# extensions, quotients and generalized pseudo-anosov maps[精选推荐pdf]

arXiv:math/0204119v1 [math.DS] 10 Apr 2002EXTENSIONS, QUOTIENTS AND GENERALIZED PSEUDO-ANOSOV MAPSANDR´E DE CARVALHODedicated to Dennis Sullivan on his 60thbirthdayAbstract. We describe a circle of ideas relating the dynamics of 2-dimensional homeomorphisms to that of 1-dimensional endomorphisms. This is used to introduce a new class of maps generalizing that of Thurston’s pseudo-Anosov homeomorphisms.1. IntroductionIn this paper we discuss a circle of ideas which is present in many different contexts in dynamicalsystems. It was first introduced by Williams in his study of expanding attractors, and has been used since by many authors. In its most basic form it can be stated as follows:Collapsing segments of stable manifolds of a homeomorphism yields a lower dimensional endo- morphism; the original homeomorphism may be recovered by taking the natural extension (inverse limit) of the quotient endomorphism.The context we focus attention on is the interplay this creates between the dynamics of 1- dimensional endomorphisms — endomorphisms of trees and graphs — and that of 2-dimensional homeomorphisms — homeomorphisms of surfaces. In recent years, this interplay between graphendomorphisms and surface homeomorphisms has been used for different, although related, pur-poses. For example, it was used to give algorithmic proofs of Thurston’s classification theorem for surface homeomorphisms up to isotopy. It also appeared in the contruction of models for families of surface homeomorphisms passing from trivial to chaotic dynamics as parameters are varied. And a combination of these results led to a conjecture about the way in which forcing organizes the braid types of horseshoe periodic orbits.Complexification should yield a closely related discussion — which will not be treated here — linking the dynamics of endomorphisms of branched (Riemann) surfaces to the dynamics of automorphisms of C2. The natural extension of an endomorphism of a branched surface is usuallya homeomorphism of a surface lamination. Such laminated spaces have already been found to be among the main objects in the study of complex H´ enon maps. In both the real and complex cases, much is known about 1-dimensional endomorphisms, at least when the space is unbranched, whereas about homeomorphisms in dimension 2 much less is known. This double interplay — between 1- and 2-dimensional dynamics and between the real and complex settings — seems to be an interesting approach to explore. In this paper we develop some of its aspects, mostly in the real setting.We also introduce a different kind of quotient, taking a surface homeomorphism to another sur- face homeomorphism by collapsing dynamically irrelevant (wandering) domains. This produces an interesting class of surface homeomorphisms which includes torus Anosovs and Thurston’s pseudo- Anosov maps. These maps — called generalized pseudo-Anosov maps — preserve a pair of invariantmeasured foliations with (possibly infinitely many) singularities, giving the underlying surface anaturally defined complex structure. Regarded from the point of view of this complex structure, the invariant foliations become the horizontal and vertical trajectories of an integrable quadraticDate: August 2001. 12ANDR´E DE CARVALHOdifferential (with possibly infinitely many poles) and the generalized pseudo-Anosov map becomes a Teichm¨ uller mapping.The construction of generalized pseudo-Anosov maps is done by first introducing generalizedtrain tracks: smooth branched 1-submanifolds, with possibly infinitely many branches, of theambient surface. We describe how to find invariant generalized train tracks for certain surface homeomorphisms which, together with measure theoretic information obtained from transition matrices, are the main ingredients in the construction of generalized pseudo-Anosov maps. This is a variant of the same circle of ideas mentioned above.The paper is organized as follows: in Section 2 the natural extension is defined, the class of thick graph maps is introduced and, in Proposition 2.1, the circle of ideas mentioned above is closed. In Section 3 the 0-entropy equivalence relation, which collapses dynamically irrelevant domains, is introduced and a theorem (Theorem 3.1) is stated about the quotient maps so obtained — this isthe first way in which generalized pseudo-Anosov maps appear. In Section 4 infinite train tracksare defined as the main tool to construct generalized pseudo-Anosov maps in the following section. The complex structure is discussed in the last subsection of Section 5.The writing style is Sullivanian as is fitting for a paper prepared for such an occasion. Many arguments are only sketched and some are omitted entirely.Acknowledgements: I have worked on or around the main subjects in this paper for many years.Sevaral people in the course of several conversations have influenced me. Dennis Sullivan, of course,was a major influence as my adviser. Others who have directly contributed to the ideas in this paper are Michael Handel, with whom I had many conversations about many things, includinginfinite train tracks and generalized pseudo-Anosovs; Fred Gardiner, who taught me Teichm¨ uller theory and how to show a puncture is a puncture; and my friend and coworker Toby Hall, whose work was the original stimulus for me to study the subject and who wrote a good part of Section 4 when we were engaged in a project which has now been put on hold.2. The natural extensionIn this paper a dynamical system will mean a continuous self-map of a topological space, atleast. As we go along, we may require more of our maps, for example, that they be differentiableor diffeomorphisms.2.1. Definition and examples. Let f : X → X be a continuous surjective map of a topological space X. If f is not invertible, there is a naturally associated invertible map: setˆX = {(x0,x1,x2,.) ∈∞Y0X;f(xi+1) = xi,for i = 0,1,2,. }and defineˆf :ˆX →ˆX settingˆf(x0,x1,x2,.) = (f(x0),x0,x1,.). This map is called the natural extension of f. The spaceˆX is also known as the inverse or projective limit space and the mapˆf as the inverse or projective limit map associated to f : X → X. Here are some prototypical examples of natural extensions which come up in a variety of contexts in the study of low-dimensional dynamical systems.Examples 1. a) Let X = S1be the unit circle in the complex plane and f : S1→ S1be the squaring map f(z) = z2in complex notation. ThenbS1is the dyadic solenoid. It is a fiber bundle over S1with fiber a dyadic Cantor set. b) Let C∗= C \ 0 and f : C∗→ C∗be again f(z) = z2. ThencC∗is the complex dyadic solenoidand is a fiber bundle over C∗with same fiber as above. c) A variant of the previous example may be obtained by restricting f to X = C \ D, the exterior of the closed unit disk. Since the action of f on X has a fundamental domain (namely, any annulus of the form A = {z ∈ C;1 ǫ forsome 0 ≤ j 1 and, for every i,j, there exists a power k such that the ij-entry of Mkis arbitrarily large (in fact, the entries of M grow like (const)×λk). If M is irreducible then λ is a simple root of the characteristic polynomial of M and if M is also aperiodic, then λ is the only eigenvalue on the circle {z ∈ C;|z| = λ}.Definitions 4. A thick graph map F : G → G with quotient f : G → G will be called Markov if itsatisfies the following additional conditions:i) G has a finite number of strips and junctions (i.e., the graph G has finitely many edges). Foreach strip s we fix a homeomorphism hs: s → [0,1] ×[0,1] from the closure of the strip to the closed unit square, so that the decomposition elements of s are of the form hs−1({x} × [0,1]), for 0 1 such thatΦ(Fs,µs)=(Fs,λµs) Φ(Fu,µu)=(Fu,1 λµu).Examples of generalized pseudo-Anosov maps include the torus Anosov maps and Thurston’s pseudo-Anosov maps. We argue below that Markov thick graph maps also give rise to generalizedpseudo-Anosovs. The definition above, however, probably includes many other maps and theseEXTENSIONS, QUOTIENTS AND GENERALIZED PSEUDO-ANOSOV MAPS9Figure 6. Pronged singularities of the invariant foliations.Markov examples should form a dense set in the space of all generalized pseudo-Anosovs.A more thorough study of these issues, including a description of a uniform structure on the set of generalized pseudo-Anosovs, is currently under way and will hopefully appear in forthcoming papers.Theorem 3.1. Let F : (S,G) → (S,G) be a Markov thick graph map with G of the homotopy type of S minus a point. Assume the associated transition matrix is irreducible and aperiodic. Then the quotient of S by the 0-entropy equivalence relation is homeomorphic to S and F projects to a generalized pseudo-Anosov homeomorphism Φ: S → S.A proof of this theorem shall appear in a forthcoming paper. A more detailed account of the 0-entropy equivalence relation and its quotients is given in [dCP01]. Below we will present an alternative construction of the generalized pseudo-Anosov quotient maps.4. Generalized train tracksIn this section we will only deal with finite thick graphs, that is, thick graphs with finitely many strips and junctions. To simplify the exposition and the statements, we also assume that if (S,G) is a thick graph then G has the homotopy type of the punctured surface S \ {p}, where p ∈ S \G.We refer to p as the point at infinity and denote it by ∞. In general, G has the homotopy type of the several times punctured surface S \{p1,. ,pk}. If this is the case, the discussion below has tobe appropriately modified, but the ideas are essentially the same. The statements made here hold for the once punctured case.4.1. Definitions. We start by fixing some notation and presenting some definitions.Suppose that f : X → X is a homeomorphism.Then f is said to be supported on a subset U of X if f is the identity on X \ U. A second homeomorphism g: X → X is isotopic to f if there is a continuous map ψ: X × [0,1] → X such that each slice map ψt: X → X defined by x 7→ ψ(x,t) is a homeomorphism, and ψ0= f and ψ1= g. The map ψ is called an isotopy from f to g. A pseudo-isotopy is a continuous map ψ: X × [0,1] → X such that, for 0 ≤ t n} anddefine a (possibly infinite) transition matrix M = [mij] setting, as above,mij= number of times φ(ej) crosses eiSince φ(I) ⊂ I, M has block formM =?Nn×n0 BΠ?The matrix N records transitions between real edges and Π records transitions between infinitesimaledges, whereas B records transitions from real to infinitesimal edges. The (possibly infinite) square matrix Π has only 0’s and 1’s in its entries. Each of its columns has exactly one non-zero entryand each row has at most finitely many non-zero entries. The matrix B (which has n columns andpossibly infinitely many rows) has at most finitely many non-zero entries in each column. These observations follow from those in Remarks 9.Example 7. For the horseshoe map, the transition matrix is infinite but is quite simple: m11= 2, mii−1= 1 and mij= 0 otherwise.Standing Assumption: It is assumed throughout this section that the matrix N is irreducible and aperiodic. This implies that its Perron-Frobenius eigenvalue λ > 1. It also follows that there is a positive integer k such that, for every real edge e, φ(e) contains all other real edges. In particular,τ is connected. This assumption is not necessary for all the results that follow, but it simplifies the discussion.We think of M as an operator acting on the space l1of summable sequences of real numbers with norm |y|1=P i≥1|yi|. From the remarks above, it follows that M is a bounded operator with kMk1≤ maxj{Pi|mij|} < ∞. Let Y be an eigenvector of N associated to λ (and therefore unique, up to scale). It is possible to complete Y to an eigenvector y = [Y Y′] of M. Since the columns of Π have at most one non-zeroentry which is 1, kΠk1≤ 1 and thus λI − Π is invertible. Setting Y′= (λI − Π)−1BY we haveM?Y Y′? =?N0 BΠ??Y Y′?=?NY BY + ΠY′?=?λY λY′?In order to see that Y′is a positive vector, notice thatY′=1 λ(I +1 λΠ +1 λ2Π2+ .)BY=1 λ(B +1 λΠB +1 λ2Π2B + .)Y14ANDR´E DE CARVALHOThe matrices ΠkB that appear above represent transitions from a real edge to an infinitesimal edge under the (k+1)-st iterate of the map φ. In fact, they represent exactly those transitions whichoccurred from a real to an infinitesimal edge in the first iterate and which then remained amonginfinitesimal edges for the next k iterates (the other ways to get from a real to an infinitesimal edge under the (k+1)-st iterate of φ are represented by matrices of the form Πk−j−1BNj). But everyinfinitesimal edge is the image, under some iterate of φ, of an infinitesimal edge of τ1and these are the intersection of φ(τ0) with V, that is, they are transitions from real to infinitesimal edges underthe first iterate of φ. This means that, for every i ≥ 1, there exists k ≥ 1 such that the i-th row of ΠkB is non-zero. Since Y is a positive vector, it follows that every entry of Y′is non-zero.Definition 10. A collection {yi= y(ei)}i≥1of non-negative real numbers, called weights, is said to satisfy the switch conditions if, for each switch q of τ, we havey(ei0) =X w(ei) + 2X w(ej)where ei0is the real edge with endpoint at q and the first and second sums range over the set ofinfinitesimal edges having one or both endpoints at q respectively.Lemma 5.1. Let M be the transition matrix associated to φ: τ → τ, λ its Perron-Frobenius eigenvalue and y = [Y Y′] = [y1y2.] an eigenvector associated to λ as constructed above. Thenthe set of weights {yi}i≥1satisfy the switch conditions.Proof (cf. [BH95]). Fix a large positive integer k. The equality Mky = λky written in coordinates states that for each i ≥ 1yi=1 λkXjyj· (number of times φk(ej) intersects ei)If φk(ej) crosses a switch, it must cross edges on both sides except at its endpoints. Thus, the contribution to both sides of the switch is the same up to a bounded amount. Letting k → ∞ yields the result.Now, let X be an eigenvector of NTassociated to the Perron-Frobenius eigenvalue λ. If we try to complete X to an eigenvector [X X′] for the adjoint M∗, we are forced to set X′= 0. The reason is that X′is a solution to the equation Π∗X′= λX′⇔ (λI −Π∗)X′= 0. Since kΠ∗k∞≤ 1, the only solution is X′= 0. This is why infinitesimal edges are called such. We now give a description of the construction of the generalized pseudo-Anosov homeomorphisms corresponding to φ: τ → τ. As was mentioned before, it follows closely that presented in [BH95]. Let x = (x1,x2,. ,xn,0,0,.) and y = (y1,y2,.) be the eigenvectors of M∗and M, re- spectively, associated to the Perron-Frobenius eigenvalue λ as just described. To each real edge ei, 1 ≤ i ≤ n of τ we associate a Euclidean rectangle Riof dimensions xi×yiendowed with foliations by horizontal and vertical line segments. Each foliation has a transverse measure induced by Lebesgue measure. The horizontal and vertical foliations will be called unstable and stable respectively and, under the map, unstable leaves will be stretched and