Lower bounds for symbolic complexity of iceberg dynamical systems

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by  A. A. Prikhod'ko

LOWER BOUNDS FOR SYMBOLIC COMPLEXITY OF ICEBERG
DYNAMICAL SYSTEMS
ALEXANDERPRIKHOD’KO
2
1 Abstract. ThesymboliccomplexityofaninfinitewordW isthefunctionpW(l)counting thenumberof
0 different subwords inW of length l∈N. In this paper our mainpurpose is to study the complexity for a
2 class of topological dynamical systems, called iceberg systems, given bythe followingsymbolic procedure.
n Startingfromagivenfinitewordw1 weconstructasequenceofwords
Ja wn+1=wnραn(1)(wn)···ραn(qn−1)(wn),
whereρα(wn)isacyclicrotations ofthewordwn,andconsideraninfinitewordw∞ extending eachwn
7 totheright. Itisshownthatforicebergsystemsgivenbytherandomizedparametersαn(j)thecomplexity
2 function satisfies the estimate pw∞(l)&l3−ε forany ε>0, and at the sametime itis observed that this

estimaterepresents uptoasmallcorrectiontheoptimallowerboundforthecomplexityfunction,namely,
S] pwn+1(ln)≤ln3 alongthesubsequenceln=|wn|+1.
The work is supported by grant NSh-8508.2010.1 for support of leading Russian scientific schools and
D
RFFIgrantNo.11-01-00759-a.
.
h
t
a
m
1. Introduction
[

ThesymboliccomplexityofaninfinitewordW isthefunctionp (l)countingthetotalnumberofdifferent
1 W
v subwords in W of length l ∈ N. In a series of recent investigations the concept of symbolic complexity is
7 appliedtothestudyoforbitstructurebothintopologicaldynamicsandergodictheoryofmeasurepreserving
5 transformations (e.g. see [2, 3, 4, 5, 9, 10]). Further, a kind of invariant of measure-theoretic isomorphism
7
extending the concept of symbolic complexity is introduced by S.Ferencziin [6]. Another generalapproach
5
to the complexity or orbitstructure introduced by A.Vershik is basedonthe notionof scalingentropy [14].
.
1 Out investigation is motivated by the idea of applying the concept of complexity and coding arguments
0
to the problems of isomorphismin ergodictheory of measure preservingtransformationswith zero entropy.
2
We wouldliketo startwithmentioningthe classicalsymbolicconstructionofarank one dynamicalsystem.
1
: Given a non-trivial finite word v1 in a finite alphabet A, we define words
v
Xi (1) vn+1 =vn0sn,0 vn0sn,1 vn ··· 0sn,rn vn,
r where symbol“0” used to create small spacers between the copies of the wordvn, and an infinite wordv∞
a having each v as a prefix. Following the standard scheme we can consider the compact set K defined as
n
the weak closure of all translates of v and endow K with the standard measure which is invariant under
∞
the shift trandformation T. The triple (T,K,µ) is called rank one transformation.
Rankonetransformationservesasaclassicalexampleofdynamicalsystemwithsimple spectrum. Inpa-
per [12] a class of ergodic maps is introduced, which is similar to rank one systems and extends this class.
Namely, generalizing the rule (1), while defining w we combinate not only exact copies but any cyclic
n+1
rotationsofthepreviouswordw . Itisobservedthatthemeasurepreservingmapassociatedwiththisnew
n
symbolic procedure, called iceberg transformation, has always1/4-localrank and, under certain conditions,
simple spectrum. It is not obvious that the new class is actually larger than the class of rank one trans-
formations. At the same time, these two classes potentially could contain transformations having spectral
typeswithsimilarproperties. Thus,itisinterestingtofindaninvarianthelpingtodistinguishtheseclasses,
and, in fact, it can be easily deduced from the results of this paper that the typical complexity of iceberg
systems is pw (l) &l3−ε, hence, a wide class of iceberg systems is certainly goes beyond the class of rank
∞
one systems, since for any rank one map there exists a finite partition of the phase space generating a
1
2 ALEXANDERPRIKHOD’KO
word v∞ with the property pv∞(ln) . 12ln2 (see [7]) in contradiction with the given lower bound, proving
the existence of iceberg systems which are not rank one. Though, one can observe that this approach is
essentiallynon-spectral,hence,therearenoobstaclestoexistenceofapair(T ,T )ofspectrallyisomorphic
1 2
maps,whereT isrankoneandT isanicebergmap,butnotrankone. Thus,thisclassofsymbolicsystems,
1 2
more generalthan rank one, can be consideredas a source of new examples demonstrating that rank is not
a spectral invariant.
It is interesting to compare the lower boud for symbolic complexity of iceberg systems with the asymp-
totics of the Pascal adic transformation complexity, p(l)∼l3/6 established by X.M´ela and K.Petersen
[10].
2. Iceberg dynamical systems
We start with the a series of constructions and definitions.
Definition. Consider a finite alphabet A and a word w =a ...a in A. We use the notation |w| for
1 N
the length of the wor w. Given an integer number α, 0<α<|w|, let us define the cyclic rotation ρ (w) of
α
the word w as follows
ρ (w)=w w , whenever w =w w , |w |=α.
α 2 1 1 2 1
Observe that if b ∈ A is a letter and u is a word then ρ (bu)=ub and ρ =(ρ )α. Let us extend this
1 α 1
definition to all values of α∈Z setting
ρ (w)=w and ρ (w)=ρ (w), k∈Z.
0 α α+k|w|
Thenextdefinitionisthestartingpointoftheconstructionoficebergsystems. Westartwithdefiningan
infinite word w∞, then we consider the closure Kw∞ of the sequence {w∞} in weak topology, and finally
usingthesandardprocedureweendowthecompactsetKw withaninvariantmeasureµ(w∞) andwecome
∞
to an ergodic transformation T acting as left shift on the space (Kw∞,µ(w∞)) (see [12]).
Definition. Let w be a fixed finite word in A. We suppose that w is non-trivialand contains a pair of
1 1
different letters. Let us construct a sequence of words w , such that each next word w in the sequence
n n+1
is a concatenation of cyclic rotations of w , namely,
n
(2) w =ρ (w )ρ (w )···ρ (w ),
n+1 αn(0) n αn(1) n αn(qn−1) n
where q ≥2 is the number of entries ρ (w ). Now suppose, for simplicity, that α (0)=0 for any n so
n αn(j) n n
thatw extends w to the right,andafterthe infinite sequenceofsteps wecometo aninfinite wordw
n+1 n ∞
such that any wn is prefix of w∞. Let Kw∞ be the closure of {w∞} in weak topology, i.e. the set of all
sequences x = (xj)j∈Z such that any finite subword of x can be found as a subword of some wn. The
compact set Kw∞ is invariant under the left shift map
(3) T: (...,x ,x ,...,x ,...)7→(...,x ,x ,...,x ,...).
0 1 j 1 2 j+1
We call (T,Kw ) an iceberg topological dynamical system (without spacers).
∞
2.1. Symbolic complexity. Given an infinite word w let us consider the language L(w) defined as the
set of all finite subwords of the word w. We can partition the set L(w) into a sequence of subsets L(w,l)
according to the length of a subword,
L(w,l)={u∈L(w): |u|=l}.
Symbolic complexity p (l) of the word w is the function counting the number of different subwords in w of
w
length l or, in other term, the function measuring the complexity of L(w),
p (l)=#L(w,l),
w
where #A is the cardinality of A.
Our paper is devoted to the proof of the following observation.
LOWER BOUNDS FOR SYMBOLIC COMPLEXITY OF ICEBERG DYNAMICAL SYSTEMS 3
Theorem 1. Consider an iceberg system given bythe independent uniformely distributed parameters α (j),
n
and assume that q →∞ sufficiently fast, q ≥hγ with γ ≥1, where h =|w |. Then almost surely
n n n n n
(4) pw (l)≥l3−ε
∞
for any ε>0 and l≥l (ε).
0
2.2. Measure-theoretic point of view. Throughout this paper we will continuously compare pure com-
binatorial observations with the corresponding effects in the context of measure-theoretic ergodic theory.
Infinite words are interpreted in this case as discrete sample path of stationary random processes.
The concept of iceberg transformation can be extended if we endow the compact space Kw with the
∞
natural invariant measure µ=µ(w∞) generated by the set of empirical distributions µ definied as follows:
l
givenafinitewordw ∈L(w ,l),awordoflengthl,thevalueµ (w)istheasymptoticfrequencyofoccurence
∞ l
of w inside the infinite word w . This procedure leads to an ergodic dynamical system given by the left
∞
shiftmapT actingonthe probabilityspace(Kw ,B,µ(w∞)),whereB isthe sigma-algebraofBorelsets. It
∞
is shownin paper [12] that aniceberg map canbe obtainedas a result ofcutting-and-stackingconstruction
in a way independent on the symbolic formalism we discuss (see fig.1).
Themainideaofourpaperistoprovideacombinatorialbackgroundtofurtherstudyofatypicalsymbolic
complexity of ergodic iceberg systems with respect to the distribution µ(w∞) (cf. [6] and [14]).
Whenstudyingourdynamicalsystemasanergodictransformationitisnaturallytoaskiftheunderlying
propertyoftheicebergconstructioncanbeformulatedinaninvariantform,asakindofdynamicalproperty
preserved under any measure-theoretic isomorphism?
Indeed, let us consider the classical concept of rank one approximation.
Definition. We say that a measure preserving map T on a probability space (X,µ) is of rank one if
for any measurablepartitionP ={P ,...,P } and any ε>0 there exists a setΩ of µ-measure1−ε such
1 m ε
that for any x∈Ω the orbit of x represented in a form of an infinite word(x ), where Tjx∈P , satisfies
ε j xj
the following property: (x ) is ε-covered by disjoint words W˜ which are ε-close to some fixed word W :
j j (ε)
(5) (x )=···W˜ S W˜ S ··· S W˜ ··· ,
j 1 1 2 2 j−1 j
where W˜ concides with W up to at most ε|W | wrong letters.
j (ε) (ε)
Definition. LetT be ameasurepreservingtransformation. We saythatT admitsiceberg approximation
if for any finite measurable partition P and any ε>0 one can find a set Ω of measure 1−ε such that for
ε
anypointx∈Ω the orbit(x )ofxisε-coveredbywordsW˜ whichareε-closeto cyclicrotationsofafixed
ε j j
W :
(ε)
(6) (x )=...W˜ S W˜ S ... S W˜ ..., d¯ W˜ ,ρ (W ) <ε,
j 1 1 2 2 j−1 j j ϕj (ε)
where d¯(u,v) measures the fraction of non-matching letters in u a(cid:0)nd v. (cid:1)
It is important to say that a priory we cannot skip spacers in this definition. Thus, generally speaking
a common construction of an iceberg map should contain spacers as well, but, at the same time, it is
interestingto observethatfromthe complexity pointofview addingspacersbetweencyclic rotationswedo
not influence significatly to the asymptotics of pw (see the effect discussed below in remark 3).
∞
3. Upper bound for symbolic complexity along a subsequence
We start with the estimation of the symbolic complexity for the sequence of special kind
l =h +1=|w |+1. The idea of this calculation is to show that the lower bound discussed in the
n n n
next section is, in a sense, optimal.
Lemma 1. pwn+1(hn+1)≤h3n for any iceberg system.
Proof. Let us consider a subwords v of length h +1=|w |+1 inside w , and consider w as
n n n+1 n+1
a sequence of rotations ρ (w ). Suppose that v starts at position k so that v includes the letters
αn(j) n
4 ALEXANDERPRIKHOD’KO
f
e e
d d d
c c c c
b b b b b
a a a a a a
f f f f f
e e e e
d d d
c c
b
Figure 1. Sample ofaniceberg. Acolumnis anordinaryRokhlintowercorrespondingto
a cyclic rotation of the word “abcdef”. Transformation T lifts each set drawn as a square
to the upper square. Whenever a point reaches the top set in a column, it is mapped by T
to an arbitrary bottom set of another (or the same) column.
ραn(j−1)(wn) ραn(j)(wn) ραn(j+1)(wn) ραn(j+2)(wn)
w w
1 2
ξ
Figure 2. Estimationof pwn+1(l)for l with l =|wn|. The rectangle restrictsa subwordv
of length l. The vertical lines mark up the boundaries of blocks ρ (w ).
αn(j) n
at positions k..k+h and it covers exactly two adjacent subwords: ρ (w ) and ρ (w ). It is
n αn(j) n αn(j+1) n
easy to see that the subword v is uniquely determined by the value of the triple (k¯,α (j),α (j)), where
n n
k¯ =k (mod h +1), k¯ 6=0, and the total number of such triples is estimated as h3. This ides is illustrated
n n
on fig. 2. Let us summarize the estimates in the following line:
pwn+1(hn)≤h3n−h2n+hn, pwn+1(hn+1)≤h3n. (cid:3)
Now let us consider an iceberg system with spacers given by the scheme
wn+1 =wn0sn,0ραn(1)(wn)0sn,1 ··· 0sn,qn−2ραn(qn−1)(wn)0sn,qn−1,
and observe that spacers do not affect to the asymptotics along the sequence h +1. We write w (cid:22)w if
n 1
w is a subword of w.
1
Remark (Melting word effect). Consider a subword
v =w Sw , w (cid:22)ρ (w ), w (cid:22)ρ (w ),
1 2 1 ϕ1 n 2 ϕ2 n
of length l =|w |+1 that covers a part w of the block ρ (w ), a part w of the block ρ (w ),
n n 1 αn(j) n 2 αn(j+1) n
anda spacerS =0s. Let us imagine that the spacer S is growingto the left and becomes ν symbols larger.
It canbe observedthat the growthof the spacer can be compensated by the opposite cyclic rotationin the
LOWER BOUNDS FOR SYMBOLIC COMPLEXITY OF ICEBERG DYNAMICAL SYSTEMS 5
first block ρ (w ) by −ν positions. In other terms, the original word v coincides up to ν letters with
αn(j) n
a new word v given by the configuration
e v =w1Sw2, w1 (cid:22)ρϕ1−ν(wn) and S =0s+ν,
and a priori there are no reeasonsetoesay if v ecoincide with v or not? eIf we interpret the area covered by v
as a window we monitor the sequence ρ (w ) throughthen it looks like the block ρ (w ) is “melting”
from the right side. We erase ν lettersαinn(jt)he snubword w ewhile extending the spacer.ϕ1Furtnhermore, from
1
the statistical point of view the words v and v˜ are asymptotically d¯-close with µ-probability 1−ε , where
n
ε →0, if spacers are supposed to be asymptotically small.
n
Definition. Let us consider the order on finite words: w ≥ w if w differs from w by replacing
1 2 1 2
some letters to the spacer symbol “0”. We call saturated complexity of an infinite word w the function
p¯ (l) counting the the total number of maximal points with respect to the order “≥” in the set L(w,l) of
w
subwords in w of length l.
Theorem 2. p¯wn+1(hn+1)≤h3n for any iceberg system with spacers.
This theorem follows directly from the discussion of the privious remark.
4. Lower bounds for symbolic complexity
Now we pass to the discussion of lower estimates for pw (l). The idea is to observe that generically, for
∞
iceberg systems with sufficiently rich combinatorialstructure almost all configurations
v =w w (cid:22)ρ (w )ρ (w ), w (cid:22)ρ (w ), w (cid:22)ρ (w ),
1 2 ϕ1 n ϕ2 n 1 ϕ1 n 2 ϕ2 n
and, respectively, almost all triples (|w |,φ ,φ ) are observed inside the word w . The following lemma
1 1 2 n+1
is a simple combinatorial observation.
Lemma 2. Let us consider two large intervals on the integer line [k,k +m] and [l,l +m] of the same
length m, and assume that l−k6=0. There exists a subset A⊂[k,k+m] satisfying conditions
1
A∩(l+k−A)=∅ and #A≥ m(1+o(1)).
2
For the sequel it is convenient to find and fix a value m such that #A ≥ 1m in the above lemma
0 3
whenever m≥m .
0
Let us consider an iceberg system given by the independent random cyclic rotation parameters α (j)
n
uniformely distributed on the integer interval [0..h ), where h =|w |. Let us use notation u(cid:22) w in the
n n n ◦
case if u is a subword of some cyclic rotation of w, i.e. u(cid:22)ρ (w) for some φ. This new class of dynamical
φ
systems generalizes the class or randomized rank one dynamical systems introduced by D.Ornstein ([11],
see also [1]).
Definition. We say that our symbolic system satisfies property D (β) with β >0 if any two subwords
m
u(cid:22) w andv(cid:22) w ofthe samelengthm≥β|w | arealwaysidenticalwheneverthey areequal. Herewe
◦ n ◦ n n
use the word identical to define the case when two subwords has the same starting position in w, i.e., in
fact, they corresponds to one entrance of a subword. For example, two entrances of the word u=v =cat
in a lerger word w=littlecatandbigcatstarting at position 6, and position 15 are equal but not identical.
In the sequel we will use symbol P(A) for the probability of the set A accordingto the probability space
hosting the random parameters α (j). It is convinient to consider two letter alphabet A={0,1}. Assume
n
thatthe initialwordinthe icebergconstructionw containsν·|w | onesand(1−ν)·|w |. Itfollowseasily
1 1 1
from the definition that any word w contains exactly ν-fraction of ones. The following lemma mainly
n
concerns the n’th step of the construction when we build the new word w combining a sequence of q
n+1 n
random cyclic rotations of w .
n
6 ALEXANDERPRIKHOD’KO
Lemma 3. Consider a pair of independent cyclic rotations, for example,
W =ρ (w ) and W =ρ (w )
1 αn(j) n 2 αn(j+1) n
and a pair of intervals I ,I ⊂{0,1,...,|w |−1}. Let v =W | and v =W | be the two subwords
1 2 n 1 1 I1 2 2 I2
corresponding to the subsets of indexes I and I . Then the probability of matching v =v is less than
1 2 1 2
1−2ν+2ν2.
Proof. Itisenoughtoshowthattheconditionalprobabilityofmatchingv =v withrespecttothefinite
1 2
setof randomparametersthat influencedto the structureof w . The probility ofmatching is less than the
n
probabilityofmatchingoflettersatonlyoneposotioninI . The uniformrandomrotationgeneratesatany
1
position the random variable which is distributed as ‘0’ with probability 1−ν and ‘1’ with probability ν.
Thus, the one-letter matching process is equivalent to matching of two independent varibles given by the
vector {1−ν, ν}. (cid:3)
Remarkthatthislemmacanbesignificantlystrengthened,andourstrategyinthispaperistoavoidcom-
plicated investigationof randomprocessesincluding large deviationmechanism andto use only elementary
probability technique.
Lemma 4. Assume that β >max{36, m +1}·q−1. Then
0 n
P D (β)|D (1/2) ≥1−q3 ·h−βqn/12.
n+1 n n n
Proof. Suppose that D (1/2(cid:0)) is satisfied. Cons(cid:1)ider two subwords v and v of w (or its cyclic
m 1 2 n+1
rotation,further we will omit suchkindof remark)startingat positionk andk respectively. Assume also
1 2
that
|v |=|v |=M ≥β|w |=βq h .
1 2 n+1 n n
Both v and v are of the form:
1 2
v =v(head)ρ (w )···ρ (w )v(tail).
i i φi,1 n φi,N1 n i
In other words, it starts and ends with two incomplete blocks, enclosing the sequence of complete blocks
ρ (w ). In order to understand if v and v are matched, v =v , let us write both v and v starting
φi,j n 1 2 1 2 1 2
at zero position. At this point we must consider two cases which are essentially different. The first case
is related to the pair of subwords which are very “close”, that is the first positions y and y of v and v
1 2 1 2
satisfy inequality
1
|y −y |≤ |w |.
1 2 n
2
And the secondcase deals with the situation when matching v and v actually we match pairs of indepen-
1 2
dently rotated blocks.
InthefirstcasewecansimplyapplyaxiomD (1/2)tothesituationwhenawordw =ρ (w )ismatched
n φ n
with itsels shifted by non-zero amount of positions, since the size of overlapping is equal or grather than
|w |−|y −y |≥|w |/2.
n 1 2 n
Concerning the second case it can be easily seen that v and v contain a sequence of overlapping block
1 2
pairs
(ρ (w ),ρ (w )), j =j ,...,j +m−1,
αn(j) n αn(j+s) n 0 0
satisfying the following conditions:
1 1 M 1
ρ (w )∩ρ (w ) ≥ |w |, m≥ ≥ βq .
αn(j) n αn(j+s) n 2 n 2|w | 2 n
(cid:12) (cid:12) n
(cid:12) (cid:12)
In this formulas we shortlywrite j+s insteadof j+s (mod q ), where q is the totalnumber of subblocks
n n
ρ (w ) in the rotated word w . Now let us apply lemma 2 to the intervals [j ,j +m− 1] and
αn(j) n n+1 0 0
[j +s,j +s+m−1] on Z and find a set J such that #J ≥1/3·m and J ∩(s+J) = ∅. Observe that
0 0
the pair of random variables (α (j),α (j+s)) for j ∈J are globally independent, since all sets {j,j+s},
n n
j ∈J, are disjoint. Now applying axiomD (1/2)we come to the followingconclusion. The only possibility
n
for a pair of overlapping blocks ρ (w ) and ρ (w ) to match is to encounter the situation when
αn(j) n αn(j+s) n
LOWER BOUNDS FOR SYMBOLIC COMPLEXITY OF ICEBERG DYNAMICAL SYSTEMS 7
these two blocks are proper rotated and the word in the overlapping enter identically in both blocks, and
the probability of such events is exactlty h−1. Thus, counting all the pairs of subwords v and v under
n 1 2
testing, we have
P D (β)|D (1/2) ≥1−h3 ·h−#J ≥
n+1 n n+1 n
(cid:0) (cid:1) ≥1−h3 ·h−βqn/6 ≥1−q3 ·h−βqn/12.
n+1 n n n
(cid:3)
Supposethatsomedecreasingsequenceβ →0isgiven. Applying thepreviouslemmawecaneasilyfind
n
a sequence q such that for any n the probability discussed in the lemma is positive and we can take for
n
any step of the construction an appropriate configuration (α (0),...,α (q −1)) such that D (β ) is true
n n n n n
for all n.
Though, remark that this reasoning still has a strong logical gap. In fact, to start this process we
need a starting word w satisfying D (1/2) “a priori”, and actually we can do it by choosing such “never
1 1
matching” word like
10, 100, 1001, ...,
but in view offuther applications to the isomorphismproblemwe need a versionof this lemma that can be
applied to any starting word w .
1
Lemma 5. Suppose that β > max{36, m +1}·q−1 and assume that the initial word of the construction
0 n
contains ν-fraction of symbol 1. Then
P D (1/2) ≥1−2q3 ·η−βqn/12,
n n
(cid:0) (cid:1)
where η =1−2ν+2ν2.
Proof. Letus examine oncemorethe firstcaseinthe proofoflemma 4takinginto accountthatnowwe
cannot say nothing about the matching of large blocks in w . An example of such word poorly amenable
n
to decoding is 1010101010...10.
Without loss of generality we can assume that in a pair of matched subwords v and v the second
1 2
subwords starts just one position to the right of (the beggining of) v . In other words, if y =y +1. The
1 2 1
only iformation we have is just that the last symbol of any full block ρ (w ) in v is compared to the
αn(j) n 2
firstsymbolofthe next full block ρ (w ) included in v , and the probability ofthis eventis less than
αn(j+1) n 1
η (see lemma 3). The additional summand to the probability of matching is estimated as
h2 h ·η#J ≤q2 ·η−βqn/12.
n+1 n n
Further, if |y −y |>|w |/2 then the estimate is given by the value
2 1 n
h3 h ·η#J ≤q3 ·η−βqn/12
n+1 n n
and the result follows. (cid:3)
The idea of the forthcoming discussion is to examine, modulo axiom D (β ), how many subwords v of
n n
lengthh canappearasymptoticallyinthewordw . MatchingaxiomsD implythatcountingsubwords
n n+1 n
is,inasense,equivalenttocountingconfiguration(|w |,φ ,φ )describinghowv coverstwoadjacentblocks
1 1 2
ρ (w )ρ (w ) (cid:22) w . Let us consider triples (ξ,φ ,φ ) indexing such kind of configurations.
αn(j) n αn(j+1) n n+1 1 2
Recall that for any subword v in w of length h ,
n+1 n
v (cid:22)ρ (w )ρ (w ).
αn(j) n αn(j+1) n
Hereφ =α (j)andφ =α (j+1)arethecorrespondingcyclicrotationparametersoftheadjacentblocks
1 n 2 n
coveringv, and ξ has the meaning of point separatingthese blocks inside v. Notice that any tripple defines
in a unique way a word v that can occur as a subword of w . Let us denote this word as
n+1
v =V(ξ,φ ,φ ).
1 2
Our purpose is to find a set of configurations such that the corresponding words V(ξ,φ ,φ ) are different.
1 2
8 ALEXANDERPRIKHOD’KO
Lemma 6. Consider two tripes (ξ,φ ,φ ) and (η,ψ ,ψ ) and suppose that
1 2 1 2
β h <ξ <η <(1−β )h ,
n n n n
|ξ−η|>β h ,
n n
and axioms D (β ) are satisfied. Then V(ξ,φ ,φ ) = V(η,ψ ,ψ ) if and only if for any interval among
n n 1 2 1 2
[0,ξ), [ξ,η) and [η,h ) the corresponding subwords are matched identically, in other words, if
n
η−ξ =ψ −φ =ψ −φ =h +ψ −φ .
1 1 2 2 n 1 2
Observethat if the conditions oflemma 6 aretrue then evidently V(ξ,φ ,φ )6=V(η,ψ ,ψ ), exceptone
1 2 1 2
case: φ =φ and ψ =ψ which means that the adjacent blocks in the iceberg construction are cyclicly
1 2 1 2
rotated by the same amount of positions. In this case the boundaries given by the positions ξ and η are
transparent and cannot be recognized, since ρ (ww)=ρ (w)ρ (w).
φ φ φ
Proof of lemma 6. Theideaofthis lemmaistorequirethatallthe intervalsthatappearwhile matching
the words generated by these two configurations must be sufficiently long, so that we can apply property
D (β ). Infact,forintervals[0,ξ),[ξ,η)and[η,h )weknowthatthecorrespondingsubwordsintheblocks
n n n
must be identically located subwords, and it is easy to represent this conclusion in terms of parameters ξ,
η, (φ ,φ ) and (ψ ,ψ )). (cid:3)
1 2 1 2
Lemma 7. Suppose that all the triples (ξ,φ ,φ ) are found for subwords in w of length h . Then
1 2 n+1 n
counting configurations with φ 6=φ we have
1 2
1−4β
pwn+1(hn)& β n ·(h2n−hn)
n
if β ≥h−1+δ and 0<δ <1/2.
n n
Proof. Let us choose a progressionκ, κ+a, ..., κ+(m−1)a of length m, where
κ=a=⌊β h ⌋,
n n
(1−2β )h (1−2β )h (1−2β )(1−β(1−δ)/δ) 1−4β
n n n n n n n
m= ≥ −1≥ −1≥ ,
(cid:22) a (cid:23) β h +1 β β
n n n n
sinse (1−δ)/delta≥1 for δ ≤1/2. Any pair (τ ,τ ) in the set of triples
1 2
Σ= (κ+ja,φ ,φ ): φ 6=φ , 0≤j <m
1 2 1 2
(cid:8) (cid:9)
satisfiesthe conditions oflemma 6 andinadditionwe canstate thatV(τ )6=V(τ ), since φ 6=φ . Finally,
1 2 1 2
let us count the triples in Σ:
1−4β
#Σ=m·(h2 −h )≥ n ·(h2 −h ).
n n β n n
n
(cid:3)
Theorem 3. Suppose that ε > 0 is given, and q satisfies the asymptotics q ∼ hγ with γ > 1−ε. Then
n n n
our iceberg system given by the sequence of i.i.d. random parameters α (j) with |w |=q |w | almost
n n+1 n n
surely generates an infinite word w such that for n≥n
∞ 0
pw∞(hn)≥pwn+1(hn)≥h3n−ε
for n≥n .
0
Proof. Let us take β = h−1+δ with max{1−γ, 0} < δ < ε. In order to apply lemma 5 and establish
n n
property D (β ) let us look at the probability in 5:
n n
P D (1/2) ≥1−2q3 ·η−βqn/12 =1−E
n n n
(cid:0) (cid:1)
and
En =2η3γlogηhn−hn(−1+δ+γ)/12,
LOWER BOUNDS FOR SYMBOLIC COMPLEXITY OF ICEBERG DYNAMICAL SYSTEMS 9
where β q =h−1+δ+γ and −1+δ+γ >0. Since the series E converges,we can apply Borel–Cantelli
n n n n n
lemma and see that almost surely there exists n1 such that DPn(βn) for n≥n1. The symbolic complexity is
now estimated as follows:
h2
pw∞(hn)≥pwn+1(hn)≥const·βn =const·h3n−δ ≥h3n−ε,
n
whenever n≥n ≥n . (cid:3)
0 1
This theorem can be strengthen as follows to get symbolic complexity arbitrary close to l3.
Theorem 4. Consider a sequence q ∼ hγ, γ ≥ 1, defining a set of random iceberg systems given by
n n
the independent and uniformely distributed random parameters α (j). Then almost surely the symbolic
n
complexity for this iceberg system satisfies the inequality
pw∞(hn)≥h3n−ε, n≥n0(ε).
The only difference with the proof of the previous theorem is that we have to fix some maximal value ε
0
and a universal sequence βn =h−n1+δn such that δn →0 sufficiently slow.
It was observed by S.Ferenczi [5] that the lowest rate of growth for the complexity function conserning
rank one systems can be seen along the subsequence h . Thus, this case is concerned as the most difficult
n
if we are interested in the estimate from below (see also lemma 1). Now, the above theorems can be easily
extended to all lengths of subwords l∈N. The complexity outside the sequence h becomes even larger
n
(cf. [5]), since we have more freedomcombinating a wordfrom severalrotatedcopies of w . Then we come
n
to the following result.
Theorem 5. Given a random iceberg systems with q ∼hγ, γ ≥1, the infinite word w generated by the
n n ∞
system with probility one possesses almost cubic estimate for the complexity,
pw∞(l)≥l3−ε, n≥n0(ε),
where the function n (ε) depends on the system.
0
5. Application to the calculation of rank
Using the effects discussed in section 4 we can show that certain iceberg systems are not rank one. We
give only an idea of proof.
Theorem 6. Keeping the conditions of theorem 5 almost surely the iceberg system is not rank one.
Remark that the method we use is not applicable to the multiple rank property, for example, rank two.
Sketch of the proof. Suppose that our transformation T is rank one. Then there exists a partition
P = {B ,B } and an orbit x = Tkx such that the complexity of the word (x ) is quadratic along a
0 1 k 0 k
sequence H (see [5]),
i
1
p (H )≤ H2.
(xk) i 2 i
The most difficult point in the proof is to manage infinite words which are generated by arbitrary finite
partition P. In other words, the following argument is used to show that the complexity of (x ) is l3−ε
k
providing a contradiction. Having small ζ, erasing ζ-fraction of letters the following estimate remains true:
pw (l)≥l3−ε.
n+1
Indeed,ourinfintie word(x )iscoveredupto asmallerrorζ bycyclicrotationsofw andasymptotically
k n n
itiseasytoseefromapproximationthatpassingthroughthe icebergwithindexnnamesforallpoints(but
ζ -part) are ζ -close.
n n
Applying these arguments we must take into account that ζ is small but it can go to zero arbitrary
n
slowly. (cid:3)
10 ALEXANDERPRIKHOD’KO
6. Scaling approximation and Pascal adic transformation
LetusconsideratreeassociatedwiththePascaltrianglegivenbytheverticesetV ={(k,n): 0≤k ≤n},
and the set of admissible paths in this graph:
X = (y ,n): y =y or y =y +1 .
n n+1 n n+1 n
(cid:8) (cid:9)
Next, let us consider an order on X induced by the natural order on the verticies on same level: (k,n) ≤
(k+1,n). Pascal adic transformation T is the transformation on X that maps a path x to the smallest
path grater than x (up to a small neglectible set). Pascal map T becomes ergodic measure preserving
transformation if we endow X with the natural measure describing statistics of paths (see [13]).
We mention Pascaltransformationinthis paper by tworeasons. First, it belongs to the classofmeasure
preserving maps containing iceberg systems as well and based on the concept of scaling approximation
introduced below in this section. The second reason is the following theorem establishing the exact cubic
asymptotics for the symbolic complexity of T (see [10]).
Theorem 7. (X.M´ela, K.Petersen) The symbolic complexity of the Pascal adic transformation T satisfies
p(l)∼ 1l3.
6
Definition. We say that a measure preserving invertible map satisfies the property of scaling approxi-
mation of rank one with a scaling function λ(h) (or simply scaling rank one) if for any ε>0 almost every
orbitencodedwithafinite measurablepartitioncanbe ε-coveredbysubwordsu ofafixedword|W |=h
j (ε)
such that the average length of u is greater than λ(h)(1+o(1)).
j
The property of funny scaling approximation of rank one, by analogy with the funny rank one property,
is definied in the same manner but taking instead of subwords arbitrary restrictions v = W | of the
j (ε) I(j)
words W considered as a function W : [0..|W |−1)→A.
(ε) (ε) (ε)
Figure 3. On this figure we put the collection of towers in the cutting-and-stacking pro-
cedure for Pascal adic map, drawn in such a way that all the elementary sets sharing the
same level are always labeled with the same letter. Here the 9’th step od the construction
is shown, the vertical columns are the Rokhlin towers of height C(n,k), where n=9 and
k =0,1,...,n, andthe commonheightof this generalizedtower is equal to h =28 =256.
9
Red lines correspond to symbol ‘1’ and blue lines – to symbol ‘1’. Note: you can zoom
this picture and see in details any part of the coding sequence x ,x ,...,x .
0 1 255
Theorem 8. Any rank one (respectively rank m) transformation has the property of scaling approximation
of rank one (rank m) with λ(h)=1.
Theorem 9. Any iceberg map is of scaling rank one with λ(h)=1/2.