Topological dynamics of piecewise λ-affine maps
aa r X i v : . [ m a t h . D S ] M a y TOPOLOGICAL DYNAMICS OF PIECEWISE λ -AFFINE MAPS Arnaldo Nogueira Aix-Marseille Universit´e, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, [email protected]
Benito Pires and Rafael A. Rosales Departamento de Computa¸c˜ao e Matem´atica, Faculdade de Filosofia, Ciˆencias e LetrasUniversidade de S˜ao Paulo, 14040-901, Ribeir˜ao Preto - SP, [email protected], [email protected]
Abstract.
Let − < λ < f : [0 , → R be a piecewise λ -affine map, that is,there exist points 0 = c < c < · · · < c n − < c n = 1 and real numbers b , . . . , b n suchthat f ( x ) = λx + b i for every x ∈ [ c i − , c i ). We prove that, for Lebesgue almost every δ ∈ R , the map f δ = f + δ (mod 1) is asymptotically periodic. More precisely, f δ has atmost 2 n periodic orbits and the ω -limit set of every x ∈ [0 ,
1) is a periodic orbit. Introduction
Let I = [0 ,
1) and − < λ <
1. We say that f : I → R is an n -interval piecewise λ -affine map if there exist points 0 = c < c < · · · < c n − < c n = 1 and real numbers b , . . . , b n such that f ( x ) = λx + b i for every x ∈ [ c i − , c i ) and 1 ≤ i ≤ n . We areinterested in studying the topological dynamics of the one-parameter family of piecewise λ -affine contractions (see Figure 1)(1) f δ = f + δ (mod 1) , δ ∈ R . The case in which 0 < λ < f is the continuous map x λx was explicitlyconsidered by, among others, Y. Bugeaud [2], Y. Bugeaud and J-P. Conze [4], R. Coutinho[7] and P. Veerman [13] using a rotation number approach. It is known that for each δ ∈ R ,the ω -limit set ω f δ ( x ) = T m ≥ S k ≥ m { f kδ ( x ) } is the same set for every x ∈ I : either a finiteset or a Cantor set. The second situation happens for a non-trivial Lebesgue null set ofparameters δ .Here we consider the general case where f is any piecewise λ -affine contraction havingfinitely many discontinuities. Beyond the difficulty brought by the presence of discontinu-ities, we also have to deal with the possible lack of injectivity of the map, which rules outany approach based on the theory of rotation numbers. Differently from the δ -parameterfamily x λx + δ (mod 1), the dynamics of the general case allows the coexistence of Mathematics Subject Classification.
Primary 37E05 Secondary 37C20, 37E15.
Key words and phrases.
Topological dynamics, piecewise contraction, periodic orbit, iterated functionsystem. Partially supported by BREUDS. Partially supported by FAPESP 2015/20731-5 and CNPq 303731/2013-3. λ -AFFINE MAPS several attractors of finite cardinality together with several Cantor sets. In other words, ω f δ ( x ) may depend on x .Given f : I → I and x ∈ I , if there exists k ≥ f k ( x ) = x , we say that the f -orbit of x , O f ( x ) = S k ≥ { f k ( x ) } , is a periodic orbit . We say that f is asymptoticallyperiodic if ω f ( x ) is a periodic orbit for every x ∈ I .Our first result is the following. Theorem 1.1.
Let − < λ < and f : I → R be an n -interval piecewise λ -affine map,then, for Lebesgue almost every δ ∈ R , the map f δ = f + δ (mod 1) is asymptoticallyperiodic and has at most n periodic orbits. In the statement of Theorem 1.1, the bound 2 n for the number of periodic orbitsis sharp: in fact, for n = 1 and f : x
7→ − x + we have that f δ is the map x x + + δ (mod 1), which has two periodic orbits for every δ small enough. However, thebound 2 n can be replaced by n , if in (1) the map f satisfies f ( I ) ⊂ (0 , I = R , f δ = f + δ and the bound n in the place of 2 n .Observe that being asymptotically periodic is stronger than saying that ω f δ ( x ) is a finiteset for every x ∈ I . Due to the arguments in the proof of Lemma 7.1, our approach toprove Theorem 1.1 requires f to be a constant slope map .0 101 c c c f with λ = , n = 4 c c c c f δ with δ = c c c c c f δ with δ = Figure 1.
The family f δ = f + δ (mod 1).Maps of constant slope are important because many piecewise smooth interval mapsare topologically conjugate or semiconjugate to them. In this regard, J. Milnor and W.Thurston [9] proved that any continuous piecewise monotone interval map of positiveentropy htop ( T ) is topologically semiconjugate to a map whose slope in absolute valueequals e htop . This result was generalised by L. Alsed`a and M. Misiurewicz [1] to piecewisecontinuous piecewise monotone interval maps of positive entropy. Concerning countablypiecewise continuous piecewise monotone interval maps, a necessary and sufficient condi-tion for the existence of a non-decreasing semiconjugacy to a map of constant slope wasprovided by M. Misiurewicz and S. Roth [10]. A. Nogueira and B. Pires [12] proved thatevery injective piecewise contraction is topologically conjugate to a map whose slope inabsolute value equals . It is worth observing that the types of maps we consider here IECEWISE λ -AFFINE MAPS 3 appear in the field of diophantine approximation (see [3]). Concerning the dynamics ofpiecewise contractions, we refer the reader to [5, 6, 8].Theorem 1.1 can be reduced to Theorem 1.2, a much more general result. To stateit, we need some additional notation. Let I = [0 ,
1) or I = R . Denote by I and ˚ I ,respectively, the closure and the interior of I . We say that Φ = { φ , . . . , φ n } , n ≥
2, isan
Iterated Function System (IFS) defined on I if each map φ i : I → ˚ I is a Lipschitzcontraction. Set inf R = −∞ , sup R = ∞ andΩ n − = Ω n − ( I ) = { ( x , . . . , x n − ) : inf I < x < · · · < x n − < sup I } . For each ( x , . . . , x n − ) ∈ Ω n − ( I ), let f φ ,...,φ n ,x ,...,x n − : I → I be the n -interval piecewisecontraction ( PC ) defined by(2) f φ ,...,φ n ,x ,...,x n − ( x ) = φ ( x ) if x ∈ I ∩ ( −∞ , x ) φ i ( x ) if x ∈ [ x i − , x i ) , ≤ i ≤ n − .φ n ( x ) if x ∈ I ∩ [ x n − , ∞ )All measure-theoretical statements hereafter concern the Lebesgue measure denoted by µ . In particular, W Φ ⊂ I is a full set if µ ( I \ W Φ ) = 0. Theorem 1.2.
Let I = [0 , or I = R . Let Φ = { φ , . . . , φ n } be an IFS defined on I ,then there exists a full set W Φ ⊂ I such that for every ( x , . . . , x n − ) ∈ Ω n − ( I ) ∩ W n − ,the n -interval PC f φ ,...,φ n ,x ,...,x n − defined by ( ) is asymptotically periodic and has atmost n periodic orbits. Notice that in Theorem 1.2, the IFS Φ does not need to be affine nor injective. Aweaker version of Theorem 1.2 was proved by the authors in [11] under two additionalhypothesis: the maps φ , . . . , φ n were injective and had non-overlapping ranges.This article is organized as follows. The proof of Theorem 1.2 for I = [0 ,
1) is distributedalong Sections 2, 3, 4 and 5. The first three sections are dedicated to the asymptoticstability aspect while the upper bound for the number of periodic orbits is in Section 5.The proof of Theorem 1.2 for I = R is in Section 6. Theorem 1.1 is proved in Section 7.2. Highly Contractive Iterated Function Systems
In this section we provide a version of Theorem 1.2 for highly contractive IFSs definedon I = [0 ,
1) as follows. If φ : I → I is a Lipschitz map, then Dφ ( x ) exists for almostevery x ∈ I . We say that an IFS { φ , . . . , φ n } is highly contractive if there exists 0 ≤ ρ < x ∈ I ,(3) | Dφ ( x ) | + . . . + | Dφ n ( x ) | ≤ ρ < . Theorem 2.1.
Let
Φ = { φ , . . . , φ n } be a highly contractive IFS defined on I = [0 , ,then there exists a full set W Φ ⊂ I such that, for every ( x , . . . , x n − ) ∈ Ω n − ∩ W n − , thePC f φ ,...,φ n ,x ,...,x n − defined by ( ) is asymptotically periodic. We need some preparatory lemmas to prove Theorem 2.1. Throughout this section,except in Definition 2.7 and Lemma 2.8, we assume that Φ is a highly contractive IFS.
PIECEWISE λ -AFFINE MAPS Denote by Id the identity map on ¯ I . Let C = { Id } and A = ¯ I . For every k ≥
0, let(4) C k +1 = C k +1 ( φ , . . . , φ n ) = { φ i ◦ h : 1 ≤ i ≤ n, h ∈ C k } and A k = ∪ h ∈ C k h ( ¯ I ) . Lemma 2.2.
For every k ≥ , (i) A k +1 = ∪ ni =1 φ i ( A k ) ⊂ A k ; (ii) Let W = I \ ∩ k ≥ A k , then W = ¯ I almost surely.Proof. The equality in claim (i) follows from the following equalities: A k +1 = [ g ∈ C k +1 g ( ¯ I ) = n [ i =1 [ h ∈ C k φ i (cid:0) h ( ¯ I ) (cid:1) = n [ i =1 φ i (cid:16) [ h ∈ C k h ( ¯ I ) (cid:17) = n [ i =1 φ i ( A k ) . It follows easily from (4) that C k +1 = { h ◦ φ i | ≤ i ≤ n, h ∈ C k } , thus A k +1 = [ h ∈ C k n [ i =1 h (cid:0) φ i ( ¯ I ) (cid:1) ⊂ [ h ∈ C k n [ i =1 h (cid:0) ¯ I (cid:1) = [ h ∈ C k h (cid:0) ¯ I (cid:1) = A k which concludes the proof of item (i). The proof of claim (ii) follows from the change ofvariables formula for Lipschitz maps together with claim (i) and equation (3), µ ( A k +1 ) ≤ n X i =1 µ ( φ i ( A k )) ≤ n X i =1 Z A k | Dφ i | d µ = Z A k (cid:18) n X i =1 | Dφ i | (cid:19) d µ ≤ ρµ ( A k ) . Therefore µ ( A k ) ≤ ρ k , for every k ≥
0, thus µ ( ∩ k ≥ A k ) = 0 which proves item (ii). (cid:3) If Φ is a highly contractive IFS its atractor, ∩ k ≥ A k , is a null measure set, by Lemma2.2, item (ii). Using [6, Theorem 3.1], one obtains that for every point ( x , . . . , x n − ) ∈ Ω n − ∩ W n − , the map f x ,...,x n − has finitely many periodic orbits and is asymptoticallyperiodic. However, the claim of Theorem 1.2 is stronger and holds for any contractiveIFS. Lemma 2.3.
There exists a full set W ⊂ I such that h − ( { x } ) is a finite set for every x ∈ W and h ∈ ∪ k ≥ C k .Proof. It is proved in [14] that if h : ¯ I → ¯ I is a Lipschitz map, then h − ( { x } ) is finitefor almost every x ∈ ¯ I . The lemma follows immediately from the fact that C k is a finiteset. (cid:3) Hereafter, let W and W be as in Lemma 2.2, item (ii), and Lemma 2.3. Set(5) W Φ = W ∩ W , then W Φ = I almost surely . Proposition 2.4.
For each x ∈ W Φ , [ k ≥ [ h ∈ C k h − ( { x } ) is a finite subset of ¯ I (cid:15) T k ≥ A k .Proof. Let x ∈ W Φ . Assume by contradiction that ∪ k ≥ ∪ h ∈ C k h − ( { x } ) is an infinite set.By Lemma 2.3, for every k ≥
0, the set S h ∈ C k h − ( { x } ) is finite. Therefore, for infinitelymany k ≥
0, the set S h ∈ C k h − ( { x } ) is nonempty and x ∈ A k . By item (i) of Lemma 2.2, x ∈ ∩ k ≥ A k , which contradicts the fact that x ∈ W . This proves the first claim.Let y ∈ S k ≥ S h ∈ C k h − ( { x } ), then there exist ℓ ≥ h ℓ ∈ C ℓ such that x = h ℓ ( y ).Assume by contradiction that y ∈ ∩ k ≥ A k . Then x ∈ h ℓ ( A k ) ⊂ ∪ h ∈ C ℓ h ( A k ) = A ℓ + k for IECEWISE λ -AFFINE MAPS 5 every k ≥
0, implying that x ∈ ∩ k ≥ ℓ A k = ∩ k ≥ A k , which is a contradiction. This provesthe second claim. (cid:3) Theorem 2.5.
Let ( x , . . . , x n − ) ∈ Ω n − ∩ W n − and f = f φ ,...,φ n ,x ,...,x n − , then the set (6) Q = n − [ i =1 [ k ≥ f − k ( { x i } ) is finite. Moreover, Q ⊂ I \ ∩ k ≥ A k .Proof. Let ( x , . . . , x n − ) ∈ Ω n − ∩ W n − . By Proposition 2.4, the set ∪ k ≥ ∪ h ∈ C k h − ( { x i } )is finite for every 1 ≤ i ≤ n −
1. Hence, as [ k ≥ f − k ( { x i } ) ⊂ [ k ≥ [ h ∈ C k h − ( { x i } ) , ≤ i ≤ n − , we have that Q is also a finite set. Moreover, Q ⊂ I \ ∩ k ≥ A k by Proposition 2.4. (cid:3) Next corollary assures that Theorem 2.5 holds if the partition [ x , x ), . . . , [ x n − , x n )in (2) is replaced by any partition I , . . . , I n with each interval I i having endpoints x i − and x i . Corollary 2.6.
Let f = f φ ,...,φ n ,x ,...,x n − be as in Theorem 2.5. Let ˜ f : I → I be a maphaving the following properties: (P1) ˜ f ( x ) = f ( x ) for every x ∈ (0 , \ { x , . . . , x n − } ; (P2) ˜ f ( x i ) ∈ { lim x → x i − f ( x ) , lim x → x i + f ( x ) } for every ≤ i ≤ n − .Then the set ˜ Q = S n − i =1 S k ≥ ˜ f − k ( { x i } ) is finite.Proof. The definition of f given by (2) together with the properties (P1) and (P2) assurethat there exists a partition of I into n intervals I , . . . , I n such that, for every 1 ≤ i ≤ n ,the interval I i has endpoints x i − and x i and ˜ f | I i = φ i | I i . In particular, we have that˜ Q ⊂ S n − i =1 S k ≥ S h ∈ C k h − ( { x i } ) which is a finite set by Proposition 2.4. (cid:3) We remark that, in the next definition and in Lemma 2.8, the IFS is not assumed tobe highly contractive.
Definition 2.7.
Let ( x , . . . , x n − ) ∈ Ω n − and f = f φ ,...,φ n ,x ,...,x n − be such that the set Q defined in (6) is finite. The collection P = { J ℓ } mℓ =1 of all connected components of(0 , \ Q is called the invariant quasi-partition of f . In this case, we say that f has aninvariant quasi-partition .The existence of an invariant quasi-partition plays a fundamental role in this article. Lemma 2.8.
Let f = f φ ,...,φ n ,x ,...,x n − and P = { J ℓ } mℓ =1 be as in Definition 2.7, then forevery interval J ∈ P there exists an interval J ′ ∈ P such that f ( J ) ⊂ J ′ .Proof. Assume the lemma is false, then there exists J ∈ P such that f ( J ) ∩ Q = ∅ .Hence, J ∩ f − ( Q ) = ∅ . However, f − ( Q ) ⊂ Q implying that J ∩ Q = ∅ which contradictsthe definition of P . (cid:3) PIECEWISE λ -AFFINE MAPS As the next lemma shows, the existence of an invariant quasi-partition P implies thefollowing weaker notion of periodicity. Let d : I → { , . . . , n } be the piecewise constantfunction defined by d ( x ) = i if x ∈ I i . The itinerary of the point x ∈ I is the sequenceof digits d , d , d , . . . defined by d k = d (cid:0) f k ( x ) (cid:1) . We say that the itineraries of f areeventually periodic if the sequence d , d , d , . . . is eventually periodic for every x ∈ I . Lemma 2.9.
Let ( x , . . . , x n − ) ∈ Ω n − ∩ W n − , then all itineraries of f = f φ ,...,φ n ,x ,...,x n − are eventually periodic.Proof. By Theorem 2.5, Q is finite, thus f has an invariant quasi-partition P = { J ℓ } mℓ =1 as in Definition 2.7. By Lemma 2.8, there exists a map τ : { , . . . , m } → { , . . . , m } suchthat f ( J ℓ ) ⊂ J τ ( ℓ ) for every 1 ≤ ℓ ≤ m . Let 1 ≤ ℓ ≤ m and { ℓ k } ∞ k =0 be the sequencedefined recursively by ℓ k +1 = τ ( ℓ k ) for every k ≥
0. It is elementary that the sequence { ℓ k } ∞ k =0 is eventually periodic. We have that x i ∈ Q (see (6)) for every 1 ≤ i ≤ n − η : { , . . . , m } → { , . . . , n } satisfying J ℓ ⊂ I η ( ℓ ) for every 1 ≤ ℓ ≤ m , Hence, the sequence { η ( ℓ k ) } ∞ k =0 is eventually periodic and,by definition, so is the itinerary of any x ∈ J ℓ .Now let x ∈ { } ∪ Q . If { x, f ( x ) , f ( x ) , . . . } ⊂ Q , then the orbit of x is finite and so itsitinerary is eventually periodic. Otherwise, there exist 1 ≤ ℓ ≤ m and k ≥ f k ( x ) ∈ J ℓ . By the above, the itinerary of f k ( x ) is eventually periodic and so is that of x . This proves the lemma. (cid:3) Proof of Theorem 2.1.
Let ( x , . . . , x n − ) ∈ Ω n − ∩ W n − and f = f φ ,...,φ n ,x ,...,x n − . ByTheorem 2.5, Q is finite, thus f has an invariant quasi-partition P = { J ℓ } mℓ =1 as inDefinition 2.7. Let 1 ≤ ℓ ≤ m and x ∈ J ℓ . In the proof of Lemma 2.9, it was provedthat the itinerary of x in P , { ℓ k } ∞ k =0 , is eventualy periodic. Therefore there exist aninteger s ≥ p ≥ ℓ s = ℓ s + p . As P is invariant under f , f p ( J ℓ s ) ⊂ J ℓ s + p = J ℓ s . Hence, if J ℓ s = ( c, d ), there exist 0 ≤ c ≤ c ′ ≤ d ′ ≤ d ≤ f p (( c, d )) = [ c ′ , d ′ ]. We claim that c ′ > c . Assume by contradiction that c ′ = c . As f p | ( c,d ) is a nondecreasing contractive map, there exist 0 ≤ η < ǫ such that f p (( c, c + ǫ )) = [ c, c + η ]. By induction, for every integer k ≥
1, there exist 0 ≤ η k < ǫ k such that f kp (( c, c + ǫ k )) = [ c, c + η k ]. Hence, c ∈ \ k ≥ [ h ∈ C kp h ( I ) = \ k ≥ A kp = \ k ≥ A k . This contradicts the fact that c ∈ ∂J ℓ s ⊂ { , } ∪ Q ⊂ I \ ∩ k ≥ A k (see Theorem 2.5). Weconclude therefore that c ′ > c . Analogously, d ′ < d . In this way, there exists ξ > f p (( c, d )) ⊂ ( c + ξ, d − ξ ). As f p | ( a,b ) is a continuous contraction, f p has a uniquefixed point z ∈ ( c, d ). Notice that O f ( z ) is a periodic orbit. Moreover, it is clear that ω ( x ) = O f ( z ) for every x ∈ J ℓ .Now let x ∈ I \ ∪ mℓ =1 J ℓ = { } ∪ Q . By the proof of Lemma 2.9, either O f ( x ) is containedin the finite set I \∪ mℓ =1 J ℓ (and thus is finite) or there exists k ≥ f k ( x ) ∈ ∪ mℓ =1 J ℓ .By the above, in either case, ω f ( x ) is a periodic orbit. (cid:3) Iterated Function Systems
In this section we prove the following improvement of Theorem 2.1.
IECEWISE λ -AFFINE MAPS 7 Theorem 3.1.
Let
Φ = { φ , . . . , φ n } be an IFS formed by κ -Lipschitz functions, with ≤ κ < , defined on I = [0 , , then there exists a full set W Φ ⊂ I such that for every ( x , . . . , x n − ) ∈ Ω n − ( I ) ∩ W n − , the PC f φ ,...,φ n ,x ,...,x n − has an invariant quasi-partitionand is asymptotically periodic. Theorem 3.1 will be deduced from Theorem 2.1 in the following way. First, we showthat the IFS Φ can be locally replaced by a highly contractive IFS Υ and then that thelocal substitution suffices to prove Theorem 3.1.Hereafter, let ( x , . . . , x n − ) ∈ Ω n − be fixed. Set x = 0, x n = 1,(7) δ = min ≤ i ≤ n x i − x i − V ( x , . . . , x n − ) = { ( y , . . . , y n − ) ∈ Ω n − : | y i − x i | < δ, ∀ i } . In what follows, let 0 ≤ κ < and φ , . . . , φ n : [0 , → (0 ,
1) be κ -Lipschitz contrac-tions. Let Υ = { ϕ , . . . , ϕ n } be the IFS defined by ϕ ( x ) = φ ( x ) , if x ∈ [0 , x + δ ] φ ( x + δ ) , if x ∈ [ x + δ, , ϕ n ( x ) = φ n ( x n − − δ ) , if x ∈ [0 , x n − − δ ] φ n ( x ) , if x ∈ [ x n − − δ, ϕ i ( x ) = φ i ( x i − − δ ) , if x ∈ [0 , x i − − δ ] φ i ( x ) , if x ∈ [ x i − − δ, x i + δ ] φ i ( x i + δ ) , if x ∈ [ x i + δ, , ≤ i ≤ n − . A scheme illustrating the construction of the IFS { ϕ , . . . , ϕ n } from the IFS { φ , . . . , φ n } is shown in Figure 2. Lemma 3.2.
The IFS
Υ = { ϕ , . . . , ϕ n } is highly contractive.Proof. It is clear that each ϕ i : [0 , → (0 ,
1) is a Lipschitz contraction, therefore foralmost every x ∈ I , Dϕ i ( x ) exists, and, by the definition of ϕ i , | Dϕ ( x ) | + · · · + | Dϕ n ( x ) | ≤ max ≤ i ≤ n − ( | Dφ i ( x ) | + | Dφ i +1 ( x ) | ) ≤ κ < . Therefore, { ϕ , . . . , ϕ n } is highly contractive which concludes the proof. (cid:3) In what follows, let V = V ( x , . . . , x n − ) be as in (7). Lemma 3.3.
For every ( y , . . . , y n − ) ∈ V , f ϕ ,...,ϕ n ,y ,...,y n − = f φ ,...,φ n ,y ,...,y n − . Proof.
Let ( y , . . . , y n − ) ∈ V , y = 0 and y n = 1. Set K i = [ y i − , y i ) for 1 ≤ i ≤ n . Since | y i − x i | < δ , we have that K ⊂ [0 , x + δ ] , K n ⊂ [ x n − − δ,
1] and K i ⊂ [ x i − − δ, x i + δ ] , ≤ i ≤ n − . This together with the definition of ϕ i yields ϕ i | K i = φ i | K i , for every 1 ≤ i ≤ n . (cid:3) Now we will apply the results of Section 2 to the highly contractive IFS Υ = { ϕ , . . . , ϕ n } (see Lemma 3.2). With respect to such IFS, let W Υ ⊂ I be the full set defined in equation(5). Notice that all the claims in Section 2 hold true for the IFS Υ. In particular, we havethat V ∩ W n − = V almost surely. PIECEWISE λ -AFFINE MAPS φ φ φ
01 Φ = { φ , φ , φ } x x ϕ ϕ ϕ
01 Υ = { ϕ , ϕ , ϕ } δ δ δ δ y y ϕ ϕ ϕ { ϕ , ϕ , ϕ } y y f φ ,φ ,φ ,y ,y = f ϕ ,ϕ ,ϕ ,y ,y Figure 2.
Relations between the IFS { φ , φ , φ } and { ϕ , ϕ , ϕ } . Theorem 3.4.
For every ( y , . . . , y n − ) ∈ V ∩ W n − , f = f φ ,...,φ n ,y ,...,y n − has an invariantquasi-partition and is asymptotically periodic.Proof. Let ( y , . . . , y n − ) ∈ V ∩ W n − . By Lemma 3.3, f = f ϕ ,...,ϕ n ,y ,...,y n − . By Lemma 3.2,we have that Υ fulfills the hypotheses of Theorems 2.1 and 2.5. Hence, f has an invariantquasi-partition and is asymptotically periodic. (cid:3) We stress that, in the previous results, δ = min ≤ i ≤ n ( x i − x i − ) /
3, the set V and theIFS Υ depend on the point ( x , . . . , x n − ). For this reason, in the next proof, we replace V and W Υ by V ( x , . . . , x n − ) and W ( x , . . . , x n − ), respectively. IECEWISE λ -AFFINE MAPS 9 Proof of Theorem 3.1.
Set Φ = { φ , . . . , φ n } . First we show that(8) [ ( z ,...,z n − ) ∈ Ω n − ∩ Q n − V ( z , . . . , z n − ) = Ω n − . Let ( x , . . . , x n − ) ∈ Ω n − and δ = δ ( x , . . . , x n − ). Let ( z , . . . , z n − ) ∈ Ω n − ∩ Q n − besuch that | z i − x i | < δ for every 1 ≤ i ≤ n −
1. Set z = 0 and z n = 1. We have thatfor every 1 ≤ j ≤ n , z j − z j − > ( x j − δ ) − ( x j − + δ ) ≥ δ − δ = 2 δ . In this way, δ ( z , . . . , z n − ) > δ > δ , thus ( x , . . . , x n − ) ∈ V ( z , . . . , z n − ). This proves (8).Let ( z , . . . , z n − ) ∈ Ω n − ∩ Q n − and let W ( z , . . . , z n − ) be the full set in I defined by(5). By Theorem 3.4, for every ( y , . . . , y n − ) ∈ V ( z , . . . , z n − ) ∩ W ( z , . . . , z n − ) n − , themap f φ ,...,φ n ,y ,...,y n − has an invariant quasi-partition and is asymptotically periodic. Thedenumerable intersection(9) W Φ = \ ( z ,...,z n − ) ∈ Ω n − ∩ Q n − W ( z , . . . , z n − )is a full set and, for every ( y , . . . , y n − ) ∈ V ( z , . . . , z n − ) ∩ W n − , the map f φ ,...,φ n ,y ,...,y n − has an invariant quasi-partition and is asymptotically periodic. This together with (8)concludes the proof. (cid:3) Asymptotic periodicity: the general case
In this section we prove the following improvement of Theorem 3.1.
Theorem 4.1.
Let
Φ = { φ , . . . , φ n } be an IFS formed by ρ -Lipschitz functions definedon I = [0 , with ≤ ρ < , then there exists a full set W ⊂ I such that for every ( x , . . . , x n − ) ∈ Ω n − ( I ) ∩ W n − , the PC f φ ,...,φ n ,x ,...,x n − has an invariant quasi-partitionand is asymptotically periodic. Throughout this section, let 0 ≤ ρ < φ , . . . , φ n : [0 , → (0 ,
1) be ρ -Lipschitzcontractions. Let k ≥ ρ k < . By the Chain rule for Lipschitz maps, C k = C k ( φ , . . . , φ n ) (cid:0) see (4) (cid:1) is a collection of at most n k ρ k -Lipschitz contractions.For each r ≥
2, let I r denote the collection of all IFS { ψ , . . . , ψ r } , where each ψ j ,1 ≤ j ≤ r , belongs to C k ( φ , . . . , φ n ). The collection I r consists of at most n k !( n k − r )!IFS. Notice that in an IFS, the order in which the maps are listed matters. The fact that ρ k < implies that any IFS in ∪ r ≥ I r satisfies the hypothesis of Theorem 3.1.In the statement of Theorem 3.1, the set W Φ depends on the IFS Φ = { φ , . . . , φ n } . Inthe next corollary, the set W does not depend on the IFS Ψ = { ψ , . . . , ψ r } , provided Ψis chosen within the denumerable collection ∪ r ≥ I r . Corollary 4.2.
There exists a full set W ⊂ I such that for every r ≥ , { ψ , . . . , ψ r } ∈ I r and ( y , . . . , y r − ) ∈ Ω r − ∩ W r − , the r -interval PC f ψ ,...,ψ r ,y ,...,y r − has an invariantquasi-partition and is asymptotically periodic.Proof. Let I = ∪ r ≥ I r . By Theorem 3.1, for each IFS Ψ = { ψ , . . . , ψ r } ∈ I , there ex-ists a full set W Ψ ⊂ I such that the following holds: for every ( y , . . . , y r − ) ∈ Ω r − ∩ W r − , λ -AFFINE MAPS the r -interval PC g = f ψ ,...,ψ r ,y ,...,y r − has an invariant quasi-partition and is asymptoti-cally periodic. The proof is concluded by taking W = ∩ Ψ ∈ I W Ψ . Since I is denumerable,we have that W is a full subset of I . (cid:3) Corollary 4.3.
Let { ψ , . . . , ψ r } ∈ I r and ( y , . . . , y r − ) ∈ Ω r − ∩ W r − , where r ≥ ,satisfy the hypothesis of Corollary 4.2. Let g = f ψ ,...,ψ r ,y ,...,y r − and ˜ g : I → I be any maphaving the following properties: (P1) ˜ g ( y ) = g ( y ) for every y ∈ (0 , \ { y , . . . , y r − } ; (P2) ˜ g ( y j ) ∈ { lim y → y j − g ( y ) , lim y → y j + g ( y ) } for every ≤ j ≤ r − .Then the map ˜ g has an invariant quasi-partition and is asymptotically periodic.Proof. Let x ∈ I . If O ˜ g ( x ) ⊂ { } ∪ { y , . . . , y r − } , then O ˜ g ( x ) is finite. Otherwise, thereexists ℓ ≥ O ˜ g (cid:0) ˜ g ℓ ( x ) (cid:1) ⊂ (0 , \ { y , . . . , y r − } . In this case, by (P2), wehave that O ˜ g (cid:0) ˜ g ℓ ( x ) (cid:1) = O g (cid:0) ˜ g ℓ ( x ) (cid:1) , which is finite by Corollary 4.2. This proves that ˜ g isasymptotically periodic.It remains to be shown that the set ˜ Q = ∪ r − j =1 ∪ k ≥ ˜ g − k ( { y j } ) is finite. By proceeding asin the proof of Corollary 2.6, it can be proved that the claims of Lemma 3.3, Theorems 3.4and 3.1 and therefore Corollary 4.2 hold if we replace in (2) the partition [ x , x ), . . . ,[ x n − , x n ) by any partition I , . . . , I n where each interval I i has endpoints x i − and x i .This means that in Corollary 4.2 we can replace the map g by the map ˜ g and concludethat the set ˜ Q is finite. Hence, ˜ g has an invariant quasi-partition. (cid:3) Corollary 4.2 and Corollary 4.3 will be used later on this section. Now let us come backto the original IFS { φ , . . . , φ n } .We denote by Ω ′ n − the set(10) Ω ′ n − = Ω \ n − [ i =0 n − [ j =1 [ ℓ ≥ [ h ∈ C ℓ { ( x , . . . , x n − ) ∈ Ω n − : x j = h ( x i ) } , which will be used in the forthcoming results. Lemma 4.4. Ω ′ n − = Ω n − almost surely.Proof. There are only denumerably many sets of the form { ( x , . . . , x n − ) ∈ Ω n − : x j = h ( x i ) } , where 0 ≤ i ≤ n −
1, 1 ≤ j ≤ n − h ∈ S ℓ ≥ C ℓ . Being the graph of afunction, each such set is a null set. Therefore, Ω ′ n − equals Ω n − up to a null set. (cid:3) Lemma 4.5.
Let ( x , . . . , x n − ) ∈ Ω ′ n − and f = f φ ,...,φ n ,x ,...,x n − . Let γ be a periodic orbitof f , then there exists a neighborhood U ⊂ I of γ such that f ( U ) ⊂ U and γ = ∩ ℓ ≥ f ℓ ( U ) .Moreover, ω f ( x ) = γ for every x ∈ U .Proof. Let γ be a periodic orbit of f . As ( x , . . . , x n − ) ∈ Ω ′ n − and f ( I ) ⊂ (0 , γ ∩ { x , . . . , x n − } = ∅ . Let ǫ = min {| x − x i | : x ∈ γ, ≤ i ≤ n } and set U := ∪ x ∈ γ ( x − ǫ, x + ǫ ), in particular U ⊂ I \ { x , . . . , x n − } . This together with the factthat f | [ x i − ,x i ) is a Lipschitz contraction implies that f ( U ) ⊂ U , thus γ = ∩ ℓ ≥ f ℓ ( U ). (cid:3) Lemma 4.6.
Let ( x , . . . , x n − ) ∈ Ω ′ n − and f = f φ ,...,φ n ,x ,...,x n − , then f k is left-continuousor right-continuous at each point of I . IECEWISE λ -AFFINE MAPS 11 Proof.
Let y ∈ I and S y = { y, f ( y ) , . . . , f k − ( y ) } . The fact that ( x , . . . , x n − ) ∈ Ω ′ n − assures that S y ∩ { x , . . . , x n − } is either empty or an one-point set. In the former case,we have that f is continuous on S y , hence f k is continuous at y . In the latter case, thereexists y ′ ∈ S y such that f is continuous at each point of S y \ { y ′ } and f is left-continuousor right-continuous at y ′ . Accordingly, f k is either left-continuous or right-continuous at y . (cid:3) For the next result, let W be the full set in the statement of Corollary 4.2. Lemma 4.7.
There exists a full set W ⊂ W such that if ( x , . . . , x n − ) ∈ Ω ′ n − ∩ W n − and f = f φ ,...,φ n ,x ,...,x n − , then f k | ( y j − ,y j ) = ψ j | ( y j − ,y j ) , ≤ j ≤ r, for some r ≥ , ( y , . . . , y r − ) ∈ Ω r − ∩ W r − , and ψ , . . . , ψ r ∈ C k . Moreover, f k isleft-continuous or right-continuous at each point of I .Proof. Let M = n x ∈ W (cid:12)(cid:12) ∪ k − ℓ =0 ∪ h ∈ C ℓ h − ( { x } ) W o . Notice that M ⊂ ∪ k − ℓ =0 ∪ h ∈ C ℓ h ( I \ W ), where I \ W is a null set, therefore M is a null set. By Lemma 2.3, thereexists a full set W ⊂ W \ M such that S k − ℓ =0 f − ℓ ( { x } ) ⊂ S k − ℓ =0 S h ∈ C ℓ h − ( { x } ) is afinite subset of W for every x ∈ W . Now let ( x , . . . , x n − ) ∈ Ω ′ n − ∩ W n − , thus S k − ℓ =0 f − ℓ ( { x , . . . , x n − } ) is a finite subset of W \ { } and we may list its elements inascending order 0 < y < · · · < y r − <
1. In this way, ( y , . . . , y r − ) ∈ Ω r − ∩ W r − . Let usanalyze how f k acts on the the intervals E = ( y , y ), . . . , E r = ( y r − , y r ). Fix 1 ≤ j ≤ r .Since { y , . . . , y r − } = S k − ℓ =0 f − ℓ ( { x , . . . , x n − } ), we have that for each 0 ≤ ℓ ≤ k − ≤ i ℓ ≤ n such that f ℓ ( E j ) ⊂ ( x i ℓ − , x i ℓ ). This together withthe fact that f | [ x iℓ − ,x iℓ ) = φ i ℓ yields f k | E j = ψ j | E j , where ψ j = φ i k ◦ · · · ◦ φ i ∈ C k .The claim that f k is left-continuous or right-continuous at each point of I follows fromLemma 4.6. (cid:3) Proof of Theorem 4.1.
Let ( x , . . . , x n − ) ∈ Ω ′ n − ∩ W n − and f = f φ ,...,φ n ,x ,...,x n − . ByLemma 4.7, there exist r ≥
2, ( y , . . . , y r − ) ∈ Ω r − ∩ W r − , and ψ , . . . , ψ r ∈ C k suchthat(11) f k | ( y j − ,y j ) = ψ j | ( y j − ,y j ) , ≤ j ≤ r. Let g = f ψ ,...,ψ n ,y ,...,y r − and ˜ g = f k . We claim that ˜ g satisfies (P1) and (P2) in Corollary4.3. The property (P1) follows automatically from the equation (11). The property (P2)follows from (P1) together with the fact that f k is left-continuous or right-continuous ateach point of I , as assured by Lemma 4.7. By Corollary 4.3, the map ˜ g = f k has aninvariant quasi-partition, that is to say, the set˜ Q = ∪ r − j =1 ∪ s ≥ ˜ g − s ( { y j } ) = ∪ r − j =1 ∪ s ≥ f − sk ( { y j } )is finite, implying that the set Q ′ = ∪ r − j =1 ∪ s ≥ f − s ( { y j } ) is finite. By the proof ofLemma 4.7, we have that { x , . . . , x n − } ⊂ { y , . . . , y r − } . In this way, Q := ∪ n − i =1 ∪ s ≥ f − s ( { x i } ) ⊂ ∪ r − j =1 ∪ t ≥ f − t ( { y j } ) , and Q is therefore finite. This proves that f has an invariant quasi-partition. λ -AFFINE MAPS By Corollary 4.3, the map ˜ g = f k is asymptotically periodic. We claim that f is alsoasymptotically periodic. Let x ∈ I , then there exists a periodic orbit γ k of f k such that ω f k ( x ) = γ k . Let p ∈ γ k . Notice that p is a periodic point of f , thus there exists a periodicorbit γ of f that contains p and γ k . Let U be a neighborhood of γ given by Lemma 4.5.Since ω f k ( x ) = γ k ⊂ γ , there exists an integer η ≥ f ηk ( x ) ∈ U . By Lemma 4.5, ω f ( x ) = ω f (cid:0) f ηk ( x ) (cid:1) = γ which proves the claim. Hence, f is asymptotically periodic. (cid:3) An upper bound for the number of periodic orbits
Throughout this section, let φ , . . . , φ n : [0 , → (0 ,
1) be Lipschitz contractions, W be the full set in the statement of Theorem 4.1 and Ω ′ n − be the set defined in (10).Let ( x , . . . , x n − ) ∈ Ω ′ n − ∩ W n − and f = f φ ,...,φ n ,x ,...,x n − . By Theorem 4.1, f has aninvariant quasi-partition P = ∪ mℓ =1 J ℓ with endpoints in { , } ∪ ∪ n − i =1 Q i , where each set Q i = ∪ k ≥ f − k ( { x i } ) is finite.Here we prove the following result. Theorem 5.1.
The n -interval PC f has at most n periodic orbits. We would like to distinguish some intervals in P , first those having x = 0 and x n = 1as endpoints. We denote them by F and G n , where x ∈ F and x n ∈ G n . For every1 ≤ i ≤ n −
1, let F i = ( a, x i ) and G i = ( x i , b ) be the two intervals in P whichhave x i as an endpoint. We may have G i = F i +1 for some 1 ≤ i ≤ n −
2. Amongthe intervals F , G , . . . , F n − , G n − , there are at least n and at most 2( n −
1) pairwisedistinct intervals. We will prove that among them there are 1 ≤ r ≤ n pairwise distinctintervals, say C , . . . , C r , which satisfy the following: for every J ∈ P , there exist k ≥ ≤ i ≤ r such that f k ( J ) ⊂ C i . This implies that the asymptotical behavior of any interval J ∈ P coincides with theasymptotical behavior of an interval C i .Let J, J , J ∈ P and k ≥
0. We remark that f k ( J ) ⊂ J ∪ J if, and only if, f k ( J ) ⊂ J or f k ( J ) ⊂ J . Lemma 5.2.
Let ( a, b ) ∈ P with a ∈ Q i and b ∈ Q j , where ≤ i, j ≤ n − and i = j .Then there exists ℓ ≥ such that ( at least ) one of the following statements holds (i) f ℓ ( F i ) ⊂ F j ∪ G j or f ℓ ( G i ) ⊂ F j ∪ G j ; (ii) f ℓ ( F j ) ⊂ F i ∪ G i or f ℓ ( G j ) ⊂ F i ∪ G i .Proof. The hypotheses that a ∈ Q i , b ∈ Q j and ( x , . . . , x n − ) ∈ Ω ′ n − (cid:0) see (10) (cid:1) implythat there exist unique integers ℓ i , ℓ j ≥ f ℓ i ( a ) = x i and f ℓ j ( b ) = x j . Moreover, f k ( a )
6∈ { x , . . . , x n − } for every k = ℓ i , and f m ( b )
6∈ { x , . . . , x n − } for every m = ℓ j . Let J = ( a, b ), then f ℓ i ( J ) ⊂ F i ∪ G i and f ℓ j ( J ) ⊂ F j ∪ G j . Now it is clear that the claim (i)happens if ℓ i ≤ ℓ j (then we set ℓ = ℓ j − ℓ i ) and the claim (ii) occurs if ℓ i ≥ ℓ j (then weset ℓ = ℓ i − ℓ j ). (cid:3) Lemma 5.3.
Let J ∈ P , then there exist ≤ i ≤ n − and k ≥ such that f k ( J ) ⊂ F i ∪ G i . IECEWISE λ -AFFINE MAPS 13 Proof.
It follows by the arguments used in the proof of Lemma 5.2. (cid:3)
Lemma 5.4.
There exists a permutation i , . . . , i n − of , . . . , n − and intervals ( a k − , b k ) ∈ P with a k − ∈ Q i ∪ . . . ∪ Q i k − and b k ∈ Q i k for every ≤ k ≤ n − .Proof. Since Q , . . . , Q n − are pairwise disjoint finite subsets of the interval (0 , y i = min Q i , 1 ≤ i ≤ n − i , . . . , i n − of 1 , . . . , n − y i < · · · < y i n − . Set b k = y i k ∈ Q i k ,1 ≤ k ≤ n −
1, therefore 0 ≤ b < b < · · · < b n − <
1. Notice that(12) (0 , b k ) ∩ (cid:0) Q i k ∪ · · · ∪ Q i n − (cid:1) = ∅ for every 2 ≤ k ≤ n − . For every 2 ≤ k ≤ n −
1, let S k = (0 , b k ) ∩ (cid:0) Q ∪ · · · ∪ Q n − (cid:1) . Since b k − ∈ (0 , b k ) ∩ Q i k − ,we have that S k = ∅ . Set a k − = max S k , then a k − < b k and ( a k − , b k ) ∈ P . By (12), a k − ∈ Q i ∪ · · · ∪ Q i k − , which concludes the proof. (cid:3) Using the permutation i , . . . , i n − defined in Lemma 5.4, for simplicity, set F ′ k = F i k and G ′ k = G i k , for 1 ≤ k ≤ n − Corollary 5.5.
Let ≤ k ≤ n − , then there exist ≤ j < k and ℓ ≥ such that ( atleast ) one of the following statements holds :( i ) f ℓ ( F ′ j ) ⊂ F ′ k ∪ G ′ k or f ℓ ( G ′ j ) ⊂ F ′ k ∪ G ′ k , ( ii ) f ℓ ( F ′ k ) ⊂ F ′ j ∪ G ′ j or f ℓ ( G ′ k ) ⊂ F ′ j ∪ G ′ j .Proof. Let i , . . . , i n − be the permutation of 1 , . . . , n − ≤ k ≤ n −
1, there exist 1 ≤ j < k and ( a, b ) ∈ P with a ∈ Q i j and b ∈ Q i k .The interval ( a, b ) fulfills the hypothesis of Lemma 5.2. The proof is finished by makingthe following substitutions in the claim of Lemma 5.2: i = i j , j = i k , F i = F i j = F ′ j and F j = F i k = F ′ k . (cid:3) Next we introduce an equivalence relation in the family of intervals P ′ listed as P ′ = { F , G , . . . , F n − , G n − } = { F ′ , G ′ , . . . , F ′ n − , G ′ n − } . Definition 5.6.
Let C , C ∈ P ′ . We say that C and C are equivalent if there exists C ∈ P ′ such that f ℓ ( C ) ∪ f k ( C ) ⊂ C for some ℓ, k ≥
0. If C and C are equivalent,we write C ≡ C . Lemma 5.7.
The relation ≡ is an equivalence relation with at most n equivalence classes.Proof. It is clear that ≡ is reflexive and symmetric. To prove that ≡ is transitive, let C , C , C ∈ P ′ with C ≡ C and C ≡ C . We will prove that C ≡ C .There exist C, C ′ ∈ P ′ such that f ℓ ( C ) ∪ f k ( C ) ⊂ C and f p ( C ) ∪ f q ( C ) ⊂ C ′ for some ℓ, k, p, q ≥
0. If ℓ ≥ p , then f ℓ − p ( C ′ ) ⊂ C , which means that f q + ℓ − p ( C ) ⊂ C implyingthat C ≡ C . Otherwise ℓ < p , then f p − ℓ ( C ) ⊂ C ′ , which means that f k + p − ℓ ( C ) ⊂ C ′ implying that C ≡ C . We have proved that ≡ is an equivalence relationDenote by [ C ] the equivalence class of the interval C ∈ P ′ . Now we will prove that ≡ has at most n equivalence classes.For each 1 ≤ k ≤ n −
1, let m k ≥ F ′ ] , [ G ′ ] , . . . , [ F ′ k ] , [ G ′ k ]. We have that m ≤
2. By Corollary 5.5, for each λ -AFFINE MAPS ≤ k ≤ n −
1, there exist C ∈ { F ′ , G ′ , . . . , F ′ k − , G ′ k − } and C ∈ { F ′ k , G ′ k } such that C ≡ C . Hence, m k ≤ m k − + 1 for every 2 ≤ k ≤ n −
1. By induction, m k ≤ k + 1 forevery 1 ≤ k ≤ n −
1. The proof is finished by taking k = n − (cid:3) Proof of Theorem 5.1 . The fact that ( x , . . . , x n − ) ∈ Ω ′ n − implies that the periodicorbits of f are entirely contained in the union of the intervals of the quasi-partition P .Moreover, each interval of P intersects at most one periodic orbit of f . By Lemma 5.3,every orbit of f intersects an interval of P ′ . The intervals of P ′ that intersect the sameperiodic orbit of f belong to the same equivalence class. In this way, there exists aninjective map that assigns to each periodic orbit of f an equivalence class. By Lemma5.7, the number of equivalence classes is at most n . As a result, the number of periodicorbits of f is at most n . (cid:3) Proof of Theorem 1.2
We begin with a lemma which will be used to prove Theorem 1.2 in the case I = R . Lemma 6.1.
Let φ i : R → R , ≤ i ≤ n , be ρ -Lipschitz contractions. Then there exists r = r ( φ , . . . , φ n ) > such that for every r ≥ r , the following holds: ( i ) φ i ([ − r, r ]) ⊂ ( − r, r ) for every ≤ i ≤ n ; ( ii ) For every x ∈ R , there exists k = k ( x ) ≥ such that | h ( x ) | < r , for every h ∈ C k .Proof. Let c = max i | φ i (0) | and 1 ≤ i ≤ n , then, for every x ∈ R , the following holds | φ i ( x ) | ≤ | φ i ( x ) − φ i (0) | + | φ i (0) | ≤ ρ | x | + c. (12)Set r := 2 c/ (1 − ρ ) and let r ≥ r . Note that | x | ≤ r = ⇒ | φ i ( x ) | ≤ ρ | x | + c ≤ ρr + (1 − ρ ) r < r which proves ( i ).By (12), let h = φ i ◦ φ i ◦ · · · ◦ φ i k ∈ C k , where 1 ≤ i , i , . . . , i k ≤ n . We have that | h ( x ) | = | φ i ◦ φ i ◦ · · · ◦ φ i k ( x ) | ≤ ρ k | x | + ( ρ k − + . . . + ρ + 1) c ≤ ρ k | x | + c − ρ ≤ ρ k | x | + r . Given x ∈ R , let k be so large that ρ k | x | < r/
2, then | h ( x ) | < r which proves ( ii ). (cid:3) Proof of Theorem 1.2.
It follows straightforwardly from Theorems 4.1 and 5.1 that Theo-rem 1.2 holds for I = [0 , ,
1) we consider anybounded real interval of the form [ a, b ).Now we consider the case I = R . Let φ , . . . , φ n : R → R be ρ -Lipschitz contractionsand r = r ( φ , . . . , φ n ) > k ≥
0, set I k = [ − ( r + k ) , r + k ). By the item ( i ) of Lemma 6.1 , φ i (cid:0) I k (cid:1) ⊂ ˚ I k for every 1 ≤ i ≤ n .Let φ ( k ) i := φ i | I k , then { φ ( k )1 , . . . , φ ( k ) n } is an IFS consisting of ρ -Lipschitz contractionsdefined on I k . Hence, by the first part of the proof, there exists a full subset V k of I k suchthat, for every ( x , . . . , x n − ) ∈ Ω n − ( I k ) ∩ ( V k ) n − , the map f φ ( k )1 ,...,φ ( k ) n ,x ,...,x n − : I k → ˚ I k is asymptotically periodic and has at most n periodic orbits. By the items ( i ) and ( ii ) ofLemma 6.1, for every k ≥
1, the maps f φ ,...,φ n ,x ,...,x n − : R → R and f φ ( k )1 ,...,φ ( k ) n ,x ,...,x n − : IECEWISE λ -AFFINE MAPS 15 I k → ˚ I k have the same asymptotic limits. Therefore f φ ,...,φ n ,x ,...,x n − is also asymptoticallyperiodic and has at most n periodic orbits.To conclude the proof, let W k = ( −∞ , − ( r + k )) ∪ V k ∪ ( r + k, ∞ ). Therefore W k isa full subset of R and the denumerable intersection W Φ = ∩ k ≥ W k is also a full subset of R . Let ( x , . . . , x n − ) ∈ Ω n − ( R ) ∩ ( W Φ ) n − and k be an integer largerthan max {| x | , | x n − |} . Thus, the point ( x , . . . , x n − ) also belongs to the set Ω n − ( I k ) ∩ ( V k ) n − , implying that the map f φ ,...,φ n ,x ,...,x n − is asymptotically periodic and has atmost n periodic orbits. This concludes the proof of the theorem. (cid:3) Proof of Theorem 1.1
Throughout this section, let − < λ < φ , . . . , φ n : R → R be λ -affine mapsdefined by φ i ( x ) = λx + b i , where b , . . . , b n ∈ R . Hereafter, to avoid misunderstanding,whenever a piecewise λ -affine map is defined on the whole line, we use the notation¯ f φ ,...,φ n ,x ,...,x n − in place of f φ ,...,φ n ,x ,...,x n − . Lemma 7.1 (Reduction Lemma) . Let ( c , . . . , c n − ) ∈ Ω n − ( R ) and ¯ f = ¯ f φ ,...,φ n ,c ,...,c n − .Then, for every δ ∈ R , the map ¯ f δ : R → R defined by ¯ f δ = ¯ f + δ is topologically conjugateto the map ¯ g = ¯ f φ ,...,φ n ,x ,...,x n − , where x i = c i − δ/ (1 − λ ) for every ≤ i ≤ n − .Proof. Let δ ∈ R be fixed. Set x i = c i − δ/ (1 − λ ) and ¯ g = ¯ f φ ,...,φ n ,x ,...,x n − . Let h : R → R be defined by h ( x ) = x + δ/ (1 − λ ). We claim that h ◦ ¯ g = ¯ f δ ◦ h . To show this, let I = ( −∞ , x ) , I n = [ x n − , ∞ ) and I j = [ x j − , x j ), 2 ≤ j ≤ n −
1. By the definition of h and x i , we have that h ( I ) = ( −∞ , c ), h ( I n ) = [ c n − , ∞ ) and h ( I j ) = [ c j − , c j ) for every2 ≤ j ≤ n −
1. Moreover, by (2), for every x ∈ I i , 1 ≤ i ≤ n , we have that h (¯ g ( x )) = h ( φ i ( x )) = φ i ( x ) + λδ − λ + δ = φ i (cid:18) x + δ − λ (cid:19) + δ = φ i ( h ( x )) + δ = ¯ f δ ( h ( x )) . This proves the claim. (cid:3)
Proof of Theorem 1.1.
Let I = [0 ,
1) and f : I → R be an n -interval piecewise λ -affinecontraction, then there exist λ -affine contractions φ , . . . , φ n : R → R and points 0 = d
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