points di and the programs computing the map
and its derivatives).
Proof (Sketch). Since T and T0 are computable on each
interval [di, di+1] we can compute a number k such that
1 < k 6 inf (T0 ) (see [26] Proposition 3) If we consider the
iterate TN instead of T the associated invariant density will
be the same as the one of T, and if kN > 2, then TN will satisfy
all the assumptions needed on Proposition 24. In the same
way it is possible to compute an upper bound for the related
B
. Then we have a bound on the density and we can apply
12k
Corollary 20 to compute the invariant measure. h
3.3.2. Unbounded densities and non uniformly hyperbolic
maps
The above results ensure computability of some absolutely continuous invariant measure with bounded density.
If we are interested in situations where the density is unbounded, we can consider a new norm, ‘‘killing’’ singularities.
Let us hence consider a computable function f : X ! R
and
klkf ;a ¼ sup
x2X;r>0
f ðxÞlðBðx; rÞÞ
:
ra
If a and K are computable and X is recursively compact
then
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
V a;K ¼
n
l 2 PMðXÞ : klkf ;a 6 K
o
ð4Þ
is recursively compact, and Proposition 19 also hold for the
seminorm kkf,a. If f is such that f(x) = 0 when
limr!0 lðBðx;rÞÞ
¼ 1 this can let the norm be finite when the
ra
density diverges.
As an example, where this can be applied, let us consider the Manneville Pomeau type maps on the unit interval. These are maps of the type x ? x + xz(mod1).
When 1 < z < 2 the dynamics has a unique absolutely
continuous invariant measure lz. This measure has a density ez(x) which diverges at the origin. Moreover,
ez(x) xz + 1 and is bounded elsewhere (see, e.g. [32] Section 10 and [45] Section 3). If we consider the norm kkf,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 is
computable.
3.3.3. Julia sets and the Brolin–Lyubich measure
Here we explain how our method can be applied to
compute the measure of maximal entropy (or Brolin–Lyubich measure) associated to rational maps of the Riemann
Sphere.
It is interesting to say that this measure is supported on
the Julia set (associated to the rational map) which may
happen to be uncomputable!
For sake of completeness we now attempt to briefly
introduce computability notions for invariant compact
sets, and the main results on Julia sets.
Let K be a compact subset of the plane. Informally
speaking, in order to draw the set K on a computer screen
we need a program which is able to decide, given some
precision , if pixel p has to be colored or not. By representing pixel p by a ball B(p,) where ðp; Þ 2 Q2 , the question
one would like to answer is: does B(p, e) intersects K? The
following definition captures exactly this idea:
Definition 27. A compact set K R2 is said to be computable 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 under
the definition above, then it is recursively compact in the
sense of Definition 11. The converse is however false.
The question of whether Julia sets are computable under this definition has been completely solved in a series
of papers by Binder, Braverman and Yampolsky. See
[4,5,12–14]. Here we review some of their results. For simplicity, we give the definition of the Julia set only for quadratic polynomials. Consider a quadratic polynomial
Pc ðzÞ 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 : sup jPnc ðzÞj < 1 :
n
That is, the set of points whose orbit remains bounded. For
the filled Julia set one has the following result:
Theorem 28 [14]. The filled Julia set Kc of a computable
quadratic polynomials Pc is always computable.
The Julia set can now be defined by:
J c ¼ @K c ;
where @(A) denotes the boundary of A. The Julia set is the
repeller of the dynamical system generated by Pc. For all
but finitely many points, the limit of the nth preimages
Pn
c ðzÞ coincides with the Julia set Jc. The dynamics of Pc
on the set Jc is chaotic, again rendering numerical simulation of individual orbits impractical. Yet Julia sets are
among the most drawn mathematical objects, and countless programs have been written for visualizing them.
In spite of this, the following result was shown in [12].
Theorem 29. There exist computable quadratic polynomials
Pc(z) = z2 + c such that the Julia set Jc is not computable.
This phenomenon of non-computability is rather subtle
and rare. For a detailed exposition, the reader is referred
to the monograph [13].
Thus, we cannot accurately simulate the set of limit
points of the preimages (Pc)n(z), but what about their statistical distribution? The question makes sense, as for all
z – 1 and every continuous test function w, the following
holds:
1
2n
X
w2ðP c Þn ðzÞ
wðwÞ !
n!1
Z
wdk;
where k is the Brolin–Lyubich probability measure [15,39]
supported on the Julia set Jc. We can thus ask whether the
Brolin–Lyubich measure is computable. Even if Jc = Supp(k)
is not a computable set, the answer does not a priori have
to be negative. In fact, the following result holds:
Theorem 30 [6]. The Brolin–Lyubich measure of computable
quadratic polynomial is always computable.
This result essentially follows from Theorem 17 as well.
The difficulty is once again to find a recursively compact
condition which characterizes Brolin–Lyubich measure.
This measure happens to be singular with respect to Lebesgue, so that the absolute continuity argument does not apply. The key property allowing the application of
Theorem 17 in this case is balance: a probability measure
l on C is said to be balanced with respect to a rational
function R (of degree at least 2), if for every set A C on
which R is injective, we have:
lðRðAÞÞ ¼ d lðAÞ;
where d denotes the degree of R.
Computation of the Brolin–Lyubich measure is then reduced to show that for probability measures, being balanced is a recursively compact condition. The proof of
this fact is not very difficult, but it does require to overcome some problems like the fact that the function
l ? l(A) is not computable. For more details the interested reader is referred to [6].
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
3.4. Computable systems without computable invariant
measures
We have seen several techniques to establish the computability of many important invariant measures. This
raises naturally the following question: does a computable
system necessarily have a computable invariant measure?
What about ergodic physical measures?
The following is an easy example o a simple system for
which the last answer is negative, showing that the general
question of computing invariant measures has some
subtlety.
Let us consider a system on the unit interval given as
follows. Let s 2 (0, 1) be a lower semi-computable real
number which is not computable. There is a computable
sequence of rational numbers si such that supisi = s. For
each i consider Ti(x) = max(x, si) and
TðxÞ ¼
X
2i T i ðxÞ:
iP1
The functions Ti are uniformly computable so T is also
computable.
The system ([0, 1], T) is hence a computable dynamical
system (see Fig. 1). T is non-decreasing, and T(x) > x if
and only if x < s.
This system has a physical ergodic invariant measure
which is ds, the Dirac measure placed on s. The measure
is physical because s attracts all the interval at its left.
Since s is not computable then ds is not computable. We
remark that coherently with the previous theorems ds is
not isolated.
It is easy to prove, by a simple dichotomy argument,
that a computable function from [0, 1] to itself must have
a computable fixed point. Hence it is not possible to construct a system over the interval having no computable
invariant measure at all (we always have the d over the
fixed point). With some more work one can construct such
an example on the circle. Indeed the following can be
established.
Proposition 31. There is a computable, continuous map T on
the circle having no computable invariant probability
measure.
For the description of the system and for applications to
reverse mathematics see [26] (Proposition 12).
4. Computing the speed of ergodic convergence
As recalled before, the Birkhoff ergodic theorem tells
that, if the system is ergodic, there is a full measure set
of points for which the averages of the values of the observable f along its trajectory (time averages) coincides with
the spatial average of the observable f. Similar results can
be obtained for the convergence in the L2 norm, and others.
Many, more refined results are linked to the speed of convergence of this limit. And the question naturally arise, if
there is a possibility to compute this speed of convergence
in a sense similar to Definition 5.
In the paper [2] some abstract results imply that in a
computable ergodic dynamical system, the speed of convergence of such averages can be algorithmically estimated. On the other hand it is also shown that there are
non ergodic systems where this kind of estimations are
not possible. In [27] a very short proof of this result for
ergodic 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 the
speed of convergence’’ in a pointwise a.e. convergent
sequence.
Definition 32. A random variable on (X, l) is a measurable function f : X ! R.
Definition 33. Random variables fn effectively converge
in probability to f if for each > 0, l{x : jfn(x) f(x)j < }
converges effectively to 1, uniformly in . That is, there is
a computable function n(, d) such that for all n P n(, d),
l{jfn fj P } < d.
Definition 34. Random variables fn effectively converge
almost surely to f if fn0 ¼ supkPn jfk f j effectively converge
in probability to 0.
Fig. 1. The map T.
Definition 35. A computable probability space is a pair
(X, l) where X is a computable metric space and l a computable Borel probability measure on X.
Let Y be a computable metric space. A function f :
(X, l) ? Y is almost everywhere computable (a.e. computable for short) if it is computable on an effective Gd-set of
measure one, denoted by domf and called the domain of
computability of f.
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
A morphism of computable probability spaces f :
(X, l) ? (Y, m) is a morphism of probability spaces which
is a.e. computable.
Remark 36. A sequence of functions fn is uniformly a.e.
computable if the functions are uniformly computable on
their respective domains, which are uniformly effective Gdsets. Observe that in this case, intersecting all the domains
provides an effective Gd-set on which all fn are computable.
In the following we will apply this principle to the iterates
fn = Tn of an a.e. computable function T : X ? X, which are
uniformly a.e. computable.
Remark 37. The space L1(X, l) (resp. L2(X, l)) can be made
a computable metric space, choosing some dense set of
bounded computable functions as ideal elements. We say
that an integrable function f : X ! R is L1(X, l)-computable
if its equivalence class is a computable element of the computable metric space L1(X, l). Of course, if f = g l-a.e., then f
is L1(X, l)-computable if and only if g is. Basic operations
on L1(X, l), such as addition, multiplication by a scalar,
min, max, etc. are computable. Moreover, if T : X ? X preserves 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 measure
preserving morphism and (X, l, T) is ergodic. Let kfk denote
the 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)).
1
4.2. Effective almost sure convergence
Now we use the above result to find a computable estimation for the a.s. speed of convergence.
Theorem 39. Let (X, l, T) be a computable ergodic system. If f
is L1(X, l)-computable, then the a.s. convergence is effective.
Moreover, the rate of convergence can be computed as above
starting from T, l, f.
This will be proved by the following.
Proposition 40. If f is L1(X, l) -computable as above, and
kfk1 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 ergodic theorem:
Lemma 41 (Maximal ergodic theorem). For f 2 L1(X, l) and
d > 0,
l
The idea is simple: compute some p such that kAfp k1 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 that
kAfp k 6 d=2. Applying the maximal ergodic theorem to
g :¼ Afp gives:
l
sup jAgn j > d=2
6 :
Now, Agn is not far from Afn : expanding Agn , one can check
that
Agn ¼ Afn þ
Proof. Replacing f with f fdl, we can assume that
R
fdl 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
6 pðp1Þ
kf k1
2
kAgn Afn k1 6
!
n1
X
1
pAp T pi þ kAk T pn ;
np þ k i¼0
1
ðnpkAp k þ kkAk kÞ
np þ k
kAk k
6 kAp k þ
n
kf k
:
6 kAp k þ
n
kAnpþk k 6
Let > 0. We can compute some p = p() such that
kApk < /2. Then we can compute some nðÞ P 2 kf k. The
function m() :¼ n()p() is computable and for all
m P m(), kAmk 6 . h
so
if
n P n0 P 4(p 1)kfk1/d,
kuk1
then
d=2. As a result, if jAfn ðxÞj > d for some
n P n0, then jAgn ðxÞj > d=2. From (5), we then derive
l
(
sup
nPn0
Anpþk ¼
u Tn u
;
np
u = (p 1)f + (p 2)f T + + f Tp2.
where
R
ð5Þ
n
2
The L convergence (resp. L convergence) of the Birkhoff
averages An = (f + f T + + fTn1)/n is effective. That is:
there is an algorithm n : Q ! N such that for each
R
m P nðÞ; kAm fdlk 6 . Moreover the algorithm
depends effectively on T, l, f.
1
sup jAfn j > d
6 kf k1 :
d
n
jAfn j
)!
>d
6 :
As n0 can be computed from d and
result. h
,
we get the
Remark 42. This result applies uniformly to a uniform
sequence 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 us
consider fM0 2 L1 ðX; lÞ defined as
fM0 ðxÞ ¼
minðf ; MÞ
if f ðxÞ P 0
maxðf ; MÞ if f ðxÞ 6 0:
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S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
Compute M such that kf fM0 k1 6 d. Applying Proposi
f0
tion 40 to fM0 gives some n0 such that l fsupnPn0 j AnM j >
dgÞ < . Applying Lemma 41 to fM00 ¼ f fM0 gives
f 00
l fsupn jAnM j > dg < . As a result, l fsupnPn0 jAfn j >
2dgÞ < 2. h
Remark 43. We remark that a bounded a.e. computable
function, as defined in Definition 35 is a computable element of L1(X, l) (see [30]). Conversely, if f is a computable
element of L1(X, l) then there is a sequence of uniformly
computable functions fn that effectively converge l-a.e.
to f.
5. Computing pseudorandom points, constructive
ergodic theorem
In this section we show how the previous results on
computation of invariant measures and speed of convergence allow to compute points which are statistically typical for the dynamics. Let X again be a computable metric
space and l a computable probability measure on it. Suppose X is complete.
Points satisfying the above recalled pointwise ergodic
theorem for an observable f, will be called typical for f.
Points which are typical for each f which is continuous
with 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, compared
to the whole space. For this reason, a computable point
could rarely be expected to be typical for the dynamics,
as defined before. More precisely, the Birkhoff ergodic theorem and other theorems which hold for a full measure set,
cannot help to decide if there exist a computable point
which is typical for the dynamics.
A number of theoretical questions arise naturally from
all these facts. Due to the importance of forecasting and
simulation of a dynamical system’s behavior, these questions also have some practical motivation.
Problem 44. Since simulations can only start with computable initial conditions, given some typical statistical
behavior of a dynamical system, is there some computable
initial condition realizing this behavior? How to choose
such points?
Such points could be called pseudorandom points, and a
result showing its existence and their computability from
the description of the system could be seen as a constructiveversion of the pointwise ergodic theorem.
Meaningful simulations, showing typical behaviors of
the dynamics can be performed only if computable, pseudorandom initial conditions exist. Computable points are
the only points we can use when we perform a simulation
or 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 class
of systems which satisfies certain technical assumptions
which includes systems whose decay of correlation is fas-
ter than Clog2(time). Using the results on the estimation
of the rate of convergence given in the previous section,
it is possible to remove these technical assumptions on
the speed of decay of correlations. It is also worth to remark that the result is uniform also in T and l (the pseudorandom points will be calculated from a description of the
system).
The following result ([24], Theorem 2 or [27]) shows
that from a sequence fn which converges effectively a.s.
to f and from its speed of convergence, it is possible to
compute points xi for which fn(xi) ? f(xi).
Theorem 45. Let X be a complete metric space. Let fn, f be
uniformly a.e. computable random variables. If fn effectively
converges almost surely to f then the set {x : fn(x) ? f(x)}
contains a sequence of uniformly computable points which is
dense in Supp(l). Moreover, the effective sequence found
above depends algorithmically on fn and on the function
n(d, ) giving the rate of convergence
Since by the results of the previous section n(d, ) can be
calculated starting from T, l and f, we can directly apply
the above Theorem to compute typical points for the
dynamics. Indeed, the following holds (see [27] for the
details).
Theorem 46. If (X, l, T) is a computable ergodic system and f
is L1(X, l) and a.e. computable, then there is a uniform
sequence xi of computable points which is dense on the
support of l such that for each i
lim
n!1
Z
1X
f ðT n ðxi ÞÞ ¼ f dl:
n
Moreover this sequence can be computed starting from a
description of T, l and f.
The uniformity in f of the above result together with the
existence of a r.e. collection of computable observables
which is dense in the space of compactly supported continuous functions, imply the following (see again [27] for the
details).
Theorem 47. If (X, l, T) is a computable ergodic system then
there is a uniform sequence xn of computable points which is
dense on the support of l such that for each n, xn is typical for
l. Moreover this sequence can be computed starting from a
description of T and l.
In particular, for the classes of systems where the interesting invariant measure can be computed by the description of the system (see Section 3.3.1, Theorem 26), we
obtain that in turn the pseudorandom points can be computed starting from a description of the system alone.
Corollary 48. Each piecewise expanding map, with computable derivatives, as in the assumptions of Theorem 26 has a
sequence of pseudorandom points which is dense in the
support of the measure. Moreover this sequence can be
computed starting from the description of the system.
These two statements can be seen as constructive/effective versions of the ergodic theorem.
S. Galatolo et al. / Chaos, Solitons & Fractals 45 (2012) 1–14
6. Conclusions and directions
In this article we have reviewed some recent results
about the rigorous computation of invariant measures,
invariant sets and typical points. Here, the sentence rigorous computation means ‘‘computable (up to any required
precision) by a Turing Machine’’. Thus, this can be seen
as a theoretical study of which infinite objects in dynamics
can be arbitrarily well approximated by a modern computer (in an absolute sense), and which cannot. In this line,
we presented some general principles and techniques that
allow the computation of the relevant objects in several
different situations. On the other hand, we also presented
some examples in which the computation of the relevant
objects is not possible at all, setting some theoretical limits
to the abilities of computers when used to simulate
dynamical systems.
The examples of the second kind, however, seem to be
rather rare. An important question is therefore whether
this phenomenon of non-computability is robust or prevalent in any sense, or if it is rather exceptional. For example,
one could ask whether the non-computability is destroyed
by small changes in the dynamics or whether the noncomputability occurs with small or null probability.
Besides, in this article we have not considered the efficiency (and therefore the feasibility) of any of the algorithms we have developed. An important (and more
difficult) remaining task is therefore the development of
a resource bounded version of the study presented in this
paper. In the case of Julia sets, for instance, it has been
shown in [10,43,11] that hyperbolic Julia sets, as well as
some Julia sets with parabolics, are computable in polynomial time. On the other hand, in [5] it was shown that
there exists computable Siegel quadratic Julia sets whose
time complexity is arbitrarily high.
For the purpose of computing the invariant measure
form the description of the system, in Section 3.3.1 we had
to give explicit estimations on the constants in the Lasota–
Yorke inequality. This step is important also when techniques different from ours are used (see, e.g. [37]). Similar
estimations could be done in other classes of systems, following the way the Lasota–Yorke inequality is proved in
each class (although, sometimes this is not a completely
trivial task and requires the ‘‘effectivization’’ of some step
in the proof). A more general method to have an estimation
for the constants or other ways to get information on the
regularity of the invariant measure would be useful.
Acknowledgement
We would like to thank The Abdus Salam International
Centre for Theoretical Physics (Trieste, IT) for support and
hospitality during this research.
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