Stability of iterated function systems on the circle
aa r X i v : . [ m a t h . P R ] F e b STABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE
TOMASZ SZAREK AND ANNA ZDUNIK
Abstract.
We prove that any Iterated Function System of circle homeomorphisms suchthat one of them has a dense orbit is asymptotically stable. The corresponding Perron-Frobenius operator is shown to satify the e-property, i.e. for any continuous function itsiterates are equicontinuous. The Strong Law of Large Numbers (
SLLN ) for trajectoriesstarting from an arbitrary point for such function systems is also proved. Introduction
The action of discrete groups of homeomorphisms (diffeomorphisms) of the circle hasbeen a subject of intensive studies over last decades. See [8] and [16] for a detailed descrip-tion of recent results.In this paper, we study the dynamics of discrete, finitely generated semigroups of orien-tation preserving circle homeomorphisms. After assigning a probability distribution overthe set of generators, the system becomes an Iterated Function System. Iterated FunctionSystems were extensively studied because of their close connections to fractals (see [3]). Westudy spectral properties of the corresponding Markov operator P and its dual (transfer)operator P ∗ acting on the space C ( S ) of continuous functions on the circle. Note thatsuch systems are neither contracting, nor even contracting in average, so the well-knownmethods elaborated for contracting systems cannot be applied (see [3, 13, 19, 21]).On the other hand, if, instead of an action of discrete (semi)groups, one considers randommaps with some absolutely continuous noise, several strong spectral properties of the cor-responding transfer operator, including exponential decay of correlation, can be obtained(see, e.g., [10]). However the case of action of discrete semigroups of homeomorphismsseems to be much more delicate.We consider arbitrary finitely generated semigroups of orientation preserving homeo-morphisms. The only restriction on the system which we assume for all our results is thatone of the generators has dense orbits. Such systems will be called admissible .The paper is organized as follows. In Section 2 we introduce the necessary notation, andwe define the objects we deal with. We also introduce the notions of asymptotic stabilityand the e-property. Mathematics Subject Classification.
Key words and phrases.
Iterated Function Systems, Markov operators, Semigroups of Homeomorphisms.The research partially supported by the Polish NCN grants DEC-2012/07/B/ST1/03320 (T.S.) and2014/13/B/ST1/04551 (A.Z.). T.S. was also supported by EC Grant RAQUEL.
Our first, preliminary result is devoted to the uniqueness of an invariant distribution.It is proved in Section 3. We use here some ideas of Furstenberg (see [7]) and Arnoldand Crauel (see [2] and the references therein). Recently they have been developed byKleptsyn et al. in [5] for proving uniqueness of an invariant measure for groups of circlehomeomorphisms (see also [4]).In Section 4 we show some auxiliary properties of semigroups of orientation preservinghomeomorphisms which are useful in studying asymptotic stability of our Iterated FunctionSystems.Asymptotic stability is proved in Section 5.Section 6 is devoted to the proof of the e–property. Its usefulness in the study of asymp-totic properties of Markov processes may be observed in [12].Finally, in Section 7 we show that any admissible Iterated Function System on thecircle satisfies the Strong Law of Large Numbers (
SLLN ) for trajectories starting from anarbitrary point. 2.
Notation
Let S denote the circle with the counterclockwise orientation and let d( x, y ) denote thenormalized distance between x, y ∈ S , so that the length of the circle equals 1. If I is anarc in S then | I | denotes the (normalized) length of I . Let x, y ∈ S with d( x, y ) < / x, y ] we denote the oriented (shorter) arc in S , from x to y . We define the followingrelation on S × S : we say that x < y if d( x, y ) < / x, y ] coincides with the natural orientation on the circle. Writing x < x < · · · < x M weassume that d( x , x M ) < / x i < x i +1 , i = 1 , . . . M − B ( S ) we denote the σ –algebra of Borel sets. Further, C ( S ) denotes the space of allcontinuous functions equipped with the supremum norm k · k . By H + we shall denote theset of all orientation preserving circle homeomorphisms.By M and M fin we denote the set of all Borel probability measures and Borel finitemeasures on S , respectively. By supp µ for µ ∈ M fin we denote the support of µ .An operator P : M fin → M fin is called a Markov operator if it satisfies the followingtwo conditions: • positive linearity: P ( λ µ + λ µ ) = λ P µ + λ P µ for λ , λ ≥ µ , µ ∈ M fin ; • preservation of the norm: P µ ( S ) = µ ( S ) for µ ∈ M fin .A Markov operator P is called a Feller operator if there is a linear operator P ∗ : C ( S ) → C ( S ) (dual to P ) such that Z S P ∗ f ( x ) µ (d x ) = Z S f ( x ) P µ (d x ) for f ∈ C ( S ), µ ∈ M fin .Note that, if such an operator exists, then P ∗ (11) = 11, hence P ∗ ( f ) ≥ f ≥
0. As aconsequence, || P ∗ ( f ) || ≤ || P ∗ ( | f | ) || ≤ || f || , TABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE 3 so P ∗ is a continuous operator. A measure µ ∗ is called invariant if P µ ∗ = µ ∗ . Since S is compact, every Feller operator on S has at least one invariant measure, by theKrylov–Bogolyubov theorem.A Markov operator P is called asymptotically stable if there exists a unique invariantmeasure µ ∗ ∈ M such thatlim n →∞ Z S f ( x ) P n µ (d x ) = Z S f ( x ) µ ∗ (d x )for f ∈ C ( S ) and µ ∈ M .Following [12], we say that a Feller operator P satisfies the e–property if for any x ∈ S and a continuous function f : S → R we havelim y → x sup n ∈ N | P ∗ n f ( y ) − P ∗ n f ( x ) | = 0 , i.e. if the family of iterates { P ∗ n ( f ) : n ∈ N } is equicontinuous. An equivalent notiondescribing the e-property is the so–called almost periodicity of the dual operator P ∗ . Recallthat a bounded linear operator Q : F → F of a Banach space is called almost periodic iffor every b ∈ F the sequence ( Q n ( b )) n ∈ N is relatively compact, that is, its closure in F iscompact in the norm topology. See, e.g., [14] or [17], for a description of spectral propertiesof almost periodic operators.Let Γ = { g , . . . , g k } ⊂ H + be a finite collection of homeomorphisms, and let ( p , . . . p k )be a probability distribution on { , . . . , k } . Clearly, it defines a probability distrubution p on Γ, by putting p ( g j ) = p j . We assume that all p i ’s are strictly positive. Put Σ n = { , . . . , k } n , and let Σ ∗ = S ∞ n =1 Σ n be the collection of all finite words with entries from { , . . . , k } . For a sequence i ∈ Σ ∗ , i = ( i , . . . , i n ), we denote by | i | its length (equal to n ).Finally, denote by Ω the infinite product Ω = { , . . . , k } N . Let P be the product measuredistribution on Ω generated by the initial distribution on { , . . . . , k } .We consider the action of the semigroup generated by Γ, i.e., the action of all compo-sitions g i = g i n ,i n − ,...,i = g i n ◦ g i n − ◦ · · · ◦ g i , where i = ( i , . . . i n ) ∈ Σ ∗ . The action ofΓ = { g , g , . . . , g k } is said to be equicontinuous if the family of homeomorphisms { g i } i ∈ Σ ∗ is equicontinuous. On the other hand, it is said to be contractive if for each x ∈ S , thereexists an open interval I ⊂ S containing x and a sequence ( i m ) m ∈ N of elements of Σ ∗ suchthat the length of the intervals g i m ( I ) tends to 0 as m → ∞ .The pair (Γ , p ) will be called an Iterated Function System . The Markov operator P : M fin → M fin of the form P µ = X g ∈ Γ p ( g ) µ ◦ g − , where µ ◦ g − ( A ) = µ ( g − ( A )) for A ∈ B ( S ), describes the evolution of distribution due toaction of randomly chosen homeomorphisms from the collection Γ. It is a Feller operator,i.e., the operator P ∗ : C ( S ) → C ( S ) given by the formula P ∗ f ( x ) = X g ∈ Γ p ( g ) f ( g ( x )) for f ∈ C ( S ) and x ∈ S TOMASZ SZAREK AND ANNA ZDUNIK is its dual.We will say that an Iterated Function System (Γ , p ) is asymptotically stable if the cor-responding Markov operator P is asymptotically stable.3. Uniqueness of an invariant measure
The results in this section are in the spirit of [5] (see also [7]) but we have to point out onesubstantial difference. Deroin et al. studied a group of circle homeomorphisms so that theirfamily of transformations was richer than in our case. In particular for any homeomorphismthe inverse of it, and, of course, the identity, belonged to the class. Since we do not assumethis, we have to slightly strengthen our assumption. Indeed, instead of assuming that theaction of a semigroup is minimal, we suppose that the system is admissible. The systemΓ = { g , g , . . . , g k } ⊂ H + is said to be admissible if one of homeomorphisms, say g , issuch that { g n ( x ) : n ≥ } is dense in S for some (and thus all) x ∈ S . From now on weshall assume that if Γ is admissible, then g has a dense trajectory. If Γ is admissible, thenthe pair (Γ , p ) will be called an admissible Iterated Function System.We are in a position to formulate the main result of this section. Proposition 1.
Let
Γ = { g , g , . . . , g k } ⊂ H + be admissible. Then the operator P corre-sponding to (Γ , p ) admits a unique invariant measure for any distribution p . Before giving the proof of Proposition 1 we show the following lemma.
Lemma 2.
Let
Γ = { g , g , . . . , g k } ⊂ H + be admissible. Then the action of Γ is eitherequicontinuous or contractive.Proof We start with a simple observation. Namely, if we assume that Γ is admissible,then it is topologically conjugated to some ˜Γ ⊂ H + such that an irrational rotation ˜ g ∈ ˜Γis the conjugate of g . Thus, there is no loss of generality in assuming that Γ contains anirrational rotation; say, g is an irrational rotation.Observe that if d( g i ( x ) , g i ( y )) = d( x, y ) for x, y ∈ S and i = 1 , . . . , k , then the actionof Γ is equicontinuous. Otherwise there is an arc I = [ a, b ] ⊂ S such that | g ( I ) | < | I | .Choose α ∈ (0 ,
1) such that | g ( I ) | < α | I | . Let ε > I ′ = [ a, b + 2 ε ]we have | g ( I ′ ) | ≤ α | I | . Fix x ∈ S and let J be an arbitrary arc with | J | ≤ ε . Since g is an irrational rotation, there exist m and n , . . . n m ∈ N such that g n i ( J ) ⊂ I ′ and g n i ( J ) ∩ g n j ( J ) = ∅ for i, j ∈ { , . . . , m } and, moreover, P mi =1 | g n i ( J ) | = m | J | ≥ | I | .Further g ◦ g n ( J ) ∪ . . . ∪ g ◦ g n m ( J ) ⊂ g ( I ′ ) and consequently P mj =1 | g ◦ g n j ( J ) | ≤ α | I | .Hence | g ◦ g n j ( J ) | ≤ α | J | for some j ∈ { , . . . , m } . Otherwise, we would have mα | J | < P mj =1 | g ◦ g n j ( J ) | ≤ α | I | , which is impossible. By induction we show that there is a sequence( i , i , . . . ) ∈ Σ ∞ such that | g i n ,i n − , ··· ,i ( J ) | → n → ∞ and we are done. • Proof of Proposition 1 . As in the proof of Lemma 2, we can assume that Γ contains anirrational rotation.Let Γ = { g , . . . , g k } and let g be an irrational rotation. If Γ is equicontinuous, thenthe operator P satisfies the e–property. Markov operators with the e–property have been TABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE 5 already examined even in the setting of much more general phase spaces, i.e., general Polishspaces (see [11]). In particular, it was proved that such operators may have two differentinvariant measures µ and µ only if supp µ ∩ supp µ = ∅ (see Theorem 1 in [11]). Thismay not be the case if g is an irrational rotation for the support of any invariant measureis then equal to S .Now let Γ be contractive. Assume that uniqueness does not hold. Then there existsat least two different ergodic invariant measures. Again, since Γ contains an irrationalrotation g , every invariant measure is supported on the whole circle S . Let µ and µ betwo ergodic invariant measures. We shall prove that there exists a positive constant α anda measure ν such that µ i ≥ αν for i = 1 ,
2. Hence µ = µ for the fact that two differentergodic invariant measures are mutually singular. Let J ⊂ S be such that | g i m ( J ) | → m → ∞ for some sequence ( i m ) m ∈ N of elements of Σ ∗ . Put α := min { µ ( J ) , µ ( J ) } > µ ∈ { µ , µ } . Fix f ∈ C ( S ). We define a sequence of random variables ( ξ fn ) n ∈ N bythe formula ξ fn ( ω ) = Z S f ( g i ,...,i n ( x )) µ (d x ) for ω = ( i , i , . . . ) . Since µ is an invariant measure for P , we easily check that ( ξ fn ) n ∈ N is a bounded martingale.Note that this martingale depends on the measure µ . From the Martingale ConvergenceTheorem it follows that ( ξ fn ) n ∈ N is convergent P -a.s. and since the space C ( S ) is separable,there exists a subset Ω of Ω with P (Ω ) = 1 such that ( ξ fn ( ω )) n ∈ N is convergent for any f ∈ C ( S ) and ω ∈ Ω . Therefore for any ω ∈ Ω there exists a measure ω ( µ ) ∈ M suchthat lim n →∞ ξ fn ( ω ) = Z S f ( x ) ω ( µ )(d x ) for every f ∈ C ( S ) . Now we are ready to show that for any ε > ε ⊂ Ω with P (Ω ε ) = 1satisfying the following property: for every ω ∈ Ω ε there exists an interval I of length | I | ≤ ε such that ω ( µ i )( I ) ≥ α for i = 1 , S , weobtain ( P a.s.) that there exists a point υ ( ω ) ∈ S such that ω ( µ i ) ≥ αδ υ ( ω ) for i = 1 , δ υ ( ω ) is the Diract delta measure supported at υ ( ω ). It is standard to show that thepoints υ ( ω ), ω ∈ Ω, may be chosen in such a way that the function Ω ∋ ω → υ ( ω ) ∈ S ismeasurable. This will finish our proof. Indeed, define the measure ν ∈ M by the formula ν := Z Ω δ υ ( ω ) P (d ω ) . and fix a non–negative function f ∈ C ( S ). We have Z S f ( x ) µ i (d x ) = lim n →∞ Z S f ( x ) P n µ i (d x ) = Z Ω lim n →∞ ξ fn ( ω ) P (d ω ) ≥ α Z Ω f ( υ ( ω )) P (d ω ) = α Z S f ( x ) ν (d x )for i = 1 ,
2. Since f ∈ C ( S ) was an arbitrary non–negative continuous function, we obtainthat µ i ≥ αν for i = 1 , TOMASZ SZAREK AND ANNA ZDUNIK
We now complete our proof by constructing the claimed set Ω ε . This follows some ideasof Deroin et al. (see [5]). Fix ε > l ∈ N be such that 2 /l < ε . Since | g i m ( J ) | → i m if necessary) that the arcs J m := g i m ( J ), where m ≤ l are mutually disjoint. Put n ∗ = max m ≤ l | i m | . Now observe that for any sequence j = ( j , . . . , j n ) ∈ Σ ∗ there exists m ∈ { , . . . , l } such that | g j ( J m ) | < /l < ε/
2. This shows that for any cylinder in Ω,defined by fixing the first initial n entries ( j , . . . , j n ), the conditional probability that( j , . . . , j n , . . . , j n + k ) are such that | g j ,...,j n ,...,j n + k ( J ) | ≥ ε for all k = 1 , . . . , n ∗ is less than1 − q for some q >
0. Hence there exists Ω ε ⊂ Ω with P (Ω ε ) = 1 such that for all( j , j , . . . ) ∈ Ω ε we have | g j ,...,j n ( J ) | < ε/ n ’s. Since S is compact,we may additionally assume that for infinitely many n ’s the set g j ,...,j n ( J ) is contained insome set I with | I | ≤ ε . This finishes the proof. • Auxiliary results
We start with the following lemma. Recall that we have normalized the arc length sothat the length of the circle is equal to 1.
Lemma 3.
Let h be a circle orientation preserving homeomorphism. Assume that thereexists r < such that h maps every arc of length r onto an arc of length at most r . If r is irrational, then the map h is a rotation and if r is rational, then h commutes with therotation by r (denoted by T r ).Proof First, assume that r is irrational. Denote by B the set of all β ∈ (0 ,
1) such that h maps every arc of length β onto an arc of length at most β . Then r ∈ B . It is easy tosee that if β ∈ B then { nβ } (the fractional part) is also in B . Since r is irrational, the set { n · r } is dense in [0 , h this implies that h maps every arc I onto an arc of length at most | I | . Thus, h must be an isometry – a rotation.Now, assume that r is rational, say r = p/q . Denote by T r the rotation by r . Fix some x ∈ S and the arc I = [ x , T r ( x )].Observe that T qr = id and the arcs S q − i =1 T ir ( I ) cover the circle S exactly p times. Thesame must be true for S q − i =1 h ( T ir ( I )). Since the length of each arc h ( T ir ( I )) is not biggerthan the length of T ir ( I ), it implies that h maps every arc of length r = p/q onto an arc ofthe same length r = p/q . Consequently, h and T r commute: h ◦ T r = T r ◦ h. • Proposition 4.
Let Γ be contractive. Then there exists a rational rotation R which com-mutes with the elements of Γ and such that for any x ∈ S and any closed arc I ⊂ [ x, R ( x )) there exists a sequence ( i n ) n ∈ N of elements of Σ ∗ such that | g i n ( I ) | → as n → ∞ . TABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE 7
Proof
We will call an arc I contractible if there exists a sequence ( i n ) n ∈ N of elementsof Σ ∗ such that | g i n ( I ) | → n → ∞ . For x ∈ S we define R ( x ) in the followingway. Consider all positively oriented arcs [ x, y ] such that the arc [ x, y ] is also contractible.Clearly, if [ x, y ] ⊂ [ x, y ′ ] and [ x, y ′ ] is contractible then [ x, y ] is contractible. Keeping x fixed, take the union of all contractible arcs [ x, y ]. This is an arc of length at least, say, ε >
0. One of its endpoints is x ; the other endpoint is, by definition, R ( x ). Note that itmay happen that R ( x ) = x (in this case the length of [ x, R ( x )] is equal to the length ofthe circle).For x ∈ S denote by r ( x ) the length of the (positively oriented) arc [ x, R ( x )]. Now,let x, z, w ∈ S ; we assume that x < w < z . Then r ( z ) ≥ r ( w ) − d( z, w ) and r ( w ) ≥ r ( x ) − d( x, w ). Consequently:( r ( z )+d( z, x )) − ( r ( w )+d( w, x )) = r ( z ) − r ( w )+d( z, x ) − d( w, x ) = r ( z ) − r ( w )+d( z, w ) ≥ . Putting ˜ r ( z ) := r ( z )+d( x, z ), we thus have ˜ r ( z ) ≥ ˜ r ( w ) ≥ ˜ r ( x ) = r ( x ). Clearly, this impliesthat the function r is Borel measurable. Moreover, there exists a limit lim z → x + ˜ r ( z ), whichsatisfies lim z → x + ˜ r ( z ) ≥ r ( x ), and, consequently, there exists a limitlim z → x + r ( z ) ≥ r ( x ) . Similarly, there exists a limit lim z → x − r ( z ) ≤ r ( x ). Obviously, if an arc I is contractiblethen for every g i , i ∈ { , . . . , k } , the arc g − i ( I ) is also contractible. Thus(1) r ( g − ( x )) ≥ r ( x ) , where, recall, g is an irrational rotation. Since g is ergodic with respect to the Lebesguemeasure, formula (1) implies that the function r is constant (Lebesgue) almost everywherein S . Since, for every x , we have lim z → x + r ( z ) ≥ r ( x ), and lim z → x − r ( z ) ≤ r ( x ), bothlimits must be equal to r ( x ), so the function r is continuous and, since it is g - invariant,it is constant everywhere. So r ( x ) ≡ r for some r >
0. Consequently, for every x ∈ S thelength of the arc [ x, R ( x )] is equal to r .Now, choose g ∈ Γ such that g is not a rotation. If an arc I is contractible then g − ( I )is also contractible. Hence, g − maps any arc of length r onto an arc of length at most r .Therefore, r is rational, by Lemma 3. From Lemma 3 we conclude also that either r = 1and then R = id or r < R := T r . The proof iscomplete. • Lemma 5.
Let Γ be contractive and let R be a rational rotation that commutes with Γ .Then the unique invariant measure µ ∗ is R –invariant, i.e. µ ∗ ◦ R − = µ ∗ .Proof We have P ( µ ∗ ◦ R − )( A ) = k X i =1 p i µ ∗ ( R − g − i ( A ))= k X i =1 p i µ ∗ ( g i R ) − ( A )) = k X i =1 p i µ ∗ ( g − i ( R − ( A )) = µ ∗ ( R − ( A )) = ( µ ∗ ◦ R − )( A ) TOMASZ SZAREK AND ANNA ZDUNIK for any Borel set A . Since P possesses a unique invariant measure we conclude that µ ∗ ◦ R − = µ ∗ . • The above lemma and the proof of Proposition 1 easily imply the following.
Proposition 6.
The measure ω ( µ ∗ ) for ω = ( i , i , . . . ) defined P -a.s. is R –invariant and,consequently, ω ( µ ∗ ) = 1 M M − X m =0 δ R m ( υ ( ω )) , where υ ( ω ) are the points defined in the proof of Proposition 1 and M is the order of therotation R , i.e., the smallest integer such that R M = Id . Stability
We start with an easy criterion for stability when Γ = { g , . . . , g k } is equicontinuous. Theorem 7.
Let
Γ = { g , . . . , g k } be equicontinuous and let a probability distribution p begiven. Let P be the Markov operator corresponding to (Γ , p ) and P ∗ its dual. If for any f ∈ C ( S ) , f ≥ and f , there exists r ∈ N such that P ∗ r f ( x ) > for x ∈ S , thenthe Iterated Function System (Γ , p ) is asymptotically stable. The proof follows from the results proved for almost periodic primitive operators (fordetails see Theorem 5.5.3 in [17]. See also Theorem 6 in [20]).
Remark 8.
Assume now that Γ contains two rotations g = T α and g = T β such that( α − β ) is irrational. Then there exists r ∈ N such that P ∗ r f ( x ) > x ∈ S andconsequently the corresponding Iterated Function System (Γ , p ) is asymptotically stablefor any probability distribution p . Proof
Define the set U m,x := { T kα +( m − k ) β ( x ) : k = 0 , . . . m } = { T k ( α − β ) ( T mβ ( x )) : k = 0 , . . . m } for x ∈ S and m ∈ N . Fix f ∈ C ( S ) and observe that to prove that P ∗ r f ( x ) > x ∈ S and some r ∈ N it is enough to show that for every ε > x ∈ S there is m ∈ N such that U m,x forms an ε –net in S . Indeed, since f is continuous and f
0, thereexists ε > ε –net there is y from this net such that f ( y ) >
0. Further,if U m,x is some ε –net, then f ( T iα +( m − i ) β ( x )) > i ∈ { , . . . , m } and consequently P ∗ m f ( x ) ≥ p i p m − i f ( T iα +( m − i ) β ( x )) > γ := α − β , the problem reduces to the following immediate observation.Assume that γ is irrational. Then for every ε > m ∈ N such that for every y ∈ S the set { T kγ ( y ) , k = 0 , . . . m } forms an ε –net in S . • TABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE 9
Now, assume that Γ is equicontinuous, and let g ∈ Γ be a hoemomorphism with denseorbits. After the appropriate change of variables we can assume that g is an irrationalrotation. Using Lemma 2 we see that all elements of Γ are rotations. So we can reformulateRemark 8 as follows. Remark 9.
Assume that Γ contains an element with dense orbits, and that Γ is equicon-tinuous. If, for some g i , g j ∈ Γ the homeomorphism g − i ◦ g j has dense orbits then for anyprobability distribution p the Iterated Function System (Γ , p ) is asymptotically stable.The following lower bound criterion for stability of Markov operators generalizing Doe-blin’s theorem (see [6]) will be useful in proving stability when the family Γ is contractive. Theorem 10.
Let P be an arbitrary Markov operator. Assume that for any ε > and f ∈ C ( S ) there exists α > such that for every µ , µ ∈ M ( S ) there are ν , ν ∈ M ( S ) and N ∈ N satisfying (2) P N µ i ≥ αν i for i = 1 , and lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) P n ν (d x ) − Z S f ( x ) P n ν (d x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. Then the operator P is asymptotically stable.Proof Fix ε > f ∈ C ( S ). Fix µ , µ ∈ M ( S ). We shall show that(3) (cid:12)(cid:12)(cid:12)(cid:12)Z S f d P n µ − Z S f d P n µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε for all n sufficiently large. To do this choose k ∈ N such that 2(1 − α ) k k f k < ε , where α > ε and f . Using (2) we have P N µ i = αν i +(1 − α )˜ µ i ,where ˜ µ i are some probability measures. Proceeding inductively, and using (2) at everystep we may find sequences of probability measures ν i , . . . , ν ki and ˜ µ ki for i = 1 , N , . . . , N k such that P N + ··· + N k µ i = αP N + ··· + N k ν i + α (1 − α ) P N + ··· + N k ν i + · · · + α (1 − α ) k − P N k ν ki + (1 − α ) k ˜ µ ki for i = 1 , n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) P n ν j (d x ) − Z S f ( x ) P n ν j (d x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. for j = 1 , . . . , k (for details see Theorem 5.3 in [18]). Hence, by the definition of k , condition(3) follows. Since P admits an invariant measure, the proof is complete. • Lemma 11.
Let
Γ = { g , . . . , g k } be admissible and contractive. For any n ∈ N and x, y ∈ S there exists N ∈ N and two collections of elements of Σ N : i , . . . , i n , j , . . . , j n such that g i ( x ) < g j ( y ) < g i ( x ) < g j ( y ) < · · · < g i n ( x ) < g j n ( y ) . Moreover, one can require that the length of the arcs ( g i m ( x ) , g j m ( y )) and ( g j m ( y ) , g i m +1 ( x )) for m = 1 , . . . , n is bigger than some constant, say τ , depending on n but independent of x and y .Proof After a necessary change of variables one can assume that g ∈ Γ is an irrationalrotation. Fix x, y ∈ S and let I be such an arc that x ∈ I and | g l m ( I ) | → m → ∞ for some sequence ( l m ) m ∈ N of elements of Σ ∗ . Since g is an irrational rotation, we canadditionally assume that there is an open arc I , with I \ cl I = ∅ such that g l m ( I ) ⊂ I for all m ∈ N . We shall proceed by induction on n . The case n = 1 is obvious but we showthat we may additionally require that g i ( x ) , g j ( y ) ∈ I . Indeed, set h := g l m , h = g | l m | and observe that h is an irrational rotation again. Hence there exists l ∈ N such that h l ( y ) ∈ I \ cl I and h l ( y ) > z for every z ∈ I . Since h l ( x ) ∈ I , we are done.Now let the statement of our lemma hold for some n . Denote by ˜ I = ( g i ( x ) , g j n ( y )) ⊂ I .We have h ( ˜ I ) ⊂ I . Analogously as in the previous step we find l such that h l ( x ) ∈ I \ cl I and h l ( x ) > z for every z ∈ I . Hence there exist N and ˜ i m ∈ Σ N , m = 1 , . . . , n + 1,˜ j m ∈ Σ N , m = 1 , . . . , n such that g ˜ i ( x ) < g ˜ j ( y ) < g ˜ i ( x ) < g ˜ j ( y ) < · · · < g ˜ j n ( y ) < g ˜ i n +1 ( x ) . Replacing ˜ I with ˆ I = ( g i ( x ) , g i n +1 ( x )) and x with y , and repeating the above procedurewe show our lemma for n + 1.Now we observe that the length of the arcs ( g i m ( x ) , g j m ( y )) and ( g j m ( y ) , g i m +1 ( x )) for m = 1 , . . . , n may be bigger than some constant τ (depending on n ) but independent of x, y ∈ S . Fix z ∈ S . From what we have proved above it follows that for any u, v ∈ S there exist two sequences ( p n ) n ∈ N and ( q n ) n ∈ N of elements of Σ ∗ such that | p n | = | q n | for n ∈ N and d( g p n ( u ) , g q n ( v )) → n → ∞ . Since g is an irrational rotation we mayassume additionally that g p n ( u ) → z and g q n ( v ) → z as n → ∞ . Applying now first part ofour consideration we obtain that the constant τ , chosen for x = y = z , will be also a lowerbound for the length of the arcs ( g ˜i m ( u ) , g ˜j m ( v )) and ( g ˜j m ( v ) , g ˜i m +1 ( u )) for m = 1 , . . . , n and some ˜i , . . . , ˜i n , ˜j , . . . , ˜j n (with the same length) such that g ˜i ( u ) < g ˜j ( v ) < g ˜i ( u ) < g ˜j ( v ) < · · · < g ˜i n ( u ) < g ˜j n ( v ) . This completes the proof. • Lemma 12.
Let Γ be admissible and contractive and let ( µ K ) K ∈ N and ( µ K ) K ∈ N be twosequences of probability distributions such that for any K ∈ N the measures µ K and µ K areuniformly distributed on { x , x , . . . , x K } ⊂ S and { y , y . . . , y K } ⊂ S , respectively and x < y < x < y < · · · < x K < y K . Let a probability distribution p be given and let P be the Markov operator corresponding tothe Iterated Function System (Γ , p ) . Then for an arbitrary f ∈ C ( S ) we have (4) lim K →∞ lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) P n µ K (d x ) − Z S f ( x ) P n µ K (d x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . TABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE 11
Proof
Fix f ∈ C ( S ) and let ω = ( i , i , . . . ) be such that ω ( µ ∗ ) is defined. Let z ∈ S be such ω ( µ ∗ ) = ( δ z + δ R ( z ) + . . . + δ R M − ( z ) ) /M , by Proposition 6. Fix ε > U , . . . U M − be mutually disjoint open arcs such that R i ( z ) ∈ U i and | f ( u ) − f ( v ) | ≤ ε/ u, v ∈ U i , i = 0 , . . . , M −
1. Fix K ∈ N . All the arcs ( x i , y i ) and ( y i , x i +1 ) for i = 1 , . . . , K have positive µ ∗ measure, by the fact that the support of µ ∗ is equal to S .Since µ ∗ ◦ ( g i · · · g i n ) − converges weakly to ω ( µ ∗ ) and(5) ω ( µ ∗ )( M − [ j =0 U j ) = 1 , for any arc ( u, v ) and n arbitrary large (depending on µ ∗ (( u, v ))) there exists w ∈ ( u, v )such that g i · · · g i n ( w ) ∈ S M − j =0 U j . If this is not the case, we obtain that ω ( µ ∗ )( S M − j =0 U j ) ≤ − µ ∗ (( u, v )), contrary to condition (5).Due to the above observation we find points z ni , w ni such that x < z n < y < w n < x < z n < y < w n < · · · < x K < w nK < y K and g i · · · g i n ( z ni ) , g i · · · g i n ( w ni ) ∈ M − [ j =0 U j for i = 1 , . . . , K and all n ≥ n , where n depends on the measure µ ∗ (or: on length)of the arcs ( x , y ) , ( y , x ) , . . . , ( y K − , x K ) , ( x K , y K ). In other words, it is independent oflocations of the points if the distance between them is bigger than some τ >
0. From thisit follows that there are at most 2 M pairs of ( x i , y i ) such that g i · · · g i n ( x i ) , g i · · · g i n ( y i )are not in the same U j . Hence (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) µ K ◦ ( g i · · · g i n ) − (d x ) − Z S f ( x ) µ K ◦ ( g i · · · g i n ) − (d x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε/ k f k M/K. for all n ≥ n . We may find n such that the above condition holds for all ω from someset ˜Ω with P ( ˜Ω) ≥ − ε/ (2 M ). Then we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) P n µ K (d x ) − Z S f ( x ) P n µ K (d x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε + 2 k f k M/K for all n ≥ n . Taking limit as K → ∞ completes the proof. • Remark 13.
The above proof also shows the following: Fix some f ∈ C ( S ). For every ε > τ > n = n ( τ, ε, f ) such that (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) P n µ K (d x ) − Z S f ( x ) P n µ K (d x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε + 2 k f k M/K for n ≥ n and any probability measures µ K , µ K distributed uniformly on { x , x , . . . , x K } and { y , y . . . , y K } , respectively, such that x < y < x < y < · · · < x K < y K , and the length (measure µ ∗ ) of the arcs ( x , y ) , ( y , x ) , . . . , ( y K − , x K ) , ( x K , y K ) is boundedfrom below by τ > M denotes the constant defined in Proposition 6). Theorem 14.
Let
Γ = { g , . . . , g k } be admissible and contractive and let p be a probabilitydistribution. Then the Iterated Function System (Γ , p ) is asymptotically stable.Proof Recall that we may assume that g is an irrational rotation. We are going to showthat the assumptions of Theorem 10 are satisfied. Fix ε > K ∈ N be so largethat 2 k f k M/K < ε/ . Let ( x, y ) ∈ S × S . It follows from Lemma 11 that there exists N = N x,y and i l , j l ∈ Σ N for l = 1 , . . . , K and such that g i ( x ) < g j ( y ) < g i ( x ) < g j ( y ) < · · · < g i K ( x ) < g j K ( y ) . Further, we may find open neighbourhood U x , U y of x, y , respectively, such that g i (˜ x ) < g j (˜ y ) < g i (˜ x ) < g j (˜ y ) < · · · < g i K (˜ x ) < g j K (˜ y ) for (˜ x, ˜ y ) ∈ U x × U y . Now, choose a collection of pairs ( x , y ) , . . . , ( x r , y r ) ∈ S × S such that S × S = S ri =1 U x i × U y i . Fix µ , µ ∈ M ( S ). Set p =: min i p i . Take the product measure µ × µ and observe that there exists j ∈ { , . . . , r } such that ( µ × µ )( U x j × U y j ) ≥ /r . It isnow easy to check that(6) P N xj,yj µ ≥ p N xj,yj K Z U xj m ( x ) µ (d x )and(7) P N xj,yj µ ≥ p N xj,yj K Z U yj m ( y ) µ (d y ) , where m ( x ) and m ( y ) are probability measures as in the hypothesis of Lemma 12, i.e.,the measures uniformly distributed over the points g i ( x ) , g i ( x ) , . . . , g i K ( x )and g j ( y ) , g j ( y ) , . . . , g j K ( y ) , respectively. Set ν ( · ) = R U xj m ( x )( · ) µ (d x ) µ ( U x j ) and ν ( · ) = R U yj m ( x )( · ) µ (d x ) µ ( U y j )and observe that estimates (6) and (7) can be rewritten as: P N xj,yj µ i ≥ αν i for i = 1 , α := p N Kr − , where N = max ≤ l ≤ r N x l ,y l . Since, by Lemma 12 and Remark 13 (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( z ) P n m ( x )(d z ) − Z S f ( z ) P n m ( y )(d z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε TABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE 13 for ( x, y ) ∈ U x j × U y j and n sufficiently large, we obtain thatlim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) P n ν (d x ) − Z S f ( x ) P n ν (d x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. This completes the proof. • E–property
The e–property plays an important role in proving asymptotic properties of Markovprocesses. Usually it is a necessary step when we want to justify that the studied processhas a unique invariant measure and is asymptotically stable. Here we inverted the order ofour considerations. First we showed the uniqueness of an invariant measure and its stabilityand now we shall prove independently that our operator also satisfies the e–property. Thisfact seems to be surprising because the transformations under considerations are neithercontractions, nor average contractions. Here the e–property is forced by the geometry.
Proposition 15.
Let
Γ = { g , . . . , g k } be admissible and let p be a probability distribution.Then the operator P corresponding to the Iterated Function System (Γ , p ) satisfies thee–property.Proof If Γ = { g , . . . , g k } is equicontinuous, the e–property follows immediately. So wemay assume that Γ is contractive. Fix a function f ∈ C ( S ) and ε >
0. Let K ∈ N besuch that 2 k f k M/K ≤ ε/
2. (Here again M is the constant coming from Proposition 6).Take an arbitrary point z ∈ S . From the proof of Theorem 14, applied for both x := z and y := z (see also Lemma 11), it follows that we may find α > N , . . . , N m suchthat P N + ··· + N m δ z admits two representations below:(8) P N + ··· + N m δ z = αP N + ··· + N m ν + α (1 − α ) P N + ··· + N m ν + · · · + α (1 − α ) m − P N m ν m + (1 − α ) m µ P N + ··· + N m δ z = αP N + ··· + N m ν + α (1 − α ) P N + ··· + N m ν + · · · + α (1 − α ) m − P N m ν m + (1 − α ) m µ , where, for every j = 1 , . . . m the pair of probability measures ( ν j , ν j ) is a convex combina-tion of pairs of measures such that each pair is uniformly distributed over some collectionsof points { x , . . . , x K } and { y , . . . , y K } , respectively, and x < y < x < y < · · · < x K < y K , where the length of the arcs ( x , y ) , ( y , x ) , . . . , ( y K − , x K ) , ( x K , y K ) is bounded from be-low by some τ > K . Furthermore, µ , µ are some probabilitymeasures.Now, let m ∈ N be so large that (1 − α ) m ≤ ε (4 || f || ) − . Since every element from { x , . . . , x K } and { y , . . . , y K } (on which the measures defining ν , ν , . . . , ν m , ν m are sup-ported) is of the form g j ( z ) for some j ∈ Σ ∗ , we may find η > w ∈ S with d( z, w ) < η we have P N + ··· + N m δ w = αP N + ··· + N m ˜ ν + α (1 − α ) P N + ··· + N m ˜ ν + · · · + α (1 − α ) m − P N m ˜ ν m + (1 − α ) m ˜ µ , and each pair of probability measures ( ν j , ˜ ν j ) is a convex combination of pairs of mea-sures that are uniformly distributed over some collections of points { x , . . . , x K } and { ˜ y , . . . , ˜ y K } , respectively, such that x < ˜ y < x < ˜ y < · · · < x K < ˜ y K , and the length of the arcs ( x , ˜ y ) , (˜ y , x ) , . . . , (˜ y K − , x K ) , ( x K , ˜ y K ) is bounded from belowby τ /
2. From Remark 13 it follows now that there exists n ∈ N such that for any n ≥ n and every j ∈ { , . . . , m } we have (cid:12)(cid:12)(cid:12)(cid:12)Z S f ( x ) P n ν j (d x ) − Z S f ( x ) P n ν j (d x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε/ . Hence, we obtain that for any w ∈ S such that d( w, z ) < η we have | P ∗ n f ( z ) − P ∗ n f ( w ) | ≤ ε for n ≥ N + . . . + N m + n and we are done. • Strong Law of Large Numbers
Let Γ = { g , . . . g k } ⊂ H + and let ˜Γ = { g − , . . . g − k } . We define the probability distri-bution on ˜Γ by putting p ( g − i ) = p i . Proposition 16.
Let
Γ = { g , . . . g k } ⊂ H + be admissible and let p be a probabilitydistribution. Then the Strong Law of Large Numbers for trajectories starting from anarbitrary point holds. More precisely: let φ ∈ C ( S ) . For every x ∈ S there exists a subset Ω ′ ⊂ Ω with P (Ω ′ ) = 1 such that for every ω = ( i , i , . . . ) ∈ Ω ′ (9) φ ( g i ( x )) + φ ( g i ,i ( x )) + . . . + φ ( g i n ,i n − , ··· ,i ( x )) n → Z S Z Γ φ ( y ) dp ( g ) µ ( dy ) . Proof
From Lemma 2 it follows that Γ is either equicontinuous or contractive. If Γ isequicontinuous then the theorem holds simply by the Birkhoff ergodic theorem. Indeed,since the invariant measure µ ∗ is unique it is also ergodic. From Birkhoff’s theorem itfollows then that formula (9) holds for P × µ ∗ -almost every pair ( ω, z ). Since the supportof µ ∗ is equal to S , for any x ∈ S and any k ∈ N we may find a point z k ∈ S and aset Ω k ⊂ Ω with P (Ω k ) = 1 such that condition (9) holds with x replaced with z k and | φ ( g i n ,i n − , ··· ,i ( x )) − φ ( g i n ,i n − , ··· ,i ( z k )) | < /k for n ∈ N and ( i , i , . . . ) ∈ Ω k , by the factthat Γ is equicontinuous and ϕ is uniformly continuous. Let Ω ′ = T ∞ k =1 Ω k . Since P (Ω ′ ) = 1and x satisfies condition (9) for ( i , i , . . . ) ∈ Ω ′ , we are done.Now assume that Γ is contractive and that the theorem does not hold. Then there exists x ∈ S and a set Ω x ⊂ Ω such that P (Ω x ) > ω = ( i , i , . . . ) ∈ Ω x formula TABILITY OF ITERATED FUNCTION SYSTEMS ON THE CIRCLE 15 (9) does not hold. Taking a subset of Ω x , if necessary, we may assume that(10)lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) φ ( g i ( x )) + φ ( g i ,i ( x )) + . . . + φ ( g i n ,i n − , ··· ,i ( x )) n − Z S Z Γ φ ( y )d p ( g ) µ ∗ (d y ) (cid:12)(cid:12)(cid:12)(cid:12) > ε for all ω = ( i , i , . . . ) ∈ Ω x and some ε > p . Let ˜ µ ∗ beits unique invariant measure. Since ˜ µ ∗ ( { x } ) = 0 we may find δ > x ⊂ Ω x with P ( ˜Ω x ) > ω (˜ µ ∗ )) ∩ ( x − δ, x + δ ) = ∅ for ω ∈ ˜Ω x .Let θ > | φ ( u ) − φ ( v ) | ≤ ε/ | u − v | < θ . Set γ := inf x ∈ S ˜ µ ∗ (( x − θ/ , x + θ/ γ >
0. Since g − i ◦ g − i ◦ · · · ◦ g − i n ◦ ˜ µ ∗ converges weakly to ω (˜ µ ∗ ))for P -a.s. ω = ( i , i , . . . ) ∈ ˜Ω x , we have ˜ µ ∗ (( g − i ◦ g − i ◦ · · · ◦ g − i n ) − (( x − δ, x + δ )) < γ forall n sufficiently large. From this and from the definition of γ it follows that | ( g − i ◦ g − i ◦· · · ◦ g − i n ) − (( x − δ, x + δ )) | < θ for n sufficiently large and all ω = ( i , i , . . . ) ∈ ˜Ω x . Thisgives | g i n ,i n − , ··· ,i ( u ) − g i n ,i n − , ··· ,i ( v ) | ≤ θ and consequently | φ ( g i n ,i n − , ··· ,i ( u )) − φ ( g i n ,i n − , ··· ,i ( v )) | ≤ ε/ n sufficiently large and u, v ∈ ( x − δ, x + δ ). Together with condition (10), this givesthatlim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) φ ( g i ( u )) + φ ( g i ,i ( u )) + . . . + φ ( g i n ,i n − , ··· ,i ( u )) n − Z S Z Γ φ ( y )d p ( g ) µ ∗ (d y ) (cid:12)(cid:12)(cid:12)(cid:12) > ε/ ω = ( i , i , . . . ) ∈ ˜Ω x and all u ∈ ( x − δ, x + δ ), contrary to the Birkhoff theorem. Thiscontradiction completes the proof. • Remark 17.
In [1] the authors introduce the notion of an essentially contracting system,prove this property for some special system of piecewise linear maps of an interval, andobtain several interesting consequences. The proof of Proposition 16 shows, in particular,that our system is essentially contracting. Contrary to [1], we do not need special estimates;we simply use the properties of the ”conjugate” system generated by ˜Γ.
Remark 18.
Observe that the known criteria for the Central Limit Theorem and Law ofthe Iterated Logarithm require more than it was proved in Theorem 14. To apply the resultsby M. Maxwell and M. Woodroofe (see [15]) we need to know the rate of convergence tothe invariant measure. On the other hand, finding new sufficient conditions for the CentralLimit Theorem in the setting of considered IFS’s would be an interesting question worthyof further study.
References [1] L. Alsed`a and M. Misiurewicz,
Random interval homeomorphisms , Proceedings of New Trends inDynamical Systems. Salou, 2012, Publ. Mat., 15 - 36 (2014).[2] L. Arnold,
Random Dynamical Systems , Springer-Verlag, Berlin Heidelberg, 2010.[3] M.F. Barnsley, S.G. Demko, J.H. Elton and J.S. Geronimo,
Invariant measures arising from iteratedfunction systems with place dependent probabilities , Ann. Inst. Henri Poincar´e , 367-394 (1988). [4] B. Deroin and V. Kleptsyn, Random conformal dynamical systems , GAFA , 1043-1105 (2007).[5] B. Deroin, V. Kleptsyn, and A. Navas, Sur la dynamique unidimensionnelle en r´egularit´e in-term´ediaire , Acta Math. no. 2, 199-262 (2007).[6] W. Doeblin,
Sur les propri´et´es asymptotiques de mouvement r´egis par certains types de chaˆınes sim-ples , Bull. Math. Soc. Roum. Sci. , 57-115 (1937).[7] H. Furstenberg, Boundary theory and stochastic processes on homogeneus spaces , Proc. Sympos. PureMath. , 193-229 (1973).[8] E. Ghys, Groups acting on the circle , L’Enseignement Math´ematique , 329-407 (2001).[9] A.J. Homburg and H. Zmarrou, Dynamics and bifurcations of random circle diffeomorphisms , Discreteand Continuous Dynamical Systems , 719–731 (2008).[10] A.J. Homburg and H. Zmarrou, Bifurcations of stationary measures of random diffeomorphisms ,Ergodic Theory Dynam. Systems no 5, 1651-1692 (2007).[11] R. Kapica, T. Szarek and M. ´Sl¸eczka, On a unique ergodicity of some Markov processes , PotentialAnal. , 589-606 (2012).[12] T. Komorowski, S. Peszat and T. Szarek, On ergodicity of some Markov processes , Ann. Probab. ,1401-1443 (2010).[13] A. Lasota and J. Yorke, Lower bound technique for Markov operators and iterated function systems ,Random Comput. Dynam. no. 1, 41-77 (1994).[14] M.Yu. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere , Ergod. Theor.Dyn. Syst. , 351-385 (1983).[15] Maxwell, M., Woodroofe, M., Central limit theorems for additive functionals of Markov chains , Ann.Probab.
28 (2) (2000), 713–724.[16] A. Navas,
Groups of Circle Diffeomorphisms , Chicago Lectures in Mathematics. University of ChicagoPress, 2010.[17] F. Przytycki, M. Urba´nski,
Conformal Fractals. Ergodic Theory Methods.
LMS Lect. Notes Series 371,Cambridge University Press, 2010.[18] T. Szarek,
Invariant measures for nonexpensive Markov operators on Polish spaces , DissertationesMath. (Rozprawy Mat.) , p. 62 (2003).[19] M. ´Sl¸eczka,
The rate of convergence for iterated function systems , Studia Math. , no. 3, 201-214(2011).[20] P. Walters,
Invariant measures and equilibrium states for some mappings which expand distances ,Trans. of the AMS , 121-153 (1978).[21] I. Werner,
Contractive Markov systems , J. London Math. Soc. (2), 236-258 (2005). Tomasz Szarek, Institute of Mathematics, University of Gda´nsk, Wita Stwosza 57, 80-952 Gda´nsk, Poland
E-mail address : [email protected] Anna Zdunik, Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097Warszawa, Poland
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