C*-algebras associated with reversible extensions of logistic maps
CC ∗ -ALGEBRAS ASSOCIATED WITH REVERSIBLEEXTENSIONS OF LOGISTIC MAPS B. K. KWAŚNIEWSKI
Abstract.
The construction of reversible extensions of dynamical systems pre-sented in [KL08] is enhanced, so that it applies to arbitrary mappings (not nec-essarily with open range). It is based on calculating the maximal ideal space of C ∗ -algebras that extends endomorphisms to partial automorphisms via partialisometric representations, and involves a newfound set of "parameters" (the roleof parameters play chosen sets or ideals).As model examples, we give a thorough description of reversible extensionsof logistic maps, and a classification of systems associated with compression ofunitaries generating homeomorphisms of the circle. Contents
Introduction. 11. Preliminaries. Endomorphisms of commutative C ∗ -algebras, dynamicalsystems and their extensions 32. Natural reversible extensions of dynamical systems 113. Reversible extensions of logistic maps 174. Reversible extensions of homeomorphisms of a circle 32References 38 Introduction.
A general C ∗ -method of construction of reversible extensions of irreversible dy-namical systems was developed in [KL08], where in particular the complete de-scription of the maximal ideal spaces of the arising C ∗ -algebras was given. Thesealgebras naturally spring out in the spectral analysis of weighted shift operatorsand transfer operators (see, in particular, [Kwa09], [AL94]) and as is shown in[KL08] their maximal ideal spaces are tightly related to inverse (projective) limitsof dynamical systems.As a C ∗ -algebraic basis of their construction the authors of [KL08] explored theleading concept of [LO04] – an algebra whose elements play the role of Fourier co-efficients in partial isometric extensions of C ∗ -algebras. Namely, there was studied Mathematics Subject Classification.
Key words and phrases. extensions of dynamical systems, logistic maps, partial isometry,coefficient algebra.The author expresses his gratitude to A. V. Lebedev for his active interest in preparation ofthis paper. This work was in part supported by Polish Ministry of Science and High Educationgrant number N N201 382634. a r X i v : . [ m a t h . D S ] J u l B. K. KWAŚNIEWSKI the C ∗ -algebra C ∗ ( A , U ) generated by a ∗ -algebra A ⊂ L ( H ) , ∈ A , and an op-erator U ∈ L ( H ) under the assumption that A has the following three properties: A (cid:51) a → δ ( a ) := U aU ∗ ∈ A , (1) A (cid:51) a → δ ∗ ( a ) := U ∗ aU ∈ A , (2) U a = δ ( a ) U, a ∈ A . (3)In this event, A is called coefficient algebra of C ∗ ( A , U ) , cf. [LO04, Prop. 2.4].One has to stress that such objects are the major structural elements of the mostsuccessful crossed product constructions, like the ones developed by J. Cuntz andW. Krieger [Cun77, CK80], W. L. Paschke [Pas80], G. J. Murphy [Mur96], R. Exel[Exel94] and others. In general, conditions (1), (2), (3) imply that δ ( · ) = U ( · ) U ∗ isan endomorphism of A (then U is necessarily a partial isometry ) and δ ∗ ( · ) = U ( · ) U ∗ is a unique non-degenerate transfer operator for δ : A → A (in the sense of[Exel03]), see [BL]. The general crossed-product based on relations (1), (2), (3) isdeveloped in [ABL], [KL09]. As is shown in [Kwa’], it could be viewed as the crossedproduct by a Hilbert bimodule [AEE98], and hence it is one of the fundamentalmodels of relative Cuntz-Pimsner algebras [MS98], C ∗ -algebras associated with C ∗ -correspondences [Kat04], [Kat03], and Doplicher-Roberts algebras [DR89], see[Kwa”].In this article we develop the C ∗ -formalism of [KL08] and apply it to a seriesof classical dynamical systems, in order to get the description of maximal idealspaces of C ∗ -algebras associated with their reversible extensions. We recall thatthe starting point of [KL08] was a commutative unital C ∗ -subalgebra A ⊂ L ( H ) and an endomorphism δ : A → A such that(4)
A (cid:51) a → δ ( a ) := U aU ∗ ∈ A , U ∗ U ∈ A , for a certain U ∈ L ( H ) . Then (1) and (3) hold, and the C ∗ -algebra B = C ∗ (cid:0) ∞ (cid:91) n =0 U ∗ n A U n (cid:1) generated by (cid:83) ∞ n =0 U ∗ n A U n is the smallest (still commutative) coefficient C ∗ -algebraof C ∗ ( A , U ) such that A ⊂ B . The passage from A to B corresponds to passagefrom irreversible to reversible dynamics. Namely, endomorphisms δ : A → A and δ : B → B are given, via Gelfand transform, by partial dynamical systems ( M, α ) and ( (cid:102) M , (cid:101) α ) where ( (cid:102) M , (cid:101) α ) may be viewed as a universal reversible exten-sion of ( M, α ) (we will make the latter statement precise in Theorem 2.14). Theauthors of [KL08] gave a complete description of ( (cid:102) M , (cid:101) α ) in terms of ( M, α ) andnoticed that ( (cid:102) M , (cid:101) α ) contains, as a subsystem, the inverse limit of ( M, α ) . Thisindicates that the structure of the C ∗ -algebra B is related to hyperbolic attractors,[Wil70], [BS03], [Dev89] (such as solenoids or horseshoes of Smale); irreversiblecontinua [Nad92] (the most known are Brouwer-Janiszewski-Knastera continuum,or Knaster’s pseudoarc); and systems associated with classical substitution tilings[AP98] (these include tilings of Penrose, Amman, Fibonnaci, Morse, etc.).The results of [KL08], however, have one drawback. The only seemingly tech-nical assumption U ∗ U ∈ A implies that the image of α is necessarily open, whichin turn excludes a great deal of important examples. As the first step in thepresent paper we eliminate this inconvenience. The key hint on how to overcome ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 3 the mentioned drawback is given in [KL08, Rem. 3.7]. Namely, one has to passfrom the C ∗ -algebra A to the C ∗ -algebra A + := C ∗ ( A , U ∗ U ) generated by A andthe projection U ∗ U and apply the C ∗ -method of the reversible extension construc-tion to the dynamical system generated on A + . In the starting Sections 1 and 2we provide the corresponding analysis and as a result obtain a description of theextended system ( (cid:102) M , (cid:101) α ) under the conditions (1), (3) which are weaker than (4).As we show in Theorem 2.8, these axioms embrace all endomorphisms of A , andthereby all partial dynamical systems ( M, α ) . The principal novelty here is un-covering of the fact that ( (cid:102) M , (cid:101) α ) depends not only on ( M, α ) but also on a certainset of parameters Y ⊂ X (or ideals in A ). This new observation has a numberof interesting consequences. For instance, we get nontrivial results implementingour method to (already) reversible dynamical systems, such as homeomorphismsof the circle. Nevertheless, our primary example and one of our main goals is adepictive presentation of C ∗ -algebras associated with reversible extensions of thefamily of logistic maps α λ : [0 , → [0 , :(5) α λ ( x ) = 4 λx (1 − x ) , < λ ≤ , In the process of portraying the maximal ideal spaces of arising C ∗ -algebras, apartfrom the developed formalism, we take advantage of the results concerning inverselimits of logistic maps [BI96], [BM85]. In particular, we discuss in detail how theextended systems are influenced by such phenomena as bifurcations or chaos .The paper is organized as follows. In Section 1 we introduce notation andgeneralize or adapt from [KL08] the basic concepts required for presentation of theresults of the article. Here we also discuss a general C ∗ -method of extending partialdynamical systems. The main result of [KL08], description of maximal ideal spacesof C ∗ -algebras corresponding to reversible extensions of C ∗ -dynamical systems, isrefined in Section 2, where apart from giving a purely topological definition wecharacterize such systems as universal objects. Section 3 is devoted to presentationof reversible extensions of the logistic family, and finally, in Section 4 we classifythe C ∗ -algebras associated with homeomorphisms of the circle via their rotationnumbers.1. Preliminaries. Endomorphisms of commutative C ∗ -algebras,dynamical systems and their extensions Throughout the article we let A be a unital commutative C ∗ -algebra. By usingthe Gelfand transform we assume the identification A = C ( M ) , where M = M ( A ) is the maximal ideal space (also called spectrum ) of the algebra A .1.1. Endomorphisms and partial dynamical systems.
It is well known (see,for example, [KL08, Thm 2.2]) that every endomorphism δ : A → A is of the form δ ( a ) = (cid:26) a ( α ( x )) , x ∈ ∆0 , x / ∈ ∆ , a ∈ A = C ( M ) . where α : ∆ → M is a continuous mapping defined on a closed and open (brieflyclopen) subset ∆ ⊂ M . Namely, treating points of M as functionals on A we have ∆ = { x ∈ M : x ( δ (1)) (cid:54) = 0 } and α = δ ∗ | ∆ where δ ∗ is the dual operator to δ : A → A . Therefore we will refer to α : ∆ → M as to a mapping dual to endomorphism δ . B. K. KWAŚNIEWSKI
To start with we describe the necessary for our further presentation objectsrelated to algebras and endomorphisms.For every subset I ⊂ A the set hull( I ) := { x ∈ M : x ( I ) = 0 } , is a closed subset of M and if I is an ideal in A , then I = C hull( I ) ( M ) where C K ( M ) stands for the set of continuous functions on M vanishing on K ⊂ M . Plainly, inthe above notation we have hull(ker δ ) = α (∆) . Since ∆ is closed and α is continuous it follows that α (∆) is closed as well, andevidently α (∆) is open iff the characteristic function of α (∆) belongs to C ( M ) (inwhich case it is a unit in ker δ = C α (∆) ( M ) ). Accordingly, we arrive at Proposition 1.1.
The image of the mapping α dual to an endomorphism δ : A →A is open if and only if the kernel of δ is unital. The annihilator of an ideal I in A is the set I ⊥ := { a ∈ A : aI = { }} . Clearly, I ⊥ is an ideal and it could be equivalently defined as the largest ideal in A such that I ∩ I ⊥ = { } . In particular, if the kernel of δ : A → A is unital, then A admits decomposition into the following direct sum of ideals(6) A = ker δ ⊕ (ker δ ) ⊥ , and δ yields an isomorphism between the ideal (ker δ ) ⊥ and the subalgebra δ ( A ) ⊂ A . Definition 1.2.
An endomorphism δ : A → A with unital kernel ker δ and theimage δ ( A ) being an ideal in A will be called a partial automorphism of A .Since δ ( A ) is an ideal in A iff δ ( A ) = δ (1) A , it follows that δ is a partialautomorphism of A iff the algebra A admits two decompositions into direct sumof ideals A = ker δ ⊕ (ker δ ) ⊥ , A = δ ( A ) ⊕ δ ( A ) ⊥ . If this is the case, we denote by δ ∗ : δ ( A ) → (ker δ ) ⊥ the inverse to the isomorphism δ : (ker δ ) ⊥ → δ ( A ) and prolong δ ∗ onto A by putting δ ∗ | δ ( A ) ⊥ ≡ . Clearly, δ ∗ : A → A is a partial automorphism and(7) δ ◦ δ ∗ ◦ δ = δ, δ ∗ ◦ δ ◦ δ ∗ = δ ∗ . Hence δ ∗ is the so-called generalized inverse to δ .The next proposition shows a deeper relation between the objects introducedabove. Proposition 1.3.
Endomorphism δ : A → A is a partial automorphism if andonly if its dual map α : ∆ → α (∆) is a homeomorphism and α (∆) is clopen in M .Moreover, if δ is a partial automorphism, then there is a unique partial auto-morphism δ ∗ : A → A which is a generalized inverse for δ , and it is given by δ ∗ ( a ) = (cid:26) a ( α − ( x )) , x ∈ α (∆)0 , x / ∈ α (∆) , a ∈ A = C ( M ) . ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 5 Proof. If δ : A → A is a partial automorphism, then α (∆) is clopen (by Propo-sition 1.1) and α : ∆ → α (∆) is a homeomorphism since it is a mapping dualto the isomorphism δ : (ker δ ) ⊥ → δ ( A ) . The converse implication is straightfor-ward. Suppose now that δ and δ ∗ are partial automorphisms satisfying (7). Then δ = δ ◦ δ ∗ ◦ δ implies that ker δ ∗ ∩ δ ( A ) = { } , equivalently δ ( A ) ⊂ (ker δ ∗ ) ⊥ , and δ ∗ = δ ∗ ◦ δ ◦ δ ∗ implies δ ∗ ( δ ( A )) = δ ∗ ( A ) . But since δ ∗ is a partial automorphismthe latter relation gives (ker δ ∗ ) ⊥ ⊂ δ ( A ) and therefore (ker δ ∗ ) ⊥ = δ ( A ) . By sym-metry we also have (ker δ ) ⊥ = δ ∗ ( A ) , and thus it follows that δ ∗ : δ ( A ) → (ker δ ) ⊥ coincides with the inverse for δ : (ker δ ) ⊥ → δ ( A ) . In particular, δ ∗ is uniquelydetermined by δ and its dual mapping is α − . (cid:4) The above consideration makes it natural to adopt the following definitions, cf.[KL08, Def. 2.4, 2.6].
Definition 1.4.
By a (partial) dynamical system we will mean a triple ( M, ∆ , α ) ,where M is a compact Hausdorff space, ∆ a clopen subset of M , and α : ∆ → M acontinuous map. Unless a misunderstanding can arise, we will simply write ( M, α ) . Definition 1.5.
We will say that a partial dynamical system ( M, ∆ , α ) is reversible if α (∆) is an open subset of M and the map α : ∆ → α (∆) is a homeomorphism(so that the triple ( M, α (∆) , α − ) is also a partial dynamical system).1.2. Extensions of partial dynamical systems and endomorphisms.
Themain concept of the C ∗ -method of construction of reversible extensions of irre-versible dynamical systems is given in [KL08]. The present and the next subsec-tions 1.3, 1.4 and Section 2 are devoted to description of the main structural blocksof this construction. Moreover we also give a refinement of the construction that isnecessary for complete analysis of the partial dynamical systems under investiga-tion in the paper, and establish universality and minimality of natural reversibleextensions (Theorem 2.14). Definition 1.6.
Let ( M α , ∆ α , α ) and ( M β , ∆ β , β ) be partial dynamical systems.We say that a surjective continuous map Ψ : M β → M α is a semiconjugacy (or a factor map ) if and only if the following conditions hold(8) Ψ − (∆ α ) = ∆ β , (9) α (Ψ( x )) = Ψ( β ( x )) , x ∈ ∆ β . If there is a semiconjugacy from ( M β , β ) to ( M α , α ) we say that ( M β , β ) is an extension of ( M α , α ) and ( M α , α ) is a factor of ( M β , β ) . If additionally the system ( M β , β ) is reversible we call it a reversible extension of ( M α , α ) . If the factor mapis one-to-one, then its inverse is also a factor map, and we say that ( M β , β ) and ( M α , α ) are conjugated , or equivalent .The next proposition clarifies the role of the objects introduced in the abovedefinition and shows that the class of partial dynamical systems with factor mapsas morphisms form a category dual to the category of endomorphisms of unital C ∗ -algebras where the role of morphisms is played by unital monomorphisms thatconjugate endomorphisms. Proposition 1.7.
Let δ : A → A and γ : B → B be endomorphisms of unital com-mutative C ∗ -algebras A , B , and let ( M ( A ) , α ) and ( M ( B ) , β ) be the corresponding B. K. KWAŚNIEWSKI dual partial dynamical systems. Endomorphism γ extends δ in the sense that thereexits a unital monomorphism T : A → B such that (10) T ◦ δ = γ ◦ T, if and only if the partial dynamical system ( M ( B ) , β ) is an extension of the system ( M ( A ) , α ) . Furthermore, the factor map Ψ : M ( B ) → M ( A ) is the dual map tothe unital monomorphism T satisfying (10) : T ( a ) = a ◦ Ψ , a ∈ A = C ( M ( A )) . Proof.
A mapping dual to a unital monomorphism T : A → B maps M ( B ) onto M ( A ) and hence T is an operator of composition with a surjection Ψ : M ( B ) → M ( A ) where Ψ = T ∗ | M ( B ) . For a we have ( γ ◦ T )( a )( x ) = (cid:40) a (Ψ( β ( x )) , x ∈ ∆ β , , x / ∈ ∆ β , ( T ◦ δ )( a )( x ) = (cid:40) a ( α (Ψ( x )) , Ψ( x ) ∈ ∆ α , , Ψ( x ) / ∈ ∆ α . Using these formulas one checks that (10) holds iff Ψ satisfies (8) and (9). (cid:4) Remark 1.8.
We have to stress that the extension described in Definition 1.6 is aslightly different (weaker) notion than the corresponding one described in [KL08,Def. 2.7]. Namely, note that (8) is equivalent to two relations
Ψ(∆ β ) = ∆ α and Ψ( M β \ ∆ β ) = M α \ ∆ α , and (9) implies that(11) Ψ( M β \ β (∆ β )) ⊃ M α \ α (∆ α ) . However, unlike [KL08], we allow α (∆ α ) not to be open (this is vital for theapplications considered in this article). Moreover, as the further part of the articleshow the principle situation of interest is the case when β (∆ β ) is open, thus wecan not require to have equality in (11) (as in [KL08, Def. 2.7]).Continuing the above remark one may consider the following consequence of(11)(12) Ψ( M β \ β (∆ β )) ⊃ M α \ α (∆ α ) . In particular, as it will be shown in the article, by refining (fixing) the left handpart of this inclusion one can obtain various extensions naturally arising in theanalysis of dynamical systems.
Definition 1.9.
Let ( M α , ∆ α , α ) and ( M β , ∆ β , β ) be partial dynamical systemsand let Y ⊂ M α . If there is a semiconjugacy Ψ : M β → M α such that(13) Ψ( M β \ β (∆ β )) = Y we say that ( M β , ∆ β , β ) is an extension of ( M α , ∆ α , α ) associated with Y . Remark 1.10.
Plainly, the set Y satisfying (13) is necessarily closed and contains M α \ α (∆ α ) . Conversely, for any such set Y one easily constructs an extension of ( M α , ∆ α , α ) associated with Y , see for instance Fig. 1.Let us describe now the relation between extensions of dynamical systems asso-ciated with Y and extensions of endomorphisms associated with ideals.Suppose that δ : A → A and γ : B → B are endomorphisms conjugated by T ,as in Proposition 1.7. Then (10) implies the relations(14) ker δ = T − (ker γ ) , (ker δ ) ⊥ ⊃ T − ((ker γ ) ⊥ ) , ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 7 where by taking hulls the former yields (8) and the latter gives (12). In particular, hull((ker δ ) ⊥ ) = M α \ α (∆ α ) and hull (cid:0) T − ((ker γ ) ⊥ ) (cid:1) = Ψ( M β \ β (∆ β )) , where ∆ α and ∆ β are the domains of α and β respectively, and Ψ is the dualmapping to T . Accordingly, we arrive at Theorem 1.11.
Let J be an ideal in A and set Y = hull( J ) . Under the notationof Proposition 1.7, the system ( M ( B ) , β ) is an extension of the system ( M ( A ) , α ) associated with Y if and only if there exits a unital monomorphism T : A → B such that (15) T ◦ δ = γ ◦ T, and T − ((ker γ ) ⊥ ) = J. If this is the case, then J ⊂ (ker δ ) ⊥ (equivalently Y ⊃ M ( A ) \ α (∆ α ) ), so thatendomorphism γ extending δ "shrinks the annihilator of its kernel" to J . Remark 1.12.
Assuming the identification
A ⊂ B ( A ∼ = T ( A ) ⊂ B ) we have J = (ker γ ) ⊥ ∩ A and relations (14) tell that γ may enlarge the kernel of δ but onlyoutside A , that is (ker γ ) ∩A = ker δ . In other words, the only reason why J may bestrictly smaller than (ker δ ) ⊥ is that ker γ \ A is non-empty. Thus it seems naturalto look for extensions where J = (ker δ ) ⊥ (equivalently Y = M ( A ) \ α (∆ α ) ), cf.[KL08, Def. 2.7] and Remark 1.8.1.3. C ∗ -dynamical systems and partial dynamical systems. Partial dynam-ical systems are naturally associated with C ∗ -dynamical systems and representa-tions of the latter ones in turn are defined by actions of partial isometries in Hilbertspaces. Discussion of this relationship is the theme of the present subsection. Definition 1.13.
By a (concrete) C ∗ -dynamical system we mean a pair ( A , U ) ,where A is a commutative C ∗ -subalgebra of the algebra L ( H ) of bounded operatorsin a Hilbert space H , ∈ A , and U ∈ L ( H ) is a partial isometry such that(16) U A U ∗ ⊂ A , U ∗ U ∈ A (cid:48) where A (cid:48) is the commutator of A in L ( H ) .Plainly, relations (16) imply that(17) δ ( a ) := U aU ∗ , a ∈ A , is an endomorphism of A , and we will say that δ : A → A is an endomorphismgenerated by U . Similarly, if ( M, ∆ , α ) is the partial dynamical system dual to δ we will say that ( M, ∆ , α ) is a partial dynamical system generated by U . Remark 1.14.
The above definition extends [KL08, Def. 2.11], see relations (4).In particular, relations (16) and assumption that U is a partial isometry are equiv-alent to (1) and (3), cf. [LO04, Prop. 2.2]. Remark 1.15.
We will show (in Theorem 2.8) that for an arbitrary endomorphism δ : A → A the C ∗ -algebra A may be identified with an algebra of operators actingin a certain Hilbert space H in such a way that δ is generated by an operator U ∈ L ( H ) . Hence every abstract C ∗ -dynamical system ( A , δ ) can be representedby a concrete one . B. K. KWAŚNIEWSKI
As it is indicated by Theorem 1.11, in order to investigate extensions of anendomorphism generated by U ∈ L ( H ) we need to identify the annihilator of itskernel in terms of ( A , U ) . To this end, let us consider the sets (1 − U ∗ U ) A ∩ A = { a ∈ A : U ∗ U a = 0 } , U ∗ U A ∩ A = { a ∈ A : U ∗ U a = a } , which are (mutually orthogonal) ideals in A . The next proposition shows the roleof these sets in the description of endomorphisms. Proposition 1.16.
Let ( A , U ) be a C ∗ -dynamical system, δ : A → A an endomor-phism generated by U , ( M, ∆ , α ) a partial dynamical system dual to δ : A → A ,and Y = hull( U ∗ U A ∩ A ) . Then ker δ = (1 − U ∗ U ) A ∩ A and (ker δ ) ⊥ ⊃ U ∗ U A ∩ A , that is Y ⊃ M \ α (∆) . If additionally U ∗ U ∈ A , then ker δ = (1 − U ∗ U ) A and (ker δ ) ⊥ = U ∗ U A , that is Y = M \ α (∆) . In particular, i) U ∗ U ∈ A = ⇒ ker δ is unital ( α (∆) is clopen); ii) U is an isometry = ⇒ δ : A → A is a monomorphism ( α (∆) = M ); iii) U is unitary = ⇒ δ : A → A is a unital monomorphism ( ∆ = M and α : M → M is surjective). Proof.
To see that (1 − U ∗ U ) A ∩ A coincides with ker δ let a ∈ A and note that U ∗ U a = 0 = ⇒ δ ( a ) = U aU ∗ = U ( U ∗ U a ) U ∗ = 0 ,U ∗ U a (cid:54) = 0 = ⇒ U ∗ δ ( a ) U = U ∗ U aU ∗ U = U ∗ U a (cid:54) = 0 = ⇒ δ ( a ) (cid:54) = 0 . Now, since ker δ ∩ (cid:0) U ∗ U A ∩ A (cid:1) = { } we have U ∗ U A ∩ A ⊂ (ker δ ) ⊥ , and if U ∗ U ∈ A , the projection (1 − U ∗ U ) is the unit for ker δ = (1 − U ∗ U ) A , andconsequently U ∗ U is the unit for (ker δ ) ⊥ . Thus the remaining part of propositionis straightforward, cf. [KL08, Prop. 2.3]. (cid:4) In the next Lemma 1.18 and Theorem 1.19 we give a description of (ker δ ) ⊥ and obtain restraint for U ∗ U in terms of the objects related to central carriers ofelements in von Neumann algebras (for completeness of presentation we include inLemma 1.18 the known properties i) and ii) of carriers). Definition 1.17.
By a carrier of a C ∗ -subalgebra K ⊂ L ( H ) we mean the orthog-onal projection Q ∈ L ( H ) onto the subspace KH ⊂ H . Lemma 1.18.
Let Q ∈ L ( H ) be a carrier of an ideal I in a (not necessarilycommutative) C ∗ -algebra A ⊂ L ( H ) . Then i) Q = s- lim λ ∈ Λ µ λ where { µ λ } λ ∈ Λ is an approximate unit for I (the limit istaken in the strong operator topology). ii) Q ∈ A (cid:48) . iii) I ⊂ Q A ∩ A and I ⊥ = (1 − Q ) A ∩ A . Proof.
By Vigier’s Theorem the limit Q := s - lim λ ∈ Λ µ λ exists. For a ∈ A and h ∈ H we have Qah = lim λ ∈ Λ µ λ ah = lim λ ∈ Λ µ λ ( lim λ (cid:48) ∈ Λ aµ λ (cid:48) ) h = QaQh, and similarly aQh = lim λ ∈ Λ µ λ ah = lim λ ∈ Λ µ λ ( lim λ (cid:48) ∈ Λ aµ λ (cid:48) ) h = QaQh. ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 9 Using these relations one deduces that Q is a carrier of I and Q ∈ A (cid:48) . Clearly, I ⊂ QA ∩ A and I ⊥ ⊂ (1 − Q ) A ∩ A . To see that I ⊥ ⊃ (1 − Q ) A ∩ A let a / ∈ A \ I ⊥ .Then there is b ∈ I and h ∈ H such that abh (cid:54) = 0 . Hence Qabh = abh (cid:54) = 0 andtherefore Qa (cid:54) = 0 , which is equivalent to (1 − Q ) a (cid:54) = a . (cid:4) By item iii) in the above lemma we see that complements of carriers are ’born’to deal with annihilators of ideals.
Theorem 1.19.
Under the notation of Proposition 1.16, let P = 1 − Q ∈ L ( H ) bethe complement of the carrier Q of the ideal (1 − U ∗ U ) A ∩ A = ker δ in A . Then U ∗ U ≤ P, P ∈ A (cid:48) and (ker δ ) ⊥ = P A ∩ A . Moreover, if (ker δ ) ⊥ = U ∗ U A ∩ A (equivalently Y = M \ α (∆) ), which automat-ically holds when U ∗ U = P , then the implications in items i)-iii) in Proposition1.16 are in fact equivalences. Proof.
By Lemma 1.18, P ∈ A (cid:48) and (ker δ ) ⊥ = P A ∩ A . By definition of Q , Q ≤ − U ∗ U and thus U ∗ U ≤ P . Now suppose that (ker δ ) ⊥ = U ∗ U A ∩ A ; byProposition 1.16, condition U ∗ U ∈ A gives even more, namely (ker δ ) ⊥ = U ∗ U A .We show the converses to implications in items i)-iii) of Proposition 1.16.i) If ker δ is unital, then Q is the unit for ker δ and consequently P is the unitfor (ker δ ) ⊥ = U ∗ U A ∩ A . This implies that U ∗ U = P ∈ A .ii) If ker δ = { } , then A = (ker δ ) ⊥ = U ∗ U A ∩ A , that is A = U ∗ U A andtherefore U is an isometry.iii) It suffices to combine items i) and ii). (cid:4) The main theme of [KL08] is a description of the C ∗ -method of construction ofreversible C ∗ -dynamical systems, that is systems such that not only ( A , U ) butalso ( A , U ∗ ) is a C ∗ -dynamical system, and thus both of the mappings δ ( a ) := U aU ∗ , δ ∗ ( a ) := U ∗ aU, a ∈ A are endomorphisms of A . We adopt an equivalent version of [KL08, Def. 2.15]. Definition 1.20.
By a reversible C ∗ -dynamical system we mean a pair ( A , U ) where A ⊂ L ( H ) is commutative, ∈ A , and U ∈ L ( H ) is a partial isometry suchthat(18) U A U ∗ ⊂ A , U ∗ A U ⊂ A , Clearly, relations (18) are equivalent to the condition that both ( A , U ) and ( A , U ∗ ) are C ∗ -dynamical systems. Proposition 1.21. If ( A , U ) is a reversible C ∗ -dynamical system, then endomor-phisms δ : A → A and δ ∗ : A → A generated by U and U ∗ are mutually generalizedinverse partial automorphisms and (ker δ ) ⊥ = δ ∗ ( A ) = U ∗ U A , δ ( A ) = (ker δ ∗ ) ⊥ = U U ∗ A . In particular, the partial dynamical system ( M ( A ) , α ) generated by U is reversible. Proof.
By Proposition 1.16 and the symmetry between δ and δ ∗ it suffices toshow that δ ( A ) = U U ∗ A which follows because δ ( A ) = U A U ∗ = U U ∗ U A U ∗ ⊂ U U ∗ A , and U U ∗ A = U U ∗ A U U ∗ = δ ( δ ∗ ( A )) ⊂ δ ( A ) . (cid:4) C ∗ -method of extending partial dynamical systems. Given a concrete C ∗ -dynamical system ( A , U ) we have at our disposal two mappings that are definedon the whole of the C ∗ -algebra L ( H ) :(19) δ ( a ) := U aU ∗ , δ ∗ ( a ) := U ∗ aU, a ∈ L ( H ) . Moreover, δ restricted to A is an endomorphism, and δ ∗ restricted to A is anendomorphism iff ( A , U ) is reversible. Obviously δ (and similarly δ ∗ ) may defineendomorphisms on many different subalgebras of L ( H ) , and if additionally such asubalgebra, say B , contains A it yields a natural extension δ : B → B of the initialendomorphism δ : A → A , cf. [KL08, Def. 2.17]. This in turn can be rewritten inthe language of partial dynamical systems.
Theorem 1.22.
Let ( A , U ) and ( B , U ) be C ∗ -dynamical systems and let ( M ( A ) , α ) and ( M ( B ) , β ) be partial dynamical systems generated by U on the maximal idealspaces of A and B , respectively. If A ⊂ B , then ( M ( B ) , β ) is an extension of ( M ( A ) , α ) associated with the set Y = hull((ker δ | B ) ⊥ ∩ A ) = hull( P B ∩ A ) where (ker δ | B ) ⊥ is the annihilator of the kernel of δ : B → B , and P is thecomplement of the carrier of (1 − U ∗ U ) B ∩ B . Moreover, i) U ∗ U ∈ B = ⇒ Y = hull( U ∗ U A ∩ A ) , ii) U ∗ U ∈ A = ⇒ Y = M ( A ) \ α (∆) . Proof.
Since δ : B → B is an extension of δ : A → A in the sense of Propo-sition 1.7, where T = id , the first part of assertion follows from Theorem 1.11and Proposition 1.19. If additionally U ∗ U ∈ B , then in view of Proposition 1.16we have (ker δ | B ) ⊥ ∩ A = U ∗ U B ∩ A = U ∗ U A ∩ A . Similarly, if U ∗ U ∈ A , then (ker δ | B ) ⊥ ∩ A = U ∗ U A and hull( U ∗ U A ) = M ( A ) \ α (∆) , cf. Proposition 1.1. (cid:4) Using the above method one can always obtain a reversible extension which (inthe context of coefficients algebras) was first noticed in [LO04], cf. [KL08].
Theorem 1.23.
Suppose that, under the notation of Theorem 1.22,
A ⊂ B and ( B , U ) is reversible. Then ( M ( B ) , β ) is a reversible extension of ( M ( A ) , α ) asso-ciated with the set Y = hull( U ∗ U A ∩ A ) . Moreover, for any C ∗ -dynamical system ( A , U ) there exists a minimal reversible C ∗ -dynamical system ( B , U ) such that A ⊂ B . Namely, the C ∗ -algebra (20) B = C ∗ (cid:0) ∞ (cid:91) n =0 U ∗ n A U n (cid:1) generated by (cid:83) ∞ n =0 U ∗ n A U n is commutative, and it is the smallest C ∗ -algebra thatcontains A and satisfies (18) . Proof.
The first part of assertion follows from Proposition 1.21 and Theo-rem 1.22 (since U ∗ U = U ∗ U ∈ B ). Commutativity of the algebra B , given by(20) and the property that both δ : B → B and δ ∗ : B → B are endomorphisms wasestablished in [LO04, Prop. 4.1]. The remaining part is straightforward. (cid:4) ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 11 Natural reversible extensions of dynamical systems
In this section we give a complete purely topological description of partial dy-namical systems corresponding to minimal reversible extensions of C ∗ -dynamicalsystems introduced in Theorem 1.23. Additionally, we characterize such systemsas universal objects and discuss their relation to the notion of inverse limit, whichtherefore we recall now. Definition 2.1. If α : M → M is a continuous mapping of a topological space M , then the inverse limit of the inverse sequence M α ←− M α ←− M α ←− ... is thetopological space of the form lim ←−− ( M, α ) := { ( x , x , ... ) ∈ (cid:89) n ∈ N M : α ( x n +1 ) = x n , n ∈ N } equipped with the product topology inherited from (cid:81) n ∈ N M . Furthermore, on thespace lim ←−− ( M, α ) we have a naturally defined homeomorphism (cid:101) α ( x , x , ... ) = ( α ( x ) , x , x , ... ) , ( x , x , ... ) ∈ lim ←−− ( M, α ) , called a homeomorphism induced by the mapping α : M → M .2.1. C ∗ -dynamical approach. Throughout this section we let ( A , U ) be a C ∗ -dynamical system, δ and δ ∗ the mappings given by (19), and B the C ∗ -algebrafrom Theorem 1.23. We denote by ( M, ∆ , α ) the partial dynamical system dualto δ : A → A and by ( (cid:102) M , (cid:101) ∆ , (cid:101) α ) the reversible partial dynamical system dual to δ : B → B : A ∼ = C ( M ) , B = C ∗ (cid:0) ∞ (cid:91) n =0 U ∗ n A U n (cid:1) ∼ = C ( (cid:102) M ) . The dynamical system ( (cid:102) M , (cid:101) ∆ , (cid:101) α ) plays the principal role in the paper. As wasnoted in [KL08, Rem. 3.7] to obtain description of ( (cid:102) M , (cid:101) ∆ , (cid:101) α ) in terms of ( M, ∆ , α ) one has to apply the main result of [KL08] to the C ∗ -algebra A + := C ∗ ( A , U ∗ U ) generated by A and the projection U ∗ U . Therefore we need to analyze the alge-bra A + and the partial dynamical system dual to endomorphism δ : A + → A + .Hereafter we proceed to the discussion of these objects.2.1.1. Adjoining the projection U ∗ U to the algebra A . Plainly, the C ∗ -algebra A + is the direct sum of ideals(21) A + = U ∗ U A ⊕ (1 − U ∗ U ) A where (1 − U ∗ U ) A = ker( δ | A + ) is the kernel of δ : A + → A + , see Proposition 1.16.As a simple consequence, cf. for instance [KR97, Lem. 10.1.6], we get Proposition 2.2.
Algebra A + is isomorphic the direct sum of quotient algebras (22) A + ∼ = A / ker( δ | A ) ⊕ A / (cid:0) U ∗ U A ∩ A (cid:1) where ker( δ | A ) = (1 − U ∗ U ) A ∩ A is the kernel of δ : A → A and U ∗ U A ∩ A is anideal contained in the annihilator ker( δ | A ) ⊥ of ker( δ | A ) . Under the isomorphism (22) endomorphism δ : A + → A + takes the form a + ker( δ | A ) ⊕ b + (cid:0) U ∗ U A ∩ A (cid:1) (cid:55)−→ δ ( a ) + ker( δ | A ) ⊕ δ ( a ) + (cid:0) U ∗ U A ∩ A (cid:1) . Remark 2.3.
The summand (1 − U ∗ U ) A ∼ = A / (cid:0) U ∗ U A ∩ A (cid:1) is a unital C ∗ -algebrawhich coincides with the kernel of δ : A + → A + . Thus one may interpret adjoiningthe projection U ∗ U to the algebra A as a unitization of the kernel of δ : A → A (equivalently compactification of the complement of the image of α ).Let ( M + , ∆ + , α + ) denote the partial dynamical system generated by U on themaximal ideal space of A + . In view of Theorem 1.22, ( M + , α + ) is an extension of ( M, α ) associated with the set Y = hull( U ∗ U A ∩ A ) . By passing in Proposition 2.2 to duals we get the complete description of the partialdynamical system ( M + , ∆ + , α + ) . Proposition 2.4.
Under the above notation Y is the closed set containing M \ α (∆) , and the spectrum M + of the algebra A + realizes as the following (topological)direct sum M + = α (∆) (cid:116) Y. The partial mapping α + dual to δ : A + → A + is defined on the set ∆ + = ( α (∆) ∩ ∆) (cid:116) ( Y ∩ ∆) and attains values in the first summand of M + = α (∆) (cid:116) Y acting by the formula α + ( x ) = α ( x ) , x ∈ ∆ + , see Fig. 1. In particular, U ∗ U ∈ A , that is A = A + , if and only if Y = M \ α (∆) ,and then α (∆) is clopen. Proof.
Since the maximal ideal spaces of the quotient algebras A / ker( δ | A ) and A / (cid:0) U ∗ U A ∩ A (cid:1) may be identified with the sets hull(ker( δ | A )) = α (∆) and hull( U ∗ U A ∩ A ) = Y , the first part of assertion follows from Proposition 2.2. Forthe second part apply Theorem 1.19. (cid:4) Figure 1.
Partial dynamical system ( M, α ) generated by U on A (a); partial dynamical system ( M + , α + ) generated by U on A + (b).Pictorially speaking, the space M + arise from ( M, α ) by "cutting out" α (∆) and"replacing" the set M \ α (∆) with the closed set Y that contains M \ α (∆) . Theimage of α + is the set α (∆) that has been "cut out" and thus may be identifiedwith the image of α . The domain of α + is enlarged with respect to the domain of α by the set Y ∩ (∆ ∩ α (∆)) .Now, as we have obtained the description of the system ( M + , ∆ + , α + ) we passto the main object of this subsection – the system ( (cid:102) M , (cid:101) ∆ , (cid:101) α ) . ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 13 Description of the system ( (cid:102) M , (cid:101) α ) dual to δ : B → B . Let (cid:101) x ∈ (cid:102) M be amultiplicative linear functional on B . We associate to it a sequence of functionalson A given by the formula x n ( a ) := (cid:101) x ( δ n ∗ ( a )) , a ∈ A , n ∈ N . Since δ ∗ : B → B is an endomorphism, the functionals x n : A → C are multiplica-tive linear, and thus x n ∈ M or x n ≡ . Moreover, since B = C ∗ ( (cid:83) ∞ n =0 δ n ∗ ( A )) the sequence ( x , x , ... ) determines (cid:101) x uniquely.Consequently we have the injective mapping(23) (cid:101) x (cid:55)−→ ( x , x , ..., x n , ... ) that embeds the space (cid:102) M into the Cartesian product (cid:81) n ∈ N ( M ∪ { } ) of the count-able number of copies of the space M ∪ { } .The next theorem is a refinement of the main result of [KL08] up to objectsunder consideration. Theorem 2.5 (Description of the reversible system ( (cid:102) M , (cid:101) α ) ) . The maximal idealspace of the algebra B may be identified via the mapping (23) with the followingtopological space (cid:102) M = ∞ (cid:91) N =0 M N ∪ M ∞ where M N = { ( x , x , ..., x N , , ... ) : x n ∈ ∆ , α ( x n ) = x n − , n = 1 , ..., N, x N ∈ Y } ,M ∞ = { ( x , x , ... ) : x n ∈ ∆ , α ( x n ) = x n − , n (cid:62) } , are equipped with the product topology inherited from (cid:81) n ∈ N ( M ∪ { } ) , where { } is a clopen singleton, and Y is a closed set containing M \ ∆ (namely Y =hull( U ∗ U A ∩ A ) ). Furthermore the partial homeomorphisms dual to partial au-tomorphisms δ, δ ∗ : B → B are defined respectively on the clopen sets (cid:101) ∆ = { ( x , x , ... ) ∈ (cid:102) M : x ∈ ∆ } , (cid:101) α ( (cid:101) ∆) = { ( x , x , ... ) ∈ (cid:102) M : x (cid:54) = 0 } and act according to formulae (24) (cid:101) α ( x , x , ... ) = ( α ( x ) , x , x , ... ) , (cid:101) α − ( x , x , ... ) = ( x , ... ) . Proof.
It suffices to apply [KL08, Thm. 3.5, 4.1] to the C ∗ -dynamical system ( A + , U ) and then use Proposition 2.4, cf. also [KL08, Rem. 3.7]. (cid:4) Remark 2.6.
The mapping
Φ : (cid:102) M → M dual to the inclusion A ⊂ B is given bythe formula(25) Φ( x , x , ... ) = x . It is a factor map establishing that ( (cid:102) M , (cid:101) α ) is a reversible extension of ( M, α ) associated with Y .The next result shows that in the situations when U belongs to a series ofcommonly exploited classes of operators the structure of ( (cid:102) M , (cid:101) α ) becomes moretransparent. Theorem 2.7.
Under the notation of Theorem 2.5, we have M \ α (∆) ⊂ Y and i) If U ∗ U ∈ A , then Y = M \ α (∆) , and hence a sequence ( x , x , ..., x N , , ... ) is an element of M N iff x N / ∈ α (∆) (i.e. x N does not have a preimage)and x n ∈ ∆ , α ( x n ) = x n − for n = 1 , ..., N. ii) If U is an isometry, then α : ∆ → M is a surjection, and thus (cid:102) M = M ∞ . iii) If U is unitary, then α : M → M is a surjection, (cid:102) M = lim ←−− ( M, α ) and (cid:101) α : (cid:102) M → (cid:102) M is a homeomorphism induced by α : M → M . Proof.
In view of Theorem 2.5, item i) follows from Proposition 2.4, whereasitems ii), iii) follow by Proposition 1.16. (cid:4)
Construction of operators generating arbitrary partial dynamicalsystem ( M, α ) . Let ( M, ∆ , α ) be a partial dynamical system and let Y ⊂ M be a closed set containing M \ α (∆) . Theorem 2.5 leads to a natural question:does there exist a C ∗ -dynamical system ( A , U ) generating the partial dynamicalsystem ( M, ∆ , α ) and such that its reversible extension is associated with Y . Inother words, whether all the objects described in Theorem 2.5 are realizable. Theanswer is: yes. Namely, following [KL08, 4.2], we use the explicit description ofthe reversible extension ( (cid:102) M , (cid:101) α ) associated with Y to give a simple construction ofoperators generating ( M, α ) .Let ( U f )( (cid:101) x ) = (cid:40) f ( (cid:101) α ( (cid:101) x )) , (cid:101) x ∈ (cid:101) ∆0 , (cid:101) x / ∈ (cid:101) ∆ act in the Hilbert space H = (cid:96) ( (cid:102) M ) (which may be treated as L µ ( (cid:102) M ) where µ isthe counting measure), and let A ⊂ L ( H ) be the algebra consisting of operators ofmultiplication by function from C ( (cid:102) M ) dependent only on the zeroth coordinate: A = { a ∈ C ( (cid:102) M ) : a ( (cid:101) x ) = a ( x ) , where (cid:101) x = ( x , ... ) ∈ (cid:102) M } . Clearly,
A ∼ = C ( M ) and U generates on M the partial mapping α . The projection U ∗ U is the operator of multiplication by a characteristic function of (cid:101) α ( (cid:101) ∆) , andthus U ∗ U A ∩ A = { a ∈ A : U ∗ U a = a } ∼ = C Y ( M ) . Accordingly, we get
Theorem 2.8.
Let ( M, α ) be an arbitrary partial dynamical system and let Y be arbitrary closed set containing M \ α (∆) . Then there is a Hilbert space H ,a C ∗ -algebra A ⊂ L ( H ) whose spectrum is homeomorphic to M , and a partialisometry U ∈ L ( H ) such that i) on the spectrum of A operator U generates the partial mapping α , ii) on the spectrum of B := C ∗ ( (cid:83) n ∈ N U ∗ n A U n ) operator U generates the re-versible extension ( (cid:102) M , (cid:101) α ) of ( M, α ) associated with Y . Remark 2.9.
It follows from [FMR, Ex. 1.6], see [KL], [Kwa”], that concrete C ∗ -dynamical systems generating a fixed "abstract" endomorphism δ : A → A arein one-to-one correspondence with representations of a certain C ∗ -correspondence(Hilbert bimodule) X constructed from δ . Moreover, for an ideal J in A , the ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 15 C ∗ -dynamical systems ( A , U ) for which U ∗ U A ∩ A ⊂ J correspond to represen-tations of X coisometric on J . Consequently, one could construct a pair ( A , U ) enjoying the properties described in Theorem 2.8 by taking a quotient of the Fockrepresentation of X , see [MS98], which in comparison to our construction is amuch more involved approach. In view of the foregoing observation one could call Y = hull( U ∗ U A ∩ A ) a set of cosurjectivity for ( A , U ) .2.3. Topological definition, relation with inverse limits and universality.
We finish the section with a discussion of universality of the reversible extensiondescribed in Theorem 2.5. This theorem shows, in particular, that the system ( (cid:102) M , (cid:101) α ) is independent of its operator theoretical origin. Therefore we adopt thefollowing Definition 2.10.
Let ( M, α ) be a partial dynamical system and Y a closed subsetof M containing M \ α (∆) . The partial dynamical system ( (cid:102) M , (cid:101) α ) described inTheorem 2.5 will be called the natural reversible extension of ( M, α ) associatedwith Y . Remark 2.11.
In the case when α : M → M is defined on the whole of M the natural reversible extension ( (cid:102) M , (cid:101) α ) associated with Y ⊃ M \ α ( M ) has thefollowing structure:- the system ( M ∞ , (cid:101) α ) coincides with the inverse limit system ( lim ←−− ( M, α ) , (cid:101) α ) ,- for each N ∈ N , the set M N is homeomorphic to Y and (cid:101) α carries homeo-morphically M N onto M N +1 : M (cid:101) α −→ M (cid:101) α −→ M (cid:101) α −→ ... ( M ∞ , (cid:101) α ) = ( lim ←−− ( M, α ) , (cid:101) α ) . Remark 2.12.
The system ( (cid:102) M , (cid:101) α ) has an advantage over the system ( lim ←−− ( M, α ) , (cid:101) α ) .Namely, ( (cid:102) M , (cid:101) α ) is always an extension of ( M, α ) whereas ( lim ←−− ( M, α ) , (cid:101) α ) may de-generate. The next example illustrates this observation. Example 2.13.
Let α : M → M be a constant map with the only value being anon-isolated point p ∈ M , see Fig. 2 (a). The only closed set containing M \ α (∆) = (a) (b)(c) (d) ( M, α ) ( M + , α + )( (cid:102) M, (cid:101) α ) lim ←−− ( M, α ) , (cid:101) α ) Figure 2.
Systems associated with a constant map. M \ { p } is Y = M . For this set the system ( M + , α + ) described in Proposition2.4 arise from ( M, α ) by adjoining a clopen copy of the singleton { p } , Fig. 2 (b).The space (cid:102) M is a countable family of copies of M compactified with a single point M ∞ = { p } , Fig. 2 (c). Finally, the space ( lim ←−− ( M, α ) , (cid:101) α ) degenerates to a singlepoint, Fig. 2 (d).A hard piece of evidence that ( (cid:102) M , (cid:101) α ) is an appropriate reversible counterpart of ( M, α ) is its identification as a universal minimal object that we now present. Theorem 2.14 (Universality of natural reversible extension) . Let ( M, ∆ , α ) be apartial dynamical system, ( (cid:102) M , (cid:101) ∆ , (cid:101) α ) its natural reversible extension associated witha set Y ⊂ M , and Φ the corresponding factor map (25). i) If ( M β , ∆ β , β ) is a reversible extension of ( M, α ) associated with Y , and Ψ is the corresponding semiconjugacy, then there is a unique semiconjugacy (cid:101) Ψ from ( M β , β ) to ( (cid:102) M , (cid:101) α ) such that Ψ = Φ ◦ (cid:101) Ψ , i.e. the diagram ( M β , β ) (cid:101) Ψ (cid:47) (cid:47) Ψ (cid:36) (cid:36) (cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74) ( (cid:102) M , (cid:101) α ) Φ (cid:122) (cid:122) (cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117) ( M, α ) commutes, and (cid:101) Ψ( M β \ β (∆ β )) = (cid:102) M \ (cid:101) α ( (cid:101) ∆) , so that ( M β , β ) is an extensionof ( (cid:102) M , (cid:101) α ) associated with (cid:102) M \ (cid:101) α ( (cid:101) ∆) . ii) If a partial dynamical system ( M β , β ) possess the property of ( (cid:102) M , (cid:101) α ) de-scribed in item i), then ( M β , β ) and ( (cid:102) M , (cid:101) α ) are equivalent and equivalenceis established by means of (cid:101) Ψ . Proof. i) Denote by M βN , N ∈ N , the set of points that has exactly N preimagesunder β , that is y ∈ M βN iff y, β − ( y ) , ..., β − ( N − ( y ) ∈ β (∆ β ) and β − N ( y ) / ∈ β (∆ β ) .Similarly, we denote by M β ∞ the set of points that have infinitely many preimagesunder β . We put M βN (cid:51) y (cid:55)−→ (cid:101) Ψ( y ) := (Ψ( y ) , Ψ( β − ( y )) , ..., Ψ( β − N ( y )) , , ... ) ∈ M N ,M β ∞ (cid:51) y (cid:55)−→ (cid:101) Ψ( y ) := (Ψ( y ) , Ψ( β − ( y )) , ..., Ψ( β − n ( y )) , ... ) ∈ M ∞ . By (9) and (13) this yields a well defined mapping (cid:101)
Ψ : M β → (cid:102) M . It is continuous,as for an open set (cid:101) U ⊂ (cid:102) M of the form (cid:101) U = { ( x , x , ... ) ∈ (cid:102) M : x n ∈ U } , where U is open in M, the set (cid:101) Ψ − ( (cid:101) U ) = β n (Ψ − ( U )) is also open. Condition (13) implies that (cid:101) Ψ maps M βN onto M N . In particular, (cid:101) Ψ( M β \ β (∆ β )) = (cid:101) Ψ( M β ) = M = (cid:102) M \ (cid:101) α ( (cid:101) ∆) . Toshow that (cid:101) Ψ maps M β ∞ onto M ∞ we fix (cid:101) x = ( x , x , x , ... ) ∈ M ∞ and put D n = (cid:101) Ψ − ( { (cid:101) y = ( y , y , ... ) ∈ (cid:102) M : y n = x n } ) , n ∈ N . It is evident that { D n } n ∈ N forms a decreasing sequence of non-empty compact sets,and therefore (cid:84) n ∈ N D n (cid:54) = ∅ . Taking y ∈ (cid:84) n ∈ N D n we have (cid:101) Ψ( y ) = (cid:101) x , which provessurjectivity of (cid:101) Ψ . Relations (cid:101) Ψ − ( (cid:101) ∆) = ∆ β and Ψ = Φ ◦ (cid:101) Ψ are straightforward.For the uniqueness of (cid:101) Ψ we note that reversibility of the systems ( M β , β ) , ( (cid:102) M , (cid:101) α ) and the equality (cid:101) Ψ( M β \ β (∆ β )) = (cid:102) M \ (cid:101) α ( (cid:101) ∆) imply that a semiconjugacy (cid:101) Ψ from ( M β , β ) to ( (cid:102) M , (cid:101) α ) is automatically a semiconjugacy from ( M β , β − ) to ( (cid:102) M , (cid:101) α − ) .This together with relation Ψ = Φ ◦ (cid:101) Ψ give a family of relations Ψ ◦ β − n = Φ ◦ (cid:101) α − n ◦ (cid:101) Ψ , ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 17 n ∈ N , understood in the sense that not only functions but also their naturaldomains are equal. This forces (cid:101) Ψ to act according to formulas which we used as adefinition in the first part of the proof.ii) We have two semiconjugacies (cid:101) Φ : M β → (cid:102) M and (cid:101) Ψ : (cid:102) M → M β which by theargument from item i) (with the same convention concerning domains) satisfy Ψ ◦ β − n = Φ ◦ (cid:101) α − n ◦ (cid:101) Ψ , Φ ◦ (cid:101) α − n = Ψ ◦ β − n ◦ (cid:101) Φ , n ∈ N . Thus (Φ ◦ (cid:101) α − n ) ◦ ( (cid:101) Ψ ◦ (cid:101) Φ) = Φ ◦ (cid:101) α − n and by the form of (cid:102) M it follows that (cid:101) Ψ ◦ (cid:101) Φ = id .Therefore (cid:101) Φ is a homeomorphism which yields a desired equivalence. (cid:4) Identifying, A = C ( M ) with the subalgebra { a ◦ Φ : a ∈ C ( M ) } of (cid:101) A := C ( (cid:102) M ) ,denoting by δ and (cid:101) δ endomorphisms corresponding to α and (cid:101) α respectively, andputting J = C Y ( M ) , one may interpret the above result in terms of endomorphismsas follows. Theorem 2.15.
Every partial automorphism γ : B → B that extends δ : A → A insuch a way that (ker γ ) ⊥ ∩ A = J automatically extends the partial automorphism (cid:101) δ : (cid:101) A → (cid:101) A in such a way that (ker γ ) ⊥ ∩ (cid:101) A = (ker (cid:101) δ ) ⊥ . Moreover, this propertycharacterizes the pair ( (cid:101) A , (cid:101) δ ) (up to isomorphisms conjugating endomorphisms). Reversible extensions of logistic maps
Now as the necessary preparatory work is implemented and the required C ∗ -objectsare described we pass to presentation of calculation of concrete examples of re-versible extensions of dynamical systems. Figure 3.
Graph of the logistic map α λ : [0 , → [0 , (a); graphof the injective map γ λ : R → R arising from α λ (b).In this section, we conduct a thorough analysis of reversible extensions of thefamily of logistic maps { α λ } λ ∈ (0 , , where by a logistic map with parameter λ ∈ (0 , we mean a quadratic map α λ : [0 , → [0 , given by (5), cf. Fig. 3 (a). For betterillustration of our C ∗ -method we define the operators generating α λ , λ ∈ (0 , , ina concrete fashion, even though we already know that such operators always exist,see Theorem 2.8. To this end, we let H = L ( R ) and consider the C ∗ -algebra A consisting of operators of multiplication by periodic functions a ( t ) of period ,continuous on [0 , , and possessing a limit in from below, i.e.:(26) a ( t + 1) = a ( t ) , a | [0 , ∈ C ([0 , and there exists lim t → − a ( t ) . Plainly, A is isomorphic to C ([0 , and we will identify its spectrum M with theunit interval [0 , : M = [0 , . We fix λ ∈ (0 , and define a piecewise continuous function γ λ : R → R where(27) γ λ ( x ) = (cid:40) λt (1 − t ) + 2 k, t ∈ [ k, k + ]4 λt (1 − t ) + 2 k + 1 , t ∈ [ k + , k + 1) , k ∈ Z . The graph of γ λ arises from the graph of α λ by a "propagation" of the halves ofthe parabola y = α λ ( x ) on R , so that one obtains a graph of an injective mappingon R , see Fig. 3 (b). The mapping γ λ is injective, and it is bijective iff λ = 1 .We let U λ : L ( R ) → L ( R ) be a normalized operator of composition with γ λ :(28) ( U λ f )( t ) = (cid:113) | γ (cid:48) λ ( t ) | f ( γ λ ( t )) = 2 (cid:112) | λ (1 − t ) | f ( γ λ ( t )) , f ∈ L ( R ) , so that U λ is a coisometry where the adjoint isometry is given by the formula ( U ∗ λ f )( t ) = (cid:40)(cid:113) | ( γ − λ ) (cid:48) ( t ) | f ( γ − λ ( t )) , t ∈ γ λ ( R )0 , t / ∈ γ λ ( R ) , f ∈ L ( R ) . In particular, U λ is unitary if and only if λ = 1 . Proposition 3.1.
For each λ ∈ (0 , the operator U λ generates on the spectrumof the C ∗ -algebra A the logistic map α λ Moreover the hull of the ideal J = ( U ∗ λ U λ ) A ∩ A = { a ∈ A : U ∗ λ U λ a = a } is the smallest possible, cf. Proposition 1.16, that is Y = hull( J ) = M \ α λ ( M ) = (cid:40) [ λ, if λ < , ∅ if λ = 1 . Proof.
Since U ∗ λ U λ is the operator of multiplication by the characteristic func-tion χ γ λ ( R ) of the set γ λ ( R ) , we have U ∗ λ U λ ∈ A (cid:48) and the hull of J is the set Y = hull( { a ∈ A : χ γ λ ( R ) a = a } ) = [ λ, for λ < and Y = ∅ for λ = 1 . For anyoperator a of multiplication by a ( t ) , U λ aU ∗ λ is an operator of multiplication by ( U λ aU ∗ λ )( t ) = a ( γ λ ( t )) = a ( { γ λ ( t ) } ) = a ( α λ ( { t } )) , where { t } ∈ [0 , denotes the fractional part of a number t ∈ R . Hence one seesthat U λ generates on [0 , the mapping α λ . (cid:4) In view of the above the first of the following operations δ λ ( a ) := U λ aU ∗ λ , δ ∗ ,λ ( a ) := U ∗ λ aU λ , a ∈ L ( L ( R )) preserves the algebra A and the logistic map α λ is its dual map. In particular, wemay adopt the identifications: A = C ([0 , , δ λ ( a ) = a ◦ α λ , a ∈ A . ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 19 On the other hand, for a ∈ A , δ ∗ ,λ ( a ) is the operator of multiplication by thefunction δ ∗ ,λ ( a )( t ) = ( U ∗ λ aU λ )( t ) = (cid:40) a ( γ − λ ( t )) , x ∈ γ λ ( R )0 , x / ∈ γ λ ( R ) , which is periodic but in general its period is two, not one. Therefore the mappings δ ∗ ,λ , λ ∈ (0 , , do not preserve the algebra A and the C ∗ -algebras(29) B λ := C ∗ (cid:0) (cid:91) n ∈ N U ∗ nλ A U nλ (cid:1) , are essentially bigger than A . Theorem 3.2.
Let λ ∈ (0 , and let ( (cid:102) M λ , (cid:101) α λ ) be the reversible extension of ([0 , , α λ ) associated with the set Y where Y = [ λ, , if λ < , and Y = ∅ , if λ = 1 . The algebra B λ may be identified with C ( (cid:102) M λ ) and then the partial automor-phism δ λ : B λ → B λ becomes the operator of composition with the homeomorphism (cid:101) α λ : (cid:102) M λ → (cid:101) α λ ( (cid:101) ∆) : B λ = C ( (cid:102) M λ ) , δ λ ( a ) = a ◦ (cid:101) α λ , a ∈ B λ . In particular operator U λ generates on (cid:102) M λ the partial homeomorphism (cid:101) α λ . Proof.
It follows from Proposition 3.1, Theorem 2.5 and Definition 2.10. (cid:4)
The description of the extended C ∗ -dynamical systems ( B λ , δ λ ) reduces to thedescription of the family of reversible topological systems { ( (cid:102) M λ , (cid:101) α λ ) } λ ∈ (0 , . We recall, cf. Remark 2.6, that the mapping
Φ : (cid:102) M λ → [0 , dual to the embedding A ⊂ B λ is a surjection such that the diagram (cid:102) M λ (cid:101) α λ −−−→ (cid:102) M λ Φ (cid:121) (cid:121) Φ [0 , α λ −−−→ [0 , commutes. We stress that a change of the parameter value λ ∈ (0 , does not onlyinfluence the dynamics of (cid:101) α λ but also the topology of the space (cid:102) M λ . The followingnotions of continuum theory will be indispensable in our analysis Definition 3.3 (see, for instance, [Nad92]) . By a continuum we mean a connectedand compact metric space. A continuum will be calledi) nondegenerate , if it is not a singleton,ii) reducible , if it may be presented as the sum of two its proper subcontinuua,iii) irreducible , if it is a nondegenerate continuum which is not reducible,iv) snake-like or arc-like continuum , if it is homeomorphic to an inverse limitof an inverse sequence with bonding maps being continuous maps of aninterval, cf. [Nad92, 12.19].For all λ ∈ (0 , , the spectrum (cid:102) M λ of B λ contains the snake-like continuum M ∞ = lim ←−− ([0 , , α λ ) , and it is a general dynamical principle, discovered by M.Barge and J. Martin [BM85], that one should expect this continuum to be irre-ducible, see Theorem 3.6 below. General structure of the extended systems ( (cid:102) M λ , (cid:101) α λ ) . Let λ < . By Re-mark 2.11 the space (cid:102) M λ consists of the set M ∞ being the inverse limit lim ←−− ( M, α λ ) and a countable family of arcs M N , that is sets homeomorphic to a closed interval: M N ∼ = [ λ, . Theorem 3.4.
For λ < the maximal ideal space (cid:102) M λ of algebra B λ consists ofthe snake-like continuum M ∞ = lim ←−− ( M, α λ ) and a sequence of arcs M N such that lim n →∞ M N = M ∞ , where the limit is taken in Hausdorff metric. In particular, (cid:102) M λ = (cid:91) n ∈ N M N ∪ M ∞ = (cid:91) n ∈ N M N . The mapping (cid:101) α λ generated by U λ on (cid:102) M λ carries homeomorphically arc M N ontoarc M N +1 , and on the continuum M ∞ = lim ←−− ( M, α λ ) it coincides with the homeo-morphism induced by α λ (Definition 2.1). Proof.
We only need to show the equality lim n →∞ M N = M ∞ which (see forinstance [Nad92, Thm. 4.11]) is equivalent to the following two inclusions lim sup M N = { (cid:101) x ∈ (cid:102) M λ : ∀ (cid:101) x ∈ U open ∀ k ∈ N ∃ N>k U ∩ M N (cid:54) = ∅} ⊂ M ∞ ,M ∞ ⊂ lim inf M N = { (cid:101) x ∈ (cid:102) M λ : ∀ (cid:101) x ∈ U open ∃ k ∈ N ∀ N>k U ∩ M N (cid:54) = ∅} . If we assume that (cid:101) x ∈ lim sup M N and (cid:101) x ∈ M N , for certain N ∈ N , then takingin the definition of lim sup M N , U = M N and k = N we arrive at a contradiction.This proves the first inclusion. To prove the second one take (cid:101) x = ( x , ..., x N , ... ) ∈ M ∞ and an open neighbourhood of (cid:101) x of the form U = { (cid:101) y = ( y , ..., y N , ... ) ∈ (cid:102) M λ : y N ∈ ( x N − ε, x N + ε ) } . Note that there exists n ∈ N such that α n λ ([ λ, , λ ] . Indeed, if λ ≤ , then α λ ([ λ, , λ ] , and if λ > , then for certain n , α n λ ( λ ) > , and therefore α n λ ([ λ, , λ ] . Putting k = N + n we have U ∩ M N (cid:54) = ∅ for every N > k ,and thus (cid:101) x ∈ lim inf M N . (cid:4) The above statement implies that once we have at our disposal description of ( M ∞ , (cid:101) α λ ) we may easily describe the whole system ( (cid:102) M λ , (cid:101) α λ ) : it suffices to adjointo M ∞ the sequence of arcs { M N } N ∈ N converging to M ∞ and prolong (cid:101) α λ so that itshifts homeomorphically arcs M N towards M ∞ . The importance of this commentlies in that a great deal of facts concerning the systems of ( lim ←−− ( M, α λ ) , (cid:101) α λ ) typeis known, see [BI96]. Thus we may use them to achieve our goal.3.1.1. The extended system for λ = 1 (B-J-K continuum). For λ = 1 the mapping α is a surjection and the space (cid:102) M coincides with the inverse limit lim ←−− ([0 , , α ) of the full logistic map α ( x ) = 4 x (1 − x ) , cf. Theorem 2.7 iv). Hence (cid:102) M is one ofthe most famous irreducible continuum called Brouwer-Janiszewski-Knaster con-tinuum , briefly
B-J-K continuum [BI96], [Nad92], [Wat82]. One may graphicallydepict (cid:102) M by joining the points of the Cantor set with semicircles in a mannerpresented on Fig. 4 (a).The logistic map α ( x ) = 4 x (1 − x ) is topologically conjugate to the tent map α T ( x ) = 1 −| x − | and B-J-K continuum is usually considered [Nad92], [Wat82] as ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 21 the inverse limit lim ←−− ( M, α T ) . Then there is a natural parametrization of the com-posant of the point (0 , , , ... ) of B-J-K continuum by non-negative real numbers[Wat82], see Fig. 4 (b). Within this parametrization the induced homeomorphisms (cid:101) α and (cid:101) α T fulfill the following formulae(30) (cid:101) α ( t ) = (cid:40) k + α ( { t } ) , t ∈ [ k, k + )2( k + 1) − α ( { t } ) , t ∈ [ k + , k + 1) , (cid:101) α T ( t ) = 2 t, where { t } denotes the fractional part of the number t (obviously, the systems ( (cid:102) M , (cid:101) α ) and ( (cid:102) M , (cid:101) α T ) are topologically conjugate). These comments give a certainidea about the dynamics of the system ( (cid:102) M λ , (cid:101) α λ ) for λ = 1 . (a) (b)
13 23
Figure 4.
Brouwer-Janiszewski-Knaster continuum.
Theorem 3.5.
Algebra B may be identified with the algebra C ( (cid:102) M ) of continuousfunctions on B-J-K continuum (cid:102) M , Fig. 4 (a). Then the automorphism δ : B →B becomes the operator of composition with the homeomorphism (cid:101) α : (cid:102) M → (cid:102) M ,which within the parametrization presented on Fig. 4 (b) assumes the form (30) .Furthermore irreducibility of (cid:102) M expresses as the following property of the algebra B : if I , I are ideals in B such that B /I i , for i = 1 , , does not contain non-trivial idempotents, then I ∩ I = { } = ⇒ I = { } or I = { } . Proof.
The first part the statement follows from Theorem 3.2. To show thesecond part notice that for i = 1 , we have I i = C Y i ( (cid:102) M ) , where Y i = hull( I i ) ⊂ (cid:102) M . Algebra C ( (cid:102) M ) /I i ∼ = C ( Y i ) does not contain non-trivial idempotents if and only if Y i is connected, that is if Y i is a subcontinuum of (cid:102) M . Since I ∩ I = C Y ∪ Y ( (cid:102) M ) the condition I ∩ I = { } is equivalent to the equality Y ∪ Y = (cid:102) M and thereforethe asserted property of B is equivalent to irreducibility of continuum (cid:102) M . (cid:4) Feigenbaum limit λ ∞ . Let λ = , λ = √ ≈ . , λ , ..., be thesequence of the parameter values λ corresponding to the first cascade of period-doubling bifurcation of the system ([0 , , α λ ) . This is an increasing and the value λ ∞ = lim n →∞ λ n ≈ . is called Feigenbaum limit [BI96], [CE80], [Dev89].Interval of parameter splits into two parts (0 , λ ∞ ) and ( λ ∞ , which correspond λ λ λ ∞ µ µ Figure 5.
Bifurcation diagram for λ ∈ [0 ,
74; 1] .respectively to regular and chaotic behavior of the systems ([0 , , α λ ) , see Fig. 5.This finds a splendid reflection in the structure of the algebra B λ . Theorem 3.6.
Let λ ∈ (0 , . The following conditions are equivalent i) λ > λ ∞ , ii) spectrum (cid:102) M λ of B λ contains an irreducible continuum, iii) There exists a proper ideal J in a maximal ideal in the C ∗ -algebra B λ withthe property that for every two ideals I , I in B := B λ /J such that B /I , B /I do not contain non-trivial idempotents we have I ∩ I = { } = ⇒ I = { } or I = { } . Proof.
Equivalence of i) and ii) follows from Theorem 3.4 and [BI96, Thm. 3,4, 7]. In the proof of Theorem 3.5 we have shown that condition in item iii) isequivalent to irreducibility of continua. Moreover, the statement " J is the properideal in a maximal ideal in B λ " means that "the set Z = hull( J ) contains morethan one point". This explains equivalence of ii) and iii). (cid:4) The first cascade of bifurcation: λ ∈ (0 , λ ∞ ]. For λ < λ ∞ the dynamicsof ([0 , , α λ ) is completely understood: α λ has exactly one stable orbit which isperiodic with period n , n ∈ N , exactly one repelling periodic orbit with period k ,for each k = 0 , ..., n − , and at most two repelling fixed points. Increasing λ from to λ ∞ the number n gradually increases - the system undergoes a period-doublingbifurcation . In particular the period of the stable orbit of α λ , for λ ∈ (0 , λ ∞ ) ,increases according to Sharkovskii’s order [BS03], [Dev89]: (cid:47) (cid:47) (cid:47) ... (cid:47) n (cid:47) ...... (cid:47) m (2 n + 1) (cid:47) ... (cid:47) m · (cid:47) m · (cid:47) m · (cid:47) ...... (cid:47) n + 1) (cid:47) ... (cid:47) (cid:47) (cid:47) (cid:47) ...... (cid:47) (2 n + 1) (cid:47) ... (cid:47) (cid:47) (cid:47) . Let us now imaging that we slowly move the parameter λ from to λ ∞ , andsimultaneously observe the maximal ideal space (cid:102) M λ = (cid:91) n ∈ N M N ∪ M ∞ ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 23 of the algebra B λ . Let also assume that we watch the change in (cid:102) M λ from the pointof view of the initial space M = [0 , – looking at a point (cid:101) x = ( x , x ... ) ∈ (cid:102) M λ weread of x ∈ [0 , . Such an approach will allow us to understand in detail how thechange of the parameter λ affects the topology of the space (cid:102) M λ . In other wordswe will built an image of (cid:102) M λ with the help of the factor map Φ( x , x , ... ) = x , ( x , x , ... ) ∈ (cid:102) M λ , cf. Remark 2.6. Since however Φ is not injective, it may "wind" a piece of an arc M N several times onto an interval. Thus in order to get a homeomorphic image of (cid:102) M λ one needs to modify Φ by "adjoining" copies of the corresponding arcs. In thismanner one constructs a homeomorphism from (cid:102) M λ onto a certain subset of theplane, cf. [Kwa05”]. Here, we restrict ourselves to discussion of the results of thatprocedure, the trace of which will be seen on pictures of (cid:102) M λ where a point (cid:101) x ∈ (cid:102) M λ is labeled by its zeroth coordinate Φ( (cid:101) x ) ∈ M . The crucial role in this enterprise isplayed by the orbit { q n } n ∈ N of the critical point of the mapping α λ . We set q n := α nλ (cid:16) (cid:17) , n ∈ N . (a) (b)
12 12 ω , λ λ Figure 6.
Graph of α λ for < λ ≤ (a); < λ ≤ (b). (a) (b) q (cid:114) (cid:114) (cid:114) q q (cid:114) (cid:114) (cid:114) q q (cid:114) (cid:114) (cid:114) q q (cid:114) (cid:114) (cid:114)(cid:114) (cid:114) (cid:114) (cid:114) (cid:54)(cid:73)(cid:89)(cid:54) (cid:77) (cid:67)(cid:67)(cid:67)(cid:79)(cid:54) (cid:66)(cid:66)(cid:77) (cid:66)(cid:66)(cid:77)(cid:54) (cid:66)(cid:66)(cid:77) (cid:65)(cid:65)(cid:75) (cid:7)(cid:6) (cid:4) (cid:15) M M M M M M ∞ q (cid:114) (cid:114) (cid:114) q q (cid:114) (cid:114) (cid:114) q q (cid:114) (cid:114) (cid:114) q q (cid:114) (cid:114) (cid:114)(cid:114) (cid:114) (cid:114)(cid:114) (cid:114) ω , (cid:54)(cid:77)(cid:89)(cid:54) (cid:79) (cid:79)(cid:54) (cid:67)(cid:67)(cid:79) (cid:67)(cid:67)(cid:79)(cid:54) (cid:67)(cid:67)(cid:79) (cid:67)(cid:67)(cid:79) (cid:7)(cid:6) (cid:4) (cid:15) (cid:4)(cid:5)(cid:7) (cid:87)(cid:94) M M M M M M ∞ Figure 7.
Dynamics of ( (cid:102) M λ , (cid:101) α λ ) where < λ ≤ (a); < λ ≤ (b). We start with parameter λ taking values slightly greater than zero (smaller than ), see Fig. 6 (a). The only periodic point of the mapping α λ is a stable fixedpoint . In particular we have: q > q > q > ...., lim n →∞ q n = 0 . Accordingly, the space (cid:102) M λ compose of a singleton M ∞ and a sequence of arcs M N converging to M ∞ , Fig. 7 (a). While we increase the parameter λ the length ofintervals [0 , q N ] increases. Finally, when λ surpasses the fixed point looses itsstability transferring it onto a newly born fixed point ω , > : q > q > q > ..., lim n →∞ q n = ω , > , Fig. 6 (b). The set M ∞ becomes an arc corresponding to the interval [0 , ω , ] , andit grows as λ increases, Fig. 7 (b). (a) (b) ω , λ λ ω , ω , Figure 8.
Graph of α λ for < λ ≤ (a); < λ ≤ √ (b). (a) (b) q (cid:113) (cid:113) q (cid:113) (cid:113) (cid:113) q q (cid:113) (cid:113) (cid:113) (cid:113) q q (cid:113) (cid:113) (cid:113) (cid:113)(cid:113)(cid:113) (cid:113) q q q (cid:113) (cid:113) ω , (cid:113) (cid:113) (cid:73)(cid:89)(cid:54) (cid:18) (cid:54)(cid:54) (cid:18) (cid:0)(cid:0)(cid:18)(cid:54) (cid:18) (cid:17)(cid:17)(cid:51) (cid:2)(cid:2)(cid:14) (cid:7)(cid:6)(cid:4) (cid:15) (cid:4)(cid:5)(cid:7) (cid:87)(cid:94) (cid:78) M M M M M M ∞ q (cid:113) (cid:113) q (cid:113) (cid:113) (cid:113) q q (cid:113) (cid:113) (cid:113) (cid:113) q q (cid:113) (cid:113) (cid:113) (cid:113)(cid:113)(cid:113) (cid:113) q q q (cid:113) (cid:113) ω , ω , (cid:113) (cid:113) (cid:113) (cid:113)(cid:113)(cid:113) (cid:89)(cid:89)(cid:54) (cid:18) (cid:54)(cid:54) (cid:18) (cid:17)(cid:17)(cid:51)(cid:54) (cid:18) (cid:16)(cid:16)(cid:49) (cid:7)(cid:6)(cid:4) (cid:15) (cid:121)(cid:57)(cid:94) (cid:63) M M M M M M ∞ Figure 9.
Dynamics of ( (cid:102) M λ , (cid:101) α λ ) where < λ ≤ (a); < λ ≤ √ (b).When λ reaches the value all the intervals [0 , q n ] , n ∈ N , [0 , ω , ] become equalto [0 , ] and the space (cid:102) M λ assumes the shape of a "regular ladder" (in which every ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 25 step has the same length). As we pass λ = the orbit of the critical point loosesmonotonicity: q > q > q > ... > ω , > ... > q > q , lim n →∞ q n = ω , , Fig. 8 (a). Consequently, each arc M N , N ∈ N ∪ {∞} , develops a "curl" at one ofits endpoints: for each < N < ∞ the arc M N has N − bendings, whereas M ∞ is bended infinitely many times, Fig. 9 (a).When λ exceeds the value λ = the first bifurcation occurs – the fixed point ω , gives a birth to a new stable periodic orbit { ω , , ω , } , Fig. 8 (b). Then q > q > ... > ω , > ω , > ω , > ... > q > q , lim n →∞ q n + i = ω ,i ,i = 1 , . This implies that the attracting fixed point in M ∞ grows into an arccorresponding to the interval [ ω , , ω , ] , see Fig. 9 (b). In other words, M ∞ becomes a sin( x ) -continuum .For λ lying approximately in the middle of the interval ( , √ ] = ( λ , λ ] theorbit { ω , , ω , } is superstable (it coincides with the critical point orbit). Lengthsof the "adjoint" intervals get equal: | q k +1 − q k | = ω , − ω , , k > , and the space (cid:102) M λ assumes a regular shape. Afterwards, as we pass λ = the orbit of the criticalpoint converges to the stable orbit in a more complicated manner: q > q > ... > ω , > ... > q > q q > q > ... > ω , > ... > q > q . At the level of the subspace M ∞ ⊂ (cid:102) M λ , see Fig. 10, it causes perturbations aroundthe periodic points; the arcs of the limit bar begin to curl around their endpoints. (cid:113) (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q (cid:113) q q q (cid:113) (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113)(cid:113) q q q q q q (cid:113) ω , (cid:113) ω , (cid:113) ω , Figure 10.
Snake-like continuum M ∞ ⊂ (cid:102) M λ for λ approaching thesecond bifurcation.When λ exceeds the value λ = √ the second bifurcation occurs. The orbit { ω , , ω , } becomes repelling and transfers its stability onto a newly born periodicorbit { ω , , ω , , ω , , ω , } of period . Consequently, periodic points of M ∞ growinto arcs corresponding to intervals [ ω , , ω , ] and [ ω , , ω , ] : The maximal ideal space (cid:102) M λ of algebra B λ for λ ∈ ( λ , λ ] composeof snake-like continuum M ∞ presented on Fig. 11 and a sequenceof arcs { M N } N ∈ N converging to M ∞ in Hausdorff metric. (cid:113) (cid:113) q (cid:113)(cid:113)(cid:113) (cid:113) (cid:113) (cid:113)(cid:113)(cid:113)(cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113)(cid:113)(cid:113)(cid:113) (cid:113)(cid:113) (cid:113)(cid:113) (cid:113) (cid:113) (cid:113) (cid:113)(cid:113) (cid:113)(cid:113)(cid:113)(cid:112)(cid:112) (cid:112)(cid:112) (cid:112)(cid:112) (cid:112)(cid:112) (cid:112) (cid:112) (cid:112) (cid:112)(cid:112) (cid:112)(cid:112) (cid:112)(cid:112)(cid:112)(cid:112)(cid:112) (cid:112)(cid:112) (cid:112)(cid:112) (cid:112)(cid:112) (cid:112)(cid:112) (cid:112) (cid:112) (cid:112) (cid:112)(cid:112) (cid:112)(cid:112)(cid:113) ω , (cid:113) ω , (cid:113) ω , (cid:112) (cid:112)(cid:113) (cid:113)(cid:113) (cid:113)(cid:113) (cid:113) ω (cid:113) (cid:113)(cid:113) (cid:113)(cid:113) (cid:112)(cid:112) (cid:113) ω , (cid:113) ω , (cid:113) ω , (cid:113) (cid:17)(cid:17)(cid:17)(cid:51) x (cid:81)(cid:81)(cid:81)(cid:107) y (cid:54) zO Figure 11.
Snake-like continuum M ∞ ⊂ (cid:102) M λ after the second bi-furcation (immersed into the -dimensional space Oxyz ).This process continues: as λ approaches approximately a middle of the inter-val ( λ , λ ] the orbit { ω , , ω , , ω , , ω , } becomes superstable and the space M ∞ assumes a regular shape. Afterwards continuum M ∞ develops curls around thepoints corresponding to the orbit { ω , , ω , , ω , , ω , } until we pass λ = λ whereeach of these points grows into an arc. And so on, and so forth, cf. Fig. 12.In order to state the result formally we extend the sequence λ = , λ , λ , ...,putting λ − = 0 and λ = . By a ray we mean a topological space homeomorphicto (0 , , and an endpoint of a topological space M is a point p ∈ M whose everyneighbourhood U contains an open neighbourhood V such that the boundary of V is a singleton, cf. [Nad92]. Subspace M ∞ = lim ←−− ( M, α λ ) ⊂ (cid:102) M λ for λ < λ ∞ isgiven by the following recurrence. Theorem 3.7. [BI96, Thm. 3] If λ ∈ ( λ n , λ n +1 ] , n ∈ N , then the continuum lim ←−− ( M, α λ ) is the closure of a ray R such that lim ←−− ( M, α λ ) \ R is the union of twocopies of lim ←−− ( M, α λ (cid:48) ) , where λ (cid:48) ∈ ( λ n − , λ n ] , intersecting in a common endpoint. ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 27 R R , R , R , R , R , R , R , R , R , R , R , R , R , R , I I I I I I I I Figure 12.
Subspace M ∞ ⊂ (cid:102) M λ after the forth bifurcation(schematic presentation).Now we are ready to give a full description of the system ( (cid:102) M λ , (cid:101) α λ ) for λ lying inthe interval (0 , λ ∞ ) = (cid:83) ∞ n = − ( λ n , λ n +1 ] . If λ ∈ (0 , ] , such system is presented onFig. 7, 9. Theorem 3.8.
Let U λ be the operator given by (28) and B λ the C ∗ -algebra givenby (29) . If λ ∈ ( λ n , λ n +1 ] , n > , then i) the maximal ideal space (cid:102) M λ of the algebra B λ compose of a snake-like con-tinuum M ∞ and a sequence of arcs { M N } N ∈ N converging to M ∞ , where M ∞ = R ∪ ( R , ∪ R , ) ∪ ... ∪ ( R n − , ∪ ... ∪ R n − , n − ) ∪ ( I ∪ ... ∪ I n − ) is the sum of n − rays R , R k,i , k = 1 , ..., n − , i = 1 , ..., k , and n − arcs I i , i = 1 , ..., n − , cf. Fig. 12. The closure of R gives M ∞ and R k,i = n − k − (cid:91) j =0 2 j − (cid:91) l =0 R k + j,i + l · k ∪ n − k − − (cid:91) l =0 I i + l · k , k = 1 , ..., n − , i = 1 , ..., k . ii) Partial homeomorphism (cid:101) α λ generated by U λ on (cid:102) M λ carries M N onto M N +1 , N ∈ N , and (cid:101) α λ : M ∞ → M ∞ is a homeomorphism that preserves R ,permutes cyclically arcs I i , and the rays R k,i (for each fixed k = 1 , ..., n − ): (cid:101) α λ ( I ) = I , ..., (cid:101) α λ ( I n ) = I , (cid:101) α λ ( R k, ) = R k, , ..., (cid:101) α λ ( R k, k ) = R k, . Moreover, all the rays R , R k,i and arcs I i are pairwise disjoint except of the fol-lowing intersections R k,i ∩ R k, k − + i = { (cid:101) ω k − ,i } , k = 1 , ...n − , i = 1 , ... k − , that form periodic orbits { (cid:101) ω k , , ... (cid:101) ω k , k } with period k , k = 1 , ..., n − . Middle-points of arcs I i form a periodic orbit { (cid:101) ω n − , , ... (cid:101) ω n − , n − } with period n − , andendpoints of arcs I i form a periodic orbit { (cid:101) ω n , , ... (cid:101) ω n , n } with period n . Proof.
For the description of the space (cid:102) M λ apply Theorem 3.4, 3.7. Thedynamics of (cid:101) α λ on M ∞ may deduced from the proof of [BI96, Thm. 3], see also[BI96, 6]. (cid:4) The infinite sequence of period-doubling bifurcation leaves the following imprinton the structure of (cid:102) M λ for λ attaining the Feigenbaum limit. Theorem 3.9.
For λ = λ ∞ the maximal ideal space (cid:102) M λ of the algebra B λ possessthe property that every nondegenerate subcontinuum of (cid:102) M λ is reducible. Moreover M λ contains only three topologically different nondegenerate subcontinua: arcs,copies of the space M ∞ and sums of two copies of M ∞ intersecting in the commonendpoint. Proof.
Apply Theorem 3.4 and [BI96, Thm. 7]. (cid:4)
The parameter values corresponding to chaotic dynamics.
For λ >λ ∞ the dynamics of the system ([0 , , α λ ) is chaotic and the available knowledgeconcerning mappings { α λ } λ>λ ∞ is far from being complete. However we are ableto present a number of results that shed much light onto the structure of theconsidered systems. For example, we already know that (cid:102) M λ contains irreduciblecontinua (Theorem 3.6), and for λ = 1 , (cid:102) M λ is a B-J-K continuum. We start withthe case when B-J-K continua are the only irreducible subcontinua of (cid:102) M λ .3.3.1. Cascade of B-J-K continuum doubling.
Let us consider the sequence µ = 1 , µ , µ , ..., of parameter values, in which the bifurcation diagram splits into two"copies" of itself, Fig. 5. It is a decreasing sequence converging to λ ∞ , and formally µ n could be defined as the solution of the following equation α n λ ( λ ) = ( the largest fixed point of α n − λ ) , cf. [BI96], [CE80]. The sequence { µ n } n ∈ N admits an inductive procedure surpris-ingly similar to the one presented in Theorem 3.7. Theorem 3.10. [BI96, Thm. 6]
For n > , lim ←−− ( M, α µ n ) is the closure of a ray R such that lim ←−− ( M, α µ n ) \ R is the union of two copies of the space lim ←−− ( M, α µ n − ) intersecting in a common endpoint. R R (b)(a) B B B B B B R R Figure 13.
Subspace M ∞ ⊂ (cid:102) M λ : for λ = µ (a); for λ = µ (b). ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 29 The above statement together with Theorems 3.4, 3.5 says that for λ = µ thespace (cid:102) M λ compose of a sequence of arcs { M N } N ∈ N converging to the snake-likecontinuum M ∞ which is the union of two copies B , B of B-J-K continuum anda ray R , see Fig. 13 (a). Partial homeomorphism (cid:101) α λ transforms M N onto M N +1 , N ∈ N , and the dynamics of (cid:101) α λ on M ∞ is as follows: the endpoint of the ray R is a fixed point, the remaining points of R slide on R toward the continua B , B . Points from B are carried onto B and vice versa. The intersection B ∩ B consists of a fixed point.Analogously, for λ = µ , see Fig. 13 (b), M ∞ ⊂ (cid:102) M λ compose of four copies B , B , B , B of B-J-K continuum and three arcs R , R , R . The endpoint of R is afixed point, and the remaining points of R move toward arcs R , R . Points from R are carried onto R and vice versa. The intersection R ∩ R consists of a fixedpoint. Continua B i are cyclically permuted (cid:101) α λ ( B ) = B , (cid:101) α λ ( B ) = B , (cid:101) α λ ( B ) = B , (cid:101) α λ ( B ) = B , and ( B ∩ B ) ∪ ( B ∩ B ) constitute a periodic orbit of period .In general, for λ = µ n we have the description of ( (cid:102) M λ , (cid:101) α λ ) which differs from theone presented in Theorem 3.8 only in that the arcs I i are replaced with B-J-Kcontinua, cf. Fig. 14. R R , R , R , R , R , R , R , R , R , R , R , R , R , R , B B B B B B B B B B B B B B B B Figure 14.
Subspace M ∞ ⊂ (cid:102) M λ for λ = µ (schematic presentation). Theorem 3.11.
Let U λ be the operator given by (28) and B λ the C ∗ -algebra givenby (29) . If λ = µ n , n ∈ N + , then i) the maximal ideal space (cid:102) M λ of algebra B λ compose of a snake-like continuum M ∞ and a sequence of arcs { M N } N ∈ N converging to M ∞ , where M ∞ = R ∪ ( R , ∪ R , ) ∪ ... ∪ ( R n − , ∪ ... ∪ R n − , n − ) ∪ ( B ∪ ... ∪ B n ) is the sum of n − rays R , R k,i , k = 1 , ..., n − , i = 1 , ..., k , and n B-J-Kcontinua B i , i = 1 , ..., n , cf. Fig. 14. The closure of R gives M ∞ and R k,i = n − k − (cid:91) j =0 2 j − (cid:91) l =0 R k + j,i + l · k ∪ n − k − (cid:91) l =0 B i + l · k , k = 1 , ..., n − , i = 1 , ..., k , ii) Partial homeomorphism (cid:101) α λ generated by U λ on (cid:102) M λ carries M N onto M N +1 , N ∈ N , and on (cid:101) α λ : M ∞ → M ∞ is a homeomorphism that preserves R ,permutes cyclically continua B i , and the rays R k,i (for fixed k = 1 , ..., n − ): (cid:101) α λ ( B ) = B , ..., (cid:101) α λ ( B n ) = B , (cid:101) α λ ( R k, ) = R k, , ..., (cid:101) α λ ( R k, k ) = R k, . Moreover, all the rays R , R k,i and continua B i are pairwise disjoint except of thefollowing intersections R k,i ∩ R k, k − + i = { (cid:101) ω k − ,i } , i = { , ... k − } ,B i ∩ B n − + i = { (cid:101) ω n − ,i } i = { , ... n − } , which form periodic orbits { (cid:101) ω k , , ... (cid:101) ω k , k } with period k , k = 1 , ..., n − . Proof.
It follows from Theorems 3.4, 3.5 and inductively applied Theorem 3.10;see also the proof of [BI96, Thm. 6]. (cid:4) µ η ν η η Figure 15.
Bifurcation diagram for λ ∈ [0 ,
91; 1] .3.4.
Windows of stable periodic orbits of odd period.
Let ( η n , ν n ] , n > ,be the interval of parameter values for λ where α λ has its first stable orbit of period n +1 . The sequences η n , ν n converge decreasingly to µ , see Fig. 15. Significantly,passing with λ from ( η n , ν n ] to ( η n +1 , ν n +1 ] the period of the stable periodic orbitof α λ decreases according to Sharkovskii’s order (page 22). Theorem 3.12.
Let U λ be the operator given by (28) and B λ the C ∗ -algebra givenby (29) . If λ ∈ ( η n , ν n ] , n > , then i) the maximal ideal space (cid:102) M λ of algebra B λ compose of a snake-like continuum M ∞ and a sequence of arcs { M N } N ∈ N converging to M ∞ , and M ∞ = R ∪ C n +1 , where R is a ray, C n +1 is an irreducible continuum with exactly n + 1 endpoints and whose only proper nondegenerate subcontinua are arcs. Fur-thermore, C n +1 ∩ R = ∅ and R = R ∪ C n +1 . ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 31 ii) Partial homeomorphism (cid:101) α λ generated by U λ on (cid:102) M λ carries M N onto M N +1 ,for N ∈ N , and invariates both R and C n +1 . The endpoint of R is a fixedpoint and the remaining points of R move toward C n +1 . The endpoints of C n +1 form a periodic orbit with period n + 1 . Proof.
By Theorem 3.4 it suffices to inductively apply [BI96, Thm. 8]. (cid:4)
For λ ∈ ( η , ν ] the continuum C ⊂ (cid:102) M λ is considered to be the simplest exampleof an irreducible continuum, cf. [Nad92]. It may be obtained as an attractor of acontinuous injective map T : Ω → Ω defined on a compact subset Ω = A ∪ B ∪ C of R which acts according to Fig. 16. Actually, the subsystem ( C , (cid:101) α λ ) of ( (cid:102) M λ , (cid:101) α λ ) is topologically conjugate to the system (Λ , T ) , where Λ = (cid:84) n ∈ N T n (Ω) . A C B T ( B ) (cid:64)(cid:64)(cid:82) T ( A ) (cid:17)(cid:17)(cid:51) T ( C ) (cid:16)(cid:16)(cid:49) Figure 16.
Continuum C as an attractor.3.4.1. Cascades of bifurcation that follow the windows of stability.
After each ofthe intervals ( η n , ν n ] , cf. Fig. 15, there occurs a cascade of period-doubling bi-furcations. For instance, increasing the parameter value λ in the window ( η , ν ] of the stable orbit of period three, we observe that the continuum C ⊂ (cid:102) M λ curlsaround its endpoints { (cid:101) ω , , (cid:101) ω , , (cid:101) ω , } , Fig. 17 (a). When we pass λ = ν end-points of C grow into arcs: continuum C turns into irreducible continuum C with endpoints, Fig. 17 (b). Afterwards continuum C begin to curl aroundits endpoints which finally grow into arcs whose endpoints are endpoints of anirreducible continuum C . And so on, and so forth. In particular, after four suchbifurcations we get a continuum C which arises from continuum C by replacingeach of its endpoint with a copy of the continuum presented on Fig. 12.Similar phenomena occur after every window of stability ( η n , ν n ] , n > . In order (a) (b) (cid:101) ω , (cid:101) ω , (cid:101) ω , (cid:101) ω , (cid:101) ω , (cid:101) ω , (cid:101) ω , (cid:101) ω , (cid:101) ω , R , R , R , Figure 17.
Continua C and C . to get a formal description, for each n > , we denote by λ ( n )1 = ν n , λ ( n )2 , λ ( n )3 , ... thesequence of parameter values λ that correspond to the cascade of period-doublingbifurcation of the stable orbit appearing immediately after λ ( n )0 := η n . Theorem 3.13.
Let U λ be the operator given by (28) and B λ the C ∗ -algebra givenby (29) . If λ ∈ ( λ ( n ) m , λ ( n ) m +1 ] , for n > , m ≥ , then i) the maximal ideal space (cid:102) M λ of algebra B λ compose of the snake-like contin-uum M ∞ and the sequence of arcs { M N } N ∈ N converging to M ∞ where M ∞ = R ∪ C m (2 n +1) is the union of a ray R and an irreducible continuum C m (2 n +1) with exactly m (2 n + 1) endpoints. Furthermore R = R ∪ C m (2 n +1) and C m (2 n +1) = C n +1 ∪ m − (cid:91) k =0 2 k (2 n +1) (cid:91) i =1 R k,i ∪ m − (2 n +1) (cid:91) i =1 I i is the union of (2 m − n + 1) rays R k,i and m − (2 n + 1) arcs I i . Theclosure of C n +1 coincides with C m (2 n +1) and R k,i = m − k − (cid:91) j =0 2 j − (cid:91) l =0 R k + j,i + l · k (2 n +1) ∪ m − k − (cid:91) l =0 I i + l · k (2 n +1) , for i = 1 , ..., k (2 n + 1) , k = 0 , ..., m − . ii) Partial homeomorphism (cid:101) α λ generated by U λ on (cid:102) M λ carries M N onto M N +1 , N ∈ N , and (cid:101) α λ : M ∞ → M ∞ is a homeomorphism that preserves R and C n +1 , permutes cyclically the arcs I i , and (for each fixed k = 0 , ..., m − )the rays R k,i .Moreover, all the rays R , R k,i , all the arcs I i and the continuum C n +1 are pairwisedisjoint except the following intersections C n +1 ∩ R ,i , i = 1 , ..., n + 1 ,R k,i ∩ R k,i +2 k − (2 n +1) = { (cid:101) ω k − (2 n +1) ,i } , i = 1 , ..., k − (2 n + 1) , k = 1 , ..., m − . The sets { (cid:101) ω k (2 n +1) , , ... (cid:101) ω k (2 n +1) , k (2 n +1) } form periodic orbits with periods k (2 n +1) for k = 0 , ..., m − . The middles of the arcs I i form a periodic orbit with period m − (2 n + 1) and their endpoints form a periodic orbit with period m (2 n + 1) . Proof.
In view of Theorem 3.4 it suffices to repeat the argument from theproof of [BI96, Thm. 8], see also remarks preceding [BI96, Thm. 9], as well as anintroduction to [BI96, 6]. (cid:4) Reversible extensions of homeomorphisms of a circle
The characteristic feature of the C ∗ -method developed in this paper is that itleads from irreversible dynamics to reversible dynamics. Therefore, it may seemsurprising that applying it to (already) reversible systems one may also get nontriv-ial results. Clearly, all the interesting phenomena, arising in this case, are related tothe freedom of choice of the set Y , equivalently the ideal J , see paragraph 2.1.1. Aswe show below such considerations arise naturally in investigation of compressionsof unitary operators. We start with the general structure of dynamical systems wewill here deal with. ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 33 Proposition 4.1. If α : M → M is a homeomorphism and ( (cid:102) M , (cid:101) α ) is a reversibleextension of ( M, α ) associated with a set Y ⊂ M , then (cid:102) M may be treated as aclosed subset (cid:102) M = (cid:91) N ∈ N M N ∪ M ∞ ⊂ N × M of the product space, where N = N ∪ {∞} is the one point compactification of thediscrete space N , M ∞ = {∞} × M, M N = { N } × α N ( Y ) , N ∈ N , and the partial partial homeomorphism (cid:101) α : (cid:102) M → (cid:102) M acts according to the formula (cid:101) α ( N, x ) = ( N + 1 , α ( x )) , (cid:101) α ( ∞ , x ) = ( ∞ , α ( x )) . Proof.
We define a homeomorphism Ψ of (cid:102) M = (cid:83) N ∈ N M N ∪ M ∞ onto a closedsubspace of N × M using the factor map Φ : (cid:102) M → M , see (25), by the formulae Ψ( (cid:101) x ) := ( ∞ , Φ( (cid:101) x )) , for (cid:101) x ∈ M ∞ , Ψ( (cid:101) x ) := ( N, Φ( (cid:101) x )) , dla (cid:101) x ∈ M N , N ∈ N . Since α is a homeomorphism, one readily sees that Ψ : (cid:91) N ∈ N M N ∪ M ∞ → (cid:91) N ∈ N { N } × α N ( Y ) ∪ {∞} × M is a homeomorphism and identifying (cid:102) M with Ψ( (cid:102) M ) the assertion follows. (cid:4) Compression of unitaries generating homeomorphisms of a circle.
Let α : S → S be an orientation preserving homeomorphism of the circle and let γ : R → R be its lift to R , i.e. a continuous mapping satisfying α ( e πit ) = e πiγ ( t ) , t ∈ [0 , . We recall that γ is an increasing homeomorphism such that γ ( t + 1) = γ ( t ) + 1 , t ∈ R , determined by α up to a translation by an integer constant. We define aunitary operator U ∈ L ( H ) on the space H = L ( R ) by the formula ( U f )( t ) = (cid:112) | γ (cid:48) ( t ) | f ( γ ( t )) , which (by monotonicity of γ ) make sense for almost all t in R . In particular, ( U ∗ f )( t ) = (cid:112) | ( γ − ) (cid:48) ( t ) | f ( γ − ( t )) . We let A ⊂ L ( H ) be an algebra of operators ofmultiplication by continuous periodic functions with period : A ∼ = C ( S ) . Clearly(31) U A U ∗ ⊂ A , U ∗ A U ⊂ A , and we have Proposition 4.2.
The operators U and U ∗ generate on the maximal ideal spaceof A (identified with S ) the systems ( S , α ) and ( S , α − ) , respectively. Let us now consider compressions of the introduced objects to the space H := L ([0 , ∞ )) naturally treated as a subspace of H = L ( R ) . Namely, we denote by P : H → H the projection from H onto H and we put U := P U P, A := P A P. Then the algebra
A ⊂ L ( H ) is isomorphic to C ( S ) and U ∈ L ( H ) is a partialisometry such that U ∗ = P U ∗ P ∈ L ( H ) . Proposition 4.3.
Within the above notation the following possibilities may occur. i) If γ (0) > , then U is a non-invertible coisometry in L ( H ) , U A U ∗ ⊂ A , U ∗ A U (cid:42) A , operator U generates on the spectrum of A the system ( S , α ) and hull( U ∗ U A ∩ A ) = { e πit : t ∈ [0 , γ (0)] } , ii) If γ (0) = 0 , then U is unitary. U A U ∗ ⊂ A , U ∗ A U ⊂ A ,U and U ∗ generate respectively the systems ( S , α ) and ( S , α − ) . iii) If γ (0) < , then U is non-invertible isometry: U ∗ A U ⊂ A , U A U ∗ (cid:42) A , operator U ∗ generates on the spectrum of A the system ( S , α − ) and hull( U ∗ U A ∩ A ) = { e πit : t ∈ [0 , γ − (0)] } . Proof. If γ (0) > , then ( U f )( t ) = (cid:112) | γ (cid:48) ( t ) | f ( γ ( t )) , ( U ∗ f )( t ) = (cid:40)(cid:112) | ( γ − ) (cid:48) ( t ) | f ( γ − ( t )) , t ∈ [ γ (0) , ∞ )0 , t ∈ [0 , γ (0)) . In particular, the projection U ∗ U is the operator of multiplication by the charac-teristic function of [ γ (0) , ∞ ) and thereby hull( U ∗ U A ∩ A ) = { e πit : t ∈ [0 , γ (0)] } .Item iii) is obtained by reversing the roles of γ and γ − in item i), and item ii) isstraightforward. (cid:4) We see that in a process of compression one may lose one of the relations (31).However, according to our results one may always retrieve what is lost by passingto a bigger algebra. To fix attention let us from now on assume that γ (0) > (the case when γ (0) < is completely analogous). We put B = C ∗ (cid:32) ∞ (cid:91) n =0 U ∗ n A U n (cid:33) , which by Theorem 1.23, is the smallest C ∗ -algebra containing A such that U B U ∗ ⊂ B , U ∗ B U ⊂ B . Theorem 4.4.
The spectrum of B assumes one of the forms: i) If γ (0) ≥ , then (cid:102) M = N × S , see Fig. 18 (a). ii) If γ (0) ∈ (0 , , then (cid:102) M = (cid:83) N ∈ N M N ∪ M ∞ ⊂ N × S , where M ∞ = {∞}× S is a circle, and the sets M N , N ∈ N , are arcs: M N = { N } × [ α N (1) , α N +1 (1)] , where [ α N (1) , α N +1 (1)] stands for an arc on S with the origin α N (1) andending α N +1 (1) , see Fig. 18 (b).Under the identification B = C ( (cid:102) M ) the operator U generates on (cid:102) M the mapping (cid:101) α that acts according to the formula (cid:101) α ( N, x ) = ( N + 1 , α ( x )) , (cid:101) α ( ∞ , x ) = ( ∞ , α ( x )) . ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 35 Proof.
By Proposition 4.3 and Theorem 2.5 the system ( (cid:102) M , (cid:101) α ) is the reversibleextension of ( S , α ) associated to the set Y = { e πit : t ∈ [0 , γ (0)] } which eitheris a circle S , when γ (0) ≥ , or an arc with the origin e πi and ending α (1) = e πiγ (0) . Thus it suffices to apply Proposition 4.1. (cid:4) Figure 18.
Spectrum of B related to a homeomorphism of the circle.It follows that the algebra B depends not only on the homeomorphism α butalso on the choice of the lift γ . In particular, if γ (0) ≥ , then independently of α B ∼ = C ( N × S ) . If however γ (0) ∈ (0 , , the structure of B is uniquely determined by the orbit ofthe point ∈ S . We will discuss this issue in detail below.4.2. Classification of the extended algebras via rotation numbers.
Werecall [BS03], [Dev89], that if γ : R → R is a lift of a homeomorphism α : S → S ,the limit lim n →∞ γ ( t ) n always exits, does not depend on t ∈ R and its fractional part(32) τ ( α ) := { lim n →∞ γ ( t ) n } ∈ [0 , depend only on α . The quantity τ ( α ) is called the rotation number of α . Its roleis explained by the following theorem due to H. Poincare, cf. [BS03], [Dev89]. Thesymbol Θ τ , τ ∈ [0 , , stands for the rotation by πτ : Θ τ ( z ) = z · e πiτ , z ∈ S . Theorem 4.5 (Poincare classification) . Let α : S → S be an orientation pre-serving homeomorphism of the circle. If τ ( α ) = mn , where m and n are coprime, then all periodic points for α areof period n (and there exists at least one such a point). If τ ( α ) / ∈ Q , then α does not possess periodic points and the set Ω( α ) consisting of the accumulation points of an arbitrary orbit { α ( x ) } n ∈ Z doesnot depend on a choice of x ∈ S . There are two possible subcases a) When Ω( α ) = S , that is when α is topologically transitive, then thesystem ( S , α ) is topologically conjugated to ( S , Θ τ ( α ) ) . b) When α is not topologically transitive, then Ω( α ) is a perfect nowheredense subset of S and there exits a continuous surjection φ : Ω( α ) → S which is a semiconjugacy from (Ω( α ) , α ) onto ( S , Θ τ ( α ) ) . We use the above theorem to classify the algebras B described in item ii) ofTheorem 4.4, that is we assume throughout this subsection that the lift γ : R → R satisfies(33) < γ (0) < . Such lift always exists, provided ∈ S is not a fixed point of α : S → S . Since γ is uniquely determined by α , so is the operator U and the algebra B ∼ = C ( (cid:102) M ) ,defined in the previous subsection. Thus, it makes sense to adopt the followingnotation B α := B and (cid:102) M α := (cid:102) M .
The space M α compose of a circle M ∞ and a sequence of arcs { M N } N ∈ N . Theorem 4.6.
In the situation under consideration the following cases may occur: If τ ( α ) = mn where m and n are coprime, then the limit points of theendpoints of arcs { M N } N ∈ N form a subset of M ∞ with cardinality n , and B α ∼ = B Θ mn . If τ ( α ) / ∈ Q , then the two subcases are possible: a) α is topologically transitive, and then M ∞ is the set of limit points ofthe endpoints of arcs { M N } N ∈ N and B α ∼ = B Θ τ ( α ) . b) α is not topologically transitive, and then the set of limit points of theendpoints of arcs { M N } N ∈ N form a perfect nowhere dense subset of M ∞ . In particular, B α (cid:54) = B Θ τ , τ ∈ [0 , . Proof.
The set of limit points of the endpoints of arcs { M N } N ∈ N coincideswith the set {∞} × Ω( α ) ⊂ M ∞ where Ω( α ) is the set of limit points of the orbit { α N (1) } N ∈ N . By Theorem 4.5 we only need to consider the cases listed in theassertion.1) The set Ω( α ) consists of n points that form a periodic orbit of α . Moreprecisely, there are n points x , x , ..., x n − ∈ S enumerated according to theorientation and such that lim N →∞ α Nn + k (1) = x km , k = 0 , ..., m − ., cf. [BS03]. Thus it follows that lim N →∞ M Nn + k = {∞} × [ x km ( mod n ) , x ( k +1) m ( mod n ) ] , k = 0 , ..., n − , that is the sequence of arcs { M Nm + k } N ∈ N converge in Hausdorff metric to thearc on M ∞ with the origin ( ∞ , x kn ( mod m ) ) and ending ( ∞ , x ( k +1) n ( mod m ) ) . Let φ : (cid:102) M α → (cid:102) M Θ mn be the mapping that acts "linearly" according to the scheme: {∞} × [ x k , x k +1( modn ) ] φ (cid:55)−→ {∞} × (cid:104) e πi kn , e πi k +1 n (cid:105) , k = 0 , ..., n − , { N } × [ α N (1) , α N +1 (1)] φ (cid:55)−→ { N } × (cid:104) e πi Nn , e πi N +1 n (cid:105) , N ∈ N . It is evident that φ is a homeomorphism and hence the algebras B α = C ( (cid:102) M α ) and B Θ mn = C ( (cid:102) M Θ mn ) are isomorphic. ∗ -ALGEBRAS ASSOCIATED WITH EXTENSIONS OF LOGISTIC MAPS 37 {∞} × Ω( α ) = {∞} × S = M ∞ and there exists a homeomorphism φ : S → S such that S α −−−→ S φ (cid:121) (cid:121) φ S τ ( α ) −−−→ S is commutative. Furthermore, φ may be arranged so that φ (1) = 1 , see the proofof [BS03, Thm. 7.1.9]. It follows that the mapping id × φ : (cid:102) M α −→ (cid:102) M Θ τ ( α ) : ( id × φ )( N, x ) = (
N, φ ( x )) , ( N, x ) ∈ M N , N ∈ N , is a homeomorphism. Hence B α ∼ = B Θ τ ( α ) .2b) Since {∞} × Ω( α ) is a perfect nowhere dense subset of M ∞ the space (cid:102) M α isnot homeomorphic to any of the spaces (cid:102) M Θ τ , τ ∈ [0 , . Equivalently B α (cid:54) = B Θ τ ,for all τ ∈ [0 , . (cid:4) If φ is a topological conjugacy between ( S , α ) and ( S , β ) , then either τ ( α ) = τ ( β ) (when φ is orientation preserving) or τ ( α ) + τ ( β ) = 1 (when φ changes theorientation), so the rotation number is "almost an invariant" for homeomorphismsof the circle. For the algebras B α the rotation number is an invariant sensu stricto . Theorem 4.7.
If algebras B α and B β are isomorphic, then τ ( α ) = τ ( β ) . Proof.
Suppose that B α and B β are isomorphic. There exists a homeomorphism φ : (cid:102) M α → (cid:102) M β , where (cid:102) M α = (cid:83) N ∈ N M N and (cid:102) M β = (cid:83) N ∈ N M (cid:48) N are maximal idealspaces of B α and B β respectively. Clearly, φ necessarily carries the arcs { M N } N ∈ N onto arcs { M (cid:48) N } N ∈ N , the circle M ∞ onto circle M (cid:48)∞ , and the set Ω( α ) ⊂ M ∞ oflimit points of endpoints of { M N } N ∈ N onto the set Ω( β ) ⊂ M (cid:48)∞ of limit pointsof endpoints of the arcs { M (cid:48) N } N ∈ N . For the simplicity of notation we adopt theidentification M ∞ = M (cid:48)∞ = S . We claim that φ establishes conjugacy betweenthe systems (Ω( α ) , α ) and (Ω( β ) , β ) or (Ω( α ) , α ) and (Ω( β ) , β − ) depending onwhether φ : M ∞ → M (cid:48)∞ preserves or changes the orientation. Once we prove this,the standard argument give us that either τ ( α ) = τ ( β ) or τ ( α )+ τ ( β − ) = 1 , wherein the letter case we get τ ( α ) = τ ( β ) since τ ( β − ) = 1 − τ ( β ) .To prove our claim we fix a sequence { α N k (1) } k ∈ N converging to an arbitrarilychosen point x ∈ Ω( α ) . Then the sequence { α N k +1 (1) } k ∈ N converges to α ( x ) ,and the sequence of arcs { M N k } N ∈ N converges (in Hausdorff metric) to the arc [ x , α ( x )] . In the case φ : M ∞ → M (cid:48)∞ preserves the orientation, almost allarcs from the sequence { M N k } N ∈ N are mapped in accordance with (the natural)orientation onto almost all arcs of the sequence { φ ( M N k ) } N ∈ N . Hence [ φ ( x ) , φ ( α ( x ))] = φ ([ x , α ( x )]) = φ ( lim k →∞ M N k ) = [ φ ( x ) , β ( φ ( x ))] . Thus φ ( α ( x )) = β ( φ ( x )) and consequently φ conjugates the systems (Ω( α ) , α ) , (Ω( β ) , β ) . In the case φ : M ∞ → M (cid:48)∞ changes the orientation, arguing similarly asabove one gets [ φ ( α ( x )) , φ ( x )] = φ ([ x , α ( x )]) = lim k →∞ φ ( M N k ) = [ β − ( φ ( x )) , φ ( x )] . Hence φ ( α ( x )) = β − ( φ ( x )) and consequently φ conjugates the systems (Ω( α ) , α ) and (Ω( β ) , β − ) . This proves our claim. (cid:4) Applying the classical result of A. Denjoy [Den32], [BS03], [Dev89] which statesthat every diffeomorphism of the circle with finite variation and irrational rotationnumber is topologically transitive we get that in the class of algebras B α associ-ated with such diffeomorphisms the rotation number is not only an invariant butactually a numerical equivalent. Theorem 4.8.
If one of the homeomorphisms α , β is a diffeomorphism with finitevariation, then the algebras B α and B β are isomorphic if and only if τ ( α ) = τ ( β ) . Proof.
Apply Theorems 4.6, 4.7 and Denjoy Theorem. (cid:4)
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Institute of Mathematics, University of Bialystok, ul. Akademicka 2, PL-15-267 Bialystok, Poland
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