Complexity and Randomness of Recursive. Discretizations of Dynamical. Systems. T. Krüger, P. Seibt1 and S. Troubetzkoy. Forsehungszentrum. BiBoS.
'L..\-\
1
and Randomness
Complexity
Discretizations
of Recursive
of Dynamical
Systems
T. Krüger, P. Seibt1 and S. Troubetzkoy Forsehungszentrum BiBoS Universität Bielefeld W-4800 Bielefeld 1 1 Introd uction. We begin by giving a heuristie Suppose points
(X,T) in X.
is a dynamical With
be automatically recursive
description
system with a distinguished
the definition
aT-invariant
discl'etization
of "computability"
set.
discretization?
approxirnation
of the mapping
averages,
in recovering
Lyapunov
quite different behavior.
about the dynamics
simulation?
of (X, T) can
(X, T) on a computer about
the dynamics
behavior
of (X, T), i.e. suitable discretizations
discretizations
whieh lead to the expected
Dur motivation
partly philosophical:
what kind of ergodic theory can be düne in a constructive
ical articIes.
is partly practical-do
and simulation
in ergodic
theory
of periodic
points
and classical systems inthe
simulations
work?manner?
and phys-
by periodic orbits is an old erle. For example,
it is we11 known thai [B02]. Periodic
computer
are the object of many mathematical
The study of discretization
time
can give rise to
(classical) behavior?
Both discretization
is ca11ed a
In both Gases the information
Different recursive
Can one fihd natural
ibis will
points and orbits of these points under
is the ergodic
exponents,etc.
(finite 01' infinite)
T. What information
of (X, T) ean one reeover from a computer we are interested
subset of a11 "computable"
If we try to simulate
then we consider finite subsets of computable
taken in ibis. paper.
taken in ibis article
Any of its subsets
of (X, T). What information
Olle re cover from a recursive a computer
of the approach
Gibbs measures
orbits are regarded
search für quantum
chaos.
can be described
by the set
as a linkage between
quantum
The interpretation
of statisti-
cal properties restricted to the periodic points hag led to confusion in the literature, due to the fact thai in the classical sense ibis motion cannot be understood as chaotic . [Be,f,V]. The theory we dev:lop in ibis article resolves this paradox in the sense thai any finite ob server (who observes für arbitrary, hut finite, time and with arbitrary, hut finite, precision)
is unable to distinguish
the motion on the recursive diseretization
from
the original system. For a computer hand, coneeptually 1 CPT-CNRS
expert
diseretization
most dynamical
systems
means finite discretization. are highly non-eonstructive
Luminy, Oase 907, F-13288 Marseille Cedex9
On the ether objeets
(and
2 their behcwiour
is.usually
c:liscussed in this llon-construdive
taken in this article is thai countably
of an intermediate
infinite discretizations
setting).
approximation:
with an ordering
The approach
elevelop atheory
(i) as a lower approximation
for
to the
original continuous situation, (ii) as an upper approximation to finite eliscretizations, which will be considered as truncations of these infinite discretizations. It is wielely believed thai computer The evidence is mostly heuristic owing property
simulations
Most arguments
produce
correct ergodic behavior.
are based on the pseudo-orbit
(see [BCGGS],[Bll],[BG],[CFM],[GHY1],[GHY2],[NY]).
proven the best possible theorem for uniformly hyperbolic of ergodic theory:
almost an (with respect
Blank [Bll] has
systems frorn the point of view
to infinite product
are sha,clowed by points with gODelergo die averages.
shad-
measure)
pseudo-orbits
However computer
pseudo-orbits
form a set of measure 0 in Blank's sense anel thus need not be shaelowed by generic points.
Blank hag also studieel the persistence .
period multipication
on simulation
A uniform treatment dynanl.ics. system ,
anel the effects of
[B12].
of computability
aspects is best clone in the setting of symbolic
We can code any transformation
by constructing
.
of perioelic trajectories
a generating
on a Lebesgue
partition
space to 1:1, symbolie
anel cohsic1ering the symbolic
shift
system
corresponding to it [K]. Thus it is enough to stuely the ergodic properties of subsets of recursive
points
of the syrnbolie
system.
Let ~ c
~m = {I, 2, . . . , m}Z be the
image of (X, T) uneler such a coding and let a be the BhiEt transformation. :r; =
N such thai f(2i) - = Xi X-i for an i [C]. Let ~ be the set of recursive sequences in~. ~ is a
{:r;i} ~~ E ~ recursive if illere is a reeursive funcbon f : N -
and f(2i
1) =
~
countable
a-invariantset,
(t, a) with respect need a structure
is countable, measures
hence we call it a lattice.
to an invariant
on
t
which
no non-atomic
we use densities
or any' sublattice
measure.
enables
To study ergodic properties about
statistical
prob ability measures
with respect
B C t. A Gödelization
---t
We study the ergodic p~operties of
us to speak
invariant
require thai (B, '1) is recursive. natural
to bijective
exist.
Gödelizations
finite observer can observe. finite ergoelic properties
for the lattice,
Instead
t
Sinee
of invariant
(enumerations)
of f;
for example it is
Every Gödelization
thai is ergodie properties
which a
In ibis article we stuely which lattices (B, '1) give rise to gODel
corresponding
to same fixeel measure
of (f;, a) in terms of eomplexity
already stuelied the relationship the classical framework
properties.
Note the set B neeel not be a-invariant,
gives rise to finite ergoclic properties
of (t, a) we
is given by a bijection '1 : N f-+ B. We do not
to consider sets B that consist of one point from each orbit.
ergodie properties
We eall
properties
~i. vVe want to classify
of (B, '1). Bruelno [BI'] hag
between ergoclic theory ancl algorithmic
of a11sequences
(non-recursive).
complexity
in
'
3 The structure state
all theorems.
01' a11 theorems. this article
In seetion
is as follows.
In section
2 we give fonnal
3 we give Borne exafnples
5 we state
someopen
and in section
problems.
in [KT2] 1'01'the special
Several
case 01' Cl.measure
definitions
and
4 we give proofs
01' the theorems 01' rnaxinla1
in
entropy.
and Theorems.
Let X*
=
In section
were proven
2 Definitions
[w]
01' the article
be the set 01' finite
{x E
'Z;m, : Xi
=
strings
=
Wi für i
string w. We neeel a constructive
in 'Z;m. Astring
0,1,...,
W E X*
defines
lew) -- 1} where l(w)is
concept of a probability
measure
a eylinder
the length 01' the [USS]. A probablity
meaSure Il on 'Z;mis calleel comp'U,tableif there is a recursi:ve 1'unction M : X* XN2 such that IA1(w,rn,n) - p([w])!:::;nlm for all (w,n,rn) For W E X* let A~(x)
['Lu] anel
+ N2
E X* X N2.
be the difference betweeri the average munber
Ure first '/I,iterates of x to the set
set
of visits of
the expecteel frequency ot:visiting the set [w],
that is 1 n-l
A:~(x)
=
-:;;
L
i=O
. (l[w] (0"2X) -
anci &:,8 is the set 01' x who~e first r; iteratesvisit
(2.1)
p([w]))
the set [LU]within
5 of the expecteel
frequeney:
&:,0 = {x E B : IA~~(x)1< 5}. We say that the lattice
(2.2)
(8, I) satisfies the finite ergod'ic -theorem for the measure
V5 > 0, Vw E X*, that is 1'or each 'LUE X* and 8
>
lim n-HX>
den ([:,O) .- = 1,
j1.iil (2.3)
0 a set of x of lower density Olle visit thc set [Lu]with
the expectecl frequeney::l::8. Here the lower clensity of A c B is definecl by the following
formula :
It is natural
delleA)
= lirn inf k-+oo
hut not necessary
1 -k card {x E A : 1-1 (x)
:::; Tc}
(2.4)
.
to assurne that this limit is effectively computable2.
All
exa:mples in this article are 01'this type. All proofs remain unchangecl under this stronger requirement,
thus we will not comment
ergodic averages. Orle cannot,
Backward
usinga.
itself is countable.
on it further.
Here we consider
only forward
ergodic averages can be dealt with analogously.
countable
intersection
argument,
conclude the case 8
'Note that
=
0 since 8
Such a finite ergodie theorem haB the following interpretation.
If
we
M :N
+
._-~....__...._---_._---_._.__._..__.-
2 lim an
=
0. is called effectively computable
N such that limM(n)
=
00 anel
lan -:- 0.1
fllnction.
null set
A lattice
0 such that
(2.6) den { x E ß : x E i~ [qSuniv(6,i)J} .0, 3a > 0 such that clen{x E ß: kM(x(n)) For convenience of complexity.
~ -logp([x(n)])
-log(nl-a)}
in the following we will consider Kolmogorov's
Namely consicler a partial recursive function f : X* X N
Then the Kolmogarov
complexity
Kj(x) where again inf l(0)
~ 1-
=
in the sense of equation
(2.12)
original definition
(computer)
X* .
(2.13)
inclucecl by f is definecl by the formula
= inf{l(y):
00. For this definition (2.10).
-+
f Vn ~ 1.
f(y,l(x))
= x}
there is also a universal
Again we shall choose a fixecl universal
(2.14) p.l'. computer Kolmogorov ',.
------------..
7 computer'Ll definitions
and we shall write K(:I:(n)) instead of Kv(:I;(n)). is compl~xity K(x)
vVe call a lattice complexity
KR
K lvI replaced
KJyIR and VVKR Suppose
01' liVKR
ergodie I
4.
This
the main
Equation
(2.11) OI' (2.12) with the (2.15) implies that JC~ ~
meaBures für subshifts
of maximal
of finite type are always com-
satisfies
the finite
for Pma.~'
bounds
is quite
can be faHnd 1'01'the rate 01' convergence
technical
concepts
behind
inequality
Note that
(2.15)
of finite type anel fLmax is its Ineasure
and
thus
we do not include
the theorems.
loge n 1- CI mn
[1 -
(.n~ - \) me ]
.
(3.1)
The following list exhibits this lattice for 2:;2' ...000000...000... ...
1 00000 ...
...000000...000...
000...
'..
10 ...010000...000... ...100000...000... ...110000...000... ...000000...000.., ", 001 000 ... 000... ...010000...000... ... 011
000
... 000...
... 100 000
... 000...
Another
example,
rioclic repetition be made
Lattice
of the finite block
by the same
construction
insteacl of triloiling zeros. Similar
with
constructions
pecan
for subshifts of finite type,
2.
Suppose
thai j.lis an arbitrary computable
min{f.L([x]) : denotes
the perioclic lattice, is given
f.L([x])
>
0, l(x)
will appeal' (with
ordere clin increasing to minimize for example,
Cl,
in the space
in the following
way.
a(:Ti) is a clecreasing sequence. .a(:1:2)copies of :r:2asunifonnly
We
2:;. Let an :=
construct a ~l-weighted finitelattice.Each
trailing zeros) exactly
order of l(x). The
fluctuations
with support ,
= n} anclfor:c E X* leta(x) := lf.L([:c])janJ where laJ
the greatest integer less than
string .x E X*
measure
strings of the same (I/rn) I:;7:~1
average
Suppose Take
a( x) times.
a(xd
The
strings will be
length will be ordered l[w](--y(i)). This
can
the strings of length n are nurnbered copies
of the sequence
as possible. Continue
includively.
Xl The
and
be
so as clone,
so thai
intersperse
resulting lattice
is a. f.L-distribution.
Lattice 3. Let y
=
...
0100011011 000001
010 011
This is an example
of a Champer-
nowne sequence.For thislattice r(n) will be y with the nth bit changing parity. This lattice satisfies the finite ergodic theorem for 2:;2with the Bernoulli measure.
In fact it
even satisfies it tor E = O. However, this lattice is not }( R since for this lattice illere exists a constant
C such that rl~g{x E B: ](11,4(x(n))< C} = 1 \In:::: 1. Here C can
be choosen to be SUPn J(l(y(n)).
This shows that the inverse implication
in Theorem
4
is very far horn being true. Lattice 4.
'..
11 Let y be an infinite ranelom sequence. truncated
First construct
a lattice with rCn) heilig y
after the nth place with trailing zeros. This lattice is KfIIIR. Next, we make
this into a shirt invariant Insert into the lattice . in a monotone Theorem 5.
lattice
by choosing an increasing
sequence
ni of elensity O.
along this sequence an shirts of rows of this lattice-' (for instarlce .
way). This shift invariant
lattice is a counterexample
to t,he converse of
Lattice 5. Take the perioc1ic lattice
P in lexicographic
lexicographic Göelelization r a new lattice
C
oreler for~m'
vVe elefine from the
P* , r*) by
'Y*(i) = aLPer(-y(i))/2JrCi). Here, perC x), x E P, elenotes the length of the minimal example of a 11-distributeel latticewith ~m' Note that, although
(3.2) period of x. This lattice
respect to the measure of maximal entropy p for
the lattice (P, r) is not ~l-distributed, I
nli:~ ~.n
is an
the following still holds:
mn-1
~ l[w](r(i)) i=O
(3.3)
p([w]).
~
Lattice 6.
Here we construct attractor.
All initial conditions
of iterations [0, I/l(w)]
an example of a recursive lattice (B, r) which hag a very bad ghost
anel (infinite)
to this ghost attractor
after a finite number
ergodie averages do not exist in the sense that UJ(A~(x))
=
\i(:c, w) E B x X*. However i~ all other senses Dur lattice will be gODel: (B, r)
will be a p-distribution
(see (2.17)) as weIl as invariant;
our finite ergodie theorem
sequence:r
are attracteel
E~2
furthermore
(B, r) will satisfy
ancl be K lvIR ancl fIIILR für p. First we construct
such that UJCA;~(:y;)) = [O,l/l(w)] Vw E X*.
Let f:
N
a single --+
N2 be
the Cantor pairing functiol1, i.e. the inverse of gen, Tn) = ((n + m)2 + 3n + Tn)/2 and
.fj(i) the projection of f on the j-th coordinate. Consider I the Göclelization of X* in lexicographical order ancl define h : N define x as follows :c
.
--+
N by hC1)
=
2 and h(n)
rCfzCO))rCfzCO))"'rCfzCO))rCfzC1))rCfzCI))
,
"v
h(O) times
"'"
"Y'"
h(l)
=
CL:i
E
(4.3) 111
'..
13 This theorem is a finitizeel version of the Levin-SchnoIT theorern [USS p. 155-159]. First we will show that if (13,"Y) is not JCNI R then it is .not JvILR. Thus we assurne that :::JE> 0 s.t. \je> 0 den {x E B: J(JvI(;r;(n)) \Ve (knote
-logf-L([;r(n)J
- c}
>
E for same n.
(4.4)
by De the set of all 'Ll)E X* such that J(AI('Lu) < -logf-L([w]) ~ c. Therefore (4.4) beeome
equation
0 the elensity of x
that for same
Eß
such that
xC'!'/')E De is
greater than E for ~ome n. Let xo, xl,. .. be all the minimal elements of Dc, that is if y :::;'xi and y i- xi then y r:j De. Let Pe:= Ui[xiJ. We claim thatf-L(Pe) < rn-co To see this note t.11.atby the minimality
of the xi we have that
p,(Pc):::; N.ow choose
L f-L([xi]) ::; L
m-(e+K
(4.5)
M(xD).
yi sud} that u(V'j, l(xi)) = xi anel l(vi) = '](Jvl(xi) where u is the fixeel
univeral Kolmogorov mapping. vVehave [vi]n [vi] =0 the :r:f are minimal.
Thcrefore,
Z Comparing
if 'l i- j since 'Llis monotone Bernoulli measure we have
if /\ is the uniform
m-KM(xi)
=
L
m-l(yi)
=
L A([yi])
~ m -c.
(4.7)
vVe wish to show tha.t P := nPe elefines an effective null set.
Unfortunately
the
{xi} neeel not be r.e.. However the set De is r.e. To see this let V:= {(c,x)
Q+ x X* : J(JvI(;c) set U:=
(4.6)
:::; 1.
(4.5) anel (4.6) we have the elesireel
equations
f-L(Pe)
sequence
anel
((;r,y)
0 s.t. \je>
that
is r.e. since'Ll is p.l'.. Höwever (c,x)
- c holds for same (x,y) EU.
can be checkeel recursively
Remember
E
z8, z:f, . ... such
zi clefines an effective We assumecl
E for Borne n(c, E).
(4.8) ',"
14 Let {wi} be a r.e. l( wi) > n. Then equation
of De. Let I(n)
be a positive
integer
such that Vi > I(n)
(4.8) implies I(n)
31' > 0 s.t. Ve> 0 den
Only a finite number equivalent to
{ x E B : x(n)E
(4.9)
i~[wi] }
> I' für same n.
of the [z,f] cover [lVi] thus there exists J(n)
such that
(4.9) is
J(n)
3E>OS.t./V5>Oden
{
Finally note that the set of approximating
able and enumerable.
(4.10)
i~[cP(5,i)]} >
xEB:xE
functions
Eforsomen.
für constructive
null sets is count-
A universal approximating function hag the property that there
exists a constant C such that V5VJ3L such that U!=o[cP(5,i)] C Ur=o[cPuniv(C . 5,i)]. This together
with (4.10) implies
3, > Q s.t. V8 > Q.den { x E ß
,x
E
(4.11)
~,[ ~univ(8,i)]} > , for someL
01' that B is not M LR. Next we show the other half of the theoreln,
namely that a lattice
which is not
lVfLR is not KM R. Thus we assurne that there is an effective null set A defined by the recursive function
cPsuch that the lattice (B, I) satisfies
{
3, > 0 s.t. \/8 > 0 cleri x E ß : x E
i~
[4>(8,i) J}
>
,
(4.12) far
same L.
First we need the following Lemma.
Ve > 0, 350 s.t. V5 E (0,50] allel Vn > 0 we call effectively5
tolJe mappillg
fe,6 : X*x N ---+ X* such that Kfc,6(X) satisfying that l( x) = n allel x ;:::cP(8, i) foT same i. 5 f:
procluce a mOllO-
< -logp([x])
-
c
holcls Vx
N X Q+ X N x X* ---+X* is p.T.
'..
15 Proof,of
,Lemma:
For each c,5,n let X ni:=
L-logp,([:r;J)
:= {:rE X*
: l(x) = n,x
2:
for Bornej} := {xd.
rjJ(5,J')
- cJ. Now
I:: m -Tii:::; I:: mlog
=
~L([X;])+c
I:: fl([XiJ)
mc
0 8.t. \lc> 0 elen
je
{x
EB
:
6:I(Yi):::;
-logp,([xiJ)
Finally - c. 111
(4.12).
We see
imply that
(x(n)) < -log
Kfc,/jo
(4.13 )
,
ni and such thai
Set Tl,:= maxoS;i::;LI(cjJ(80,i)) where L(8o) was clefinecl in equation that equation
Set
fl([x(n)])
- c}
>
e für Borne n(c).
(4.14) Now let Cr := 2r + 1. Use the above lemma to construct the ~orresponclingmappings fr
:= fc,.,80' Define a p.r. f by the formula f(or1x, n) := fr,(x,n)
consecutiveO's. then J{f(x(n» je>
where or means l'
Kfr(x)+r+1. Thus ifKfc,.,/jo (x(n)) < -logp,([x(n)])~cr - r. Then equation (4.14) above implies
ThenKf(x):::; < -logfl([x(n)])
0 s:t. \Ir> 0 den {x E ß: Kf(x(n))
< -logfl([x(n)])
- T} > e für same n(r).
,
No""T equation
(2.10)
completes Proof of Theorem 4:
For the proof of this theorem
,
=
f( w). To clefine flet
m
the proof.
abreviate
finite type are always computable. f
(4.15)
f-Lmaxby p. Maximal measures
für subshifts
of
For each W E X* we will use a special ccmputer
mw (x) be the number
Suppose l(x) = '17"then set dw(x) = Imw(x)
of occurences
ofw in the strihg
x E X*.
nf1([w])I. Put an orclering on the worcls
-
of length n as follows: dw(x) and 1'01'dw(x)
=
< dw(Y)
===?
x
>
Y
(4.16)
dw(Y) we put :r < y
{=::}
:r;less than Y in lexicographical
order.
(4.17)
Inforrnally, the eloser x is to having the correct ergodie frequency für the cylinder [w] the larger
i t iso f is then
clefined by f (y
if l is the length of the m-ary ordering.
Here m is the number
, n) =
expansion
the
yth
string
oflength
of the position
of symbols in ~A'
no Then
K f
(x ( n )
=
l
of x in the list given by ibis
'N
16 this orclering for the Bernoulli measure on 2:;2,for n = 4
The following list exhibits an cl [w]=[l].
Position Decimal ---
String
dw
0000
2
1
1
1
1111
2
2
10
2
0001
1
3
11
2
0010
1
4
100
3
0100
1
5
101
3
1000
1
6
110
3
0111
1
7
111
3
1011
1
8
1000
4
1101
1
9
1001
4
1110
1
10
1010
4
0011
0
11
1011
4
0101
0
12
1100
4
0110
0
13
1101
4
1001
0
14
1110
4
1010
0
15
1111
4
1100
0
16
10000
5
Binary ---
Kf(x(n))
Thus, for example, the above list should be reacl f(1110,4) = 1001. From equation (2.10) there exists a constant c such that
~
{x E B: K(x(n))
h(.o1" n -log(nl-LY)}
C
C {x E B: Kf(:c(n)) ~ htop' '/?,-log(nl-Ü')
(4.18) - c}.
For '/7E ((), 1) set
By Chebychev's
F7~(x) :-
~~-Ol (l[w](crix) nl-TJ/2 -IL([W]))
inequality
we have
IL ({x
= nTJ/2 . A~(x).
E I;A : IF~u (x) I S; t} ) ~ 1
-
(4.19)
(4.20)
var( FT~U)
,
t2
.
Now
var(F;~)
=
n}-'f/
.
n . var(.f~U) + 2
{
L
OS;i 0 choose N := NCr;,E) so large that für all n ~ N, Vii/n?7/2 < t. Then
equation
(4.23) ,implies
p({x Let
Cn
=
E I:A: IA~(x)l:::;; E}) ~ 1-
logm card{:r
E XA
: l(.r)
=
n}.
(4.24)
nI1-?7 \In ~ N.
Set gen) := 10g(n,1-CV)+ c. A simple
counting argument shows that
lcnJ-g(n)-I
card {.r E XA : leI:)
= n,
Kj(xCn)) ~ h. n
- gen)}
:::;; mcn
m~
i=I
m Cn
:::;;
Here la J is the greatest
~
-
integer 1ess than a. The greater
(
-g(n)-I
1- m
m-1
the f-comp1exity
)
< + 2. (4.25)
of astring
the greater it is in the order 0
deiined
m
and the theorem follows.
5:
{x E B: ICA1(x(n))::; in the
a tL-distribution . .
proof
-log,u([x(n)])
of theorem
2. Since
- c}. Note that Ae(n) C Pe where
fl( Pe)
O. Set G::,6 := {x E ~ : IA~(x)1 < 5}.
(4.28)
For each E > () by the Birkhoff ergodic theorem there exists n so that. ,u(G'/:,,5) > 1 - E. , Now x E E:,6
x E G'/:,,6
~
n B.
The setG'/:,,6 is a finite union of cyllnders
of length n
it follows that limm->oo(l/,m,) Z":-l o'11 o nw,o'(r(i)) = I-l(G~~,{j).Thus the density of [,:::,,6is greater than 01'equc1.lto 1- E. Since Ewas arbitrary
thus since (B,I) the theorem Proof
is a fl-distribution
m
follows.
of Theorem
8:
Fix U anel 8' > O. For E:> 0 let UE = {x EU:
B(x, E) CU}.
Then far E > 0 sufficiently
small j.1(U\ Uf) < 8'. Set 6' = min( E,8'). Uging corollary 7 we can choose N so large that für all n ~ N there is a K(n) so that forall k ~ K card({x E E~o,6/2: I(x)::; k} 2': (1 - 5/2)k. Then für fixedn 2': N, using the continuity of T (and of Tn) we choose Tc so elose to T that dist(Ti;r;, T;x) ::; 5/2 far a11 x E Xc anel für a11 0 ::; i ::; n. It fo11ows . U6 . 1 t lat lll11n->oo Den ( En:e(n)) = 1. l1li -
5 Open
Problems.
OHr investigation leads to many interesting further. We list a few hefe. (1) Classify the set of recursive f.
=
O. For product
measure
these
problems which we plan to investigate
strings which satisfy the finite ergodie theorem für are ca11ed norrrial
numbers
01' Bernoulli
Recursive Bernou11i sequences exist, for example the Champernowne in Lattice 3 ofsection "3. (2) The lattice or lack qfrandomness [KS] have recently
(8'1)
can be thought
of (B, I) as a matrix
sequence discussed
of as an N x Z matrix.
The rancloinness
needs to be ana.lyzecl. KrÜger ancl Seiht
clefinecl what it means für an infinite matrix
classical sense of Martin-Löf.
sequences.
t,o be random
in the
All our (B, I) turn out not to be ranc1om in this sense.
Howeve~ they may have weaker ranc1om properties, such as randomness along columns. Randomness along columns is interesting because it corresponcls to space averages.
'..
19 (3) Many (more natural) chaotic dynamical systems are homeomorphic to topological Markov chains with countable alphabets ([KT1]). Develop pa.rallels to Gur theorems für the countable case. This seems not to be difficult sinte the dassical notions of ]{ ,lv:!R and MLR
can easily be extendend
(4) Consider to increasing
syntacticallength
arithmetic
lattices E
alphabets.
the dass of a11recursive strings in ~2 which are Gödelized according für same fixed formallanguage.
Alternatively
consider
dass of a11provable total recursive strings in 2:;2with ~espect to
the more constructive Peano
to symbolic spaces with countable
ordered
in increasing
syntacticallength
or proof length.
Are these
finite ergo die, J{!vI R or W J{ M R or do they satisfy the ergodie theorem
für
= O. ~5) In Theorem
8 we showed that computer simulations
can give good finite ergo~ic
averages with high probability
für randomly
choosen initial points;
probability
and the goodness
by choosing the initial point randomly
probability
distribution
which weighs complex.(with
Can Olle improve the with respect to a
respect to the computer
encodlng)
points more heavily. Acknowledgments. The third author would like to thank Deutsche Forschungsgemeinschaft
(DFG) für
their support. References
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'.
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MaTkov PaTtitions
with SingnlaTities
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anel lrregular
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'..
