Hierarchy of graph matchbox manifolds

Transcription

1 Hierarchy of graph matchbox manifolds Olga Lukina University of Leicester, UK July 2th, 2012 Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

2 Introduction I will present to you a recent project Hierarchy of graph matchbox manifolds, arxiv: v2. This work is inspired by ideas of Cantwell and Conlon, Nishimori, Hector and Tsuchiya, who developed a theory of levels for codimension 1 C 2 foliations in the 1970-ies. Ultimately, these works can be considered as generalisations of the Poincaré-Bendixson theorem, which describes limit sets of the orbits of a C 1 flow in the plane, to foliations of higher leaf dimensions. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

3 Some terminology Foliations: foliated manifolds Foliated manifold A foliation F of a smooth manifold M of dimension n is a decomposition of M into integral manifolds of a smooth distribution D such that if X 1,..., X m, m n is a local basis, then [ X i, X j ] Span( X 1,..., X n ), i, j = 1,..., m. Thus locally foliated manifolds have a product structure: for each x M there is x U M open such that there is a homeomorphism ϕ x : U x R m R n m and for any other pair (V x, φ x ) the composition φ x ϕ 1 x : ϕ x (U x V x ) φ x (U x V x ) is smooth and constant on the second component. We can then talk about local leaf and transverse directions in M. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

4 Foliations: minimal sets Some terminology A minimal set of a foliation F of M is an analogue of an ω- or α-limit set in a dynamical system. Minimal set of a foliation A minimal set in a foliated manifold M is a closed saturated subset S of M minimal with respect to these properties. Example: consider T 2 with Denjoy flow. There is x 2 a unique closed subset S of T 2 which is a union of flow lines, and which is the ω- and α-limit set of any point in T 2. The set S is the minimal set of the corresponding 1-dimensional foliation of T 2. x 2 x 1 x 1 In the relative topology from T 2, every x S has a compact neighborhood homeomorphic to [ 1, 1] Cantor set. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

5 Some terminology Foliations: matchbox manifolds Matchbox manifold A matchbox manifold is a compact connected separable topological space M such that each x M has a neighborhood x U M such that there is a homeomorphism ϕ x : U x R m Z x, where Z x is a totally disconnected topological space. Example: exceptional minimal sets of foliations, tiling spaces of tilings with finite local complexity, Vietoris solenoids, generalised solenoids of Williams 1974, suspensions of group actions on Cantor sets... Remark: the question whether a matchbox manifold can arise as a subset of a smooth foliated manifold is an open question. Remark: if Z x is a separable topological space, not necessarily totally disconnected, M is called a foliated space. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

6 Classification by accumulation properties Generalised Poincaré-Bendixson theorem Poincaré-Bendixson Theorem Consider a dynamical system given by a C 1 -flow on R 2. Then every non-empty compact ω-limit set ω(x), x R 2, that does not contain an equilibrium point, is a non-degenerate periodic orbit. Basically, the Poincaré-Bendixson theorem describes asymptotic behavior of orbits, i.e. how non-compact orbits of a C 1 -flow accumulate on compact orbits. A similar question can be asked for foliated spaces. Generalised Poincaré-Bendixson theorem Given a compact foliated space M with foliation of dimension m, classify all leaves by their accumulation properties. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

7 Classification by accumulation properties Poincaré-Bendixson for codimension 1 foliations The most successful such study has been for codimension 1, transversely C 2 -foliations of a compact manifold M. This question was studied by Cantwell and Conlon, Hector, Nishimori and Tsuchiya in the late 70-ies and early 80-ies. Cantwell and Conlon introduced the concept of a level of a leaf in a codimension 1 C 2 -foliation, which quantifies the dynamical complexity of a leaf (compact leaves and minimal sets are at level 0, leaves which have only leaves at level 0 or 1 in their closure are at level 1, and so on). Example: Let h : S 1 S 1 be a homeomorphism with 2 fixed points. The corresponding foliation of T 2 has 2 types of leaves: Compact leaves (at level 0) Non-compact leaves (at level 1) whose closure contains the compact ones. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

8 Classification by accumulation properties Theory of levels Main features of theory of levels for codimension 1 C 2 foliations: One can construct examples with leaves at any finite level. There exist examples with leaves at infinite level, and there is only one infinite level. For finite levels, there is a relation between the level and various properties of leaves (e.g. growth). The theory of levels relies heavily on the fact that the foliations considered are transversely C 2, and on the fact that the transversal is 1-dimensional; that is, given a point in the transversal, there are left and right sides. Trying to develop a similar theory without the assumption of C 2 -differentiability, or for foliations of higher codimension (Salhi) does not produce such a detailed, or universal theory. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

9 Hierarchy of graph matchbox manifolds Hierarchy of graph matchbox manifolds Hierarchy of graph matchbox manifolds The goal of the project I am going to present was to find out, how much the concepts of the theory of levels can be pushed through for the case of foliated spaces with totally disconnected transversals by topological methods. I will also restrict to a special subclass of matchbox manifolds, which I call graph matchbox manifolds. The construction of graph matchbox manifolds appeared first in the paper Laminations par surfaces de Riemann by Étienne Ghys (1999), where it is attributed to Richard Kenyon. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

10 Simple example Simple example Consider Z 2 and its Cayley graph G e e T 1 e T 2 Consider a collection of subtrees in G, containing the identity e with standard length metric d n. e T3 Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

11 Simple example Simple example Dynamical system on T = {T n } n=1? To obtain a dynamical system, we put a metric on T, and define a pseudogroup G and its action on T. Some intuition: in the picture, a ball B T2 (e, 1) is isometric to B T3 (e, 1) A ball B T1 (e, 1) is not isometric to any ball around e in T 2 or T 3 We want to say that (T 2, e) is closer to (T 3, e) than to (T 1, e) e e T 1 T 2 e T3 Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

12 Simple example Simple example Definition (Ball metric d X ) Let X be a set of pointed subtrees of G. For (T, e), (T, e) X d X (T, T ) = e r(t,t ), r(t, T ) = max{n N {0} isometry B T (e, N) B T (e, N)}. In the picture, we have d X (T 1, T 2 ) = d X (T 1, T 3 ) = 1, d X (T 2, T 3 ) = e 1. e T 1 Remark: d X defines the same topology on X as the Gromov-Hausdorff metric. e e T 2 T3 Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

13 Simple example Simple example Define γ a : T T by γ a ((T n, e)) = (T n+1, e) Then G T = γ a is a pseudogroup acting on T. For any T n T the orbit G T (T n ) = T. (T, G T ) is a pseudogroup dynamical system. The set T is not closed, as the limit point of {T n } n=1 is not in T. e e e T 1 T 2 T3 Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

14 Simple example Simple example Add limit points Want to compactify this action. Do that by adding to T a 2-ended tree (S, e), T = T (S, e) Then d X (S, T n ) = e (n 1), and (S, e) is indeed the limit point. Extend the domain of γ a by definining γ a ((S, e)) = (S, e), We obtain again a pseudogroup dynamical system (T, G T ) with two orbits. e e T 3 S Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

15 Simple example Simple example Next, applying a standard technique to this example, we obtain a compact space M with the following properties: M is a union of Riemann surfaces, called leaves, so that for each orbit of G in T there is exactly one surface. There is an embedding φ : T M, and each leaf hits φ(t ) exactly at the points of the corresponding orbit. M is given a topology compatible with the topology on T in the following sense: a leaf L 1 accumulates on a leaf L 2 if and only if the orbit corresponding to L 1 in T has limit points in the orbit corresponding to L 2. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

16 Simple example Simple example In our example, M has two leaves. One leaf is compact, and is homeomorphic to a 2-torus. Another leaf is non-compact, and is homeomorphic to a 1-ended cylinder. The cylinder leaf wraps around the toral leaf infinite number of times. Remark: construction of Kenyon and Ghys This construction was introduced by Ghys in 1999 for Z 2, and generalized by Blanc, Alcalde Cuesta et al. for the case of an arbitrary group H. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

17 Simple example Example of Kenyon and Ghys Ghys 1999: constructed a minimal foliated space M KG which contains parabolic and hyperbolic leaves simultaneously. Properties of the matchbox manifold M KG : Each leaf is dense One 4-ended leaf (hyperbolic) Every other leaf has 1 or 2 ends and is parabolic Expansive transversal dynamics Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

18 Simple example Definition of a graph matchbox manifold Let H be a finitely generated group, and (X, d X ) be the set of all pointed subtrees of its Cayley graph with the ball metric. The space (X, d X ) is closed. Let M H be the corresponding foliated space. Definition A graph matchbox manifold M is the closure L of a leaf L in M H. Examples: 1 The foliated space M obtained earlier, is a graph matchbox manifold. 2 The compact toral leaf in M is a graph matchbox manifold in its own right. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

19 Structure of a space of graph matchbox manifolds Motivation of the current work Motivation Graph matchbox manifolds (GMM) have relatively tractable transverse dynamics and geometry. It seems reasonable to use GMMs to develop general techniques to study matchbox manifolds. Current program We want to understand graph matchbox manifolds arising from the construction above, and their dynamical properties. This work We study the hierarchy of GMM s in M H, given by inclusions, in the case H = F n, a free group on n generators. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

20 Structure of a space of graph matchbox manifolds Motivation of the current work Motivation Graph matchbox manifolds (GMM) have relatively tractable transverse dynamics and geometry. It seems reasonable to use GMMs to develop general techniques to study matchbox manifolds. Current program We want to understand graph matchbox manifolds arising from the construction above, and their dynamical properties. This work We study the hierarchy of GMM s in M H, given by inclusions, in the case H = F n, a free group on n generators. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

21 Structure of a space of graph matchbox manifolds Motivation of the current work Motivation Graph matchbox manifolds (GMM) have relatively tractable transverse dynamics and geometry. It seems reasonable to use GMMs to develop general techniques to study matchbox manifolds. Current program We want to understand graph matchbox manifolds arising from the construction above, and their dynamical properties. This work We study the hierarchy of GMM s in M H, given by inclusions, in the case H = F n, a free group on n generators. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

22 Structure of a space of graph matchbox manifolds Partial order on the set of graph matchbox manifolds Let S be the set of all graph matchbox manifolds in M n. Definition Given M 1, M 2 S we say that M 1 M 2 if and only if M 1 M 2. Example: Let L 1 be a compact leaf, and L 2 be a leaf accumulating on L 1. Then are GMM s, and M 1 M 2. M 1 = L 1 = L 1 and M 2 = L 2 Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

23 Structure of a space of graph matchbox manifolds Partial order on the set of graph matchbox manifolds Proposition 1 (OL 2011) The partially ordered set (S, ) of graph matchbox manifolds in the foliated space M n has the following properties. 1 the set C = {L M n L is compact} is a dense meager subset of M n. 2 (S, ) is a directed partially ordered set, i.e. given M 1, M 2 S there exists M 3 S such that M 1 M 2 M 3. 3 (S, ) contains a unique maximal element M max = M n with a recurrent leaf, and, therefore, M n contains a residual subset of recurrent leaves. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

24 Structure of a space of graph matchbox manifolds Fusion of graph matchbox manifolds The proof of the statements 2 and 3 of the proposition is based on the fusion technique which, given graph matchbox manifolds M 1 and M 2, associates to them a third graph matchbox manifold M 3 such that M 1 M 2 M 3. Let (T i, e) be a graph corresponding to a dense leaf in M i, i = 1, 2. Fusion: the idea Take a graph T with 4 ends given by the union of two lines in G n containing vertices {g n 1, g n 2 n Z}. Attach to T copies of compact balls B Ti (e, r), r N at irregular intervals. The matchbox manifold M 3 corresponding to the resulting graph T 3, contains M 1 and M 2. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

25 Structure of a space of graph matchbox manifolds Fusion of graph matchbox manifolds The proof of the statements 2 and 3 of the proposition is based on the fusion technique which, given graph matchbox manifolds M 1 and M 2, associates to them a third graph matchbox manifold M 3 such that M 1 M 2 M 3. Let (T i, e) be a graph corresponding to a dense leaf in M i, i = 1, 2. Fusion: the idea Take a graph T with 4 ends given by the union of two lines in G n containing vertices {g n 1, g n 2 n Z}. Attach to T copies of compact balls B Ti (e, r), r N at irregular intervals. The matchbox manifold M 3 corresponding to the resulting graph T 3, contains M 1 and M 2. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

26 Structure of a space of graph matchbox manifolds Definition of a level Following Cantwell and Conlon, Hector, Nishimori and Tsuchiya for codimension 1 C 2 foliations, we introduce a quantifier of complexity of graph matchbox manifolds, called level. Definition Let (S, ) be a partially ordered set of graph matchbox manifolds. 1 M = L and all its leaves are at level 0 if M is a compact leaf or a minimal set. 2 M is at level k if the closure of the union of leaves which are not dense in M, is a proper closed subset of M; every such leaf is at level at most k 1; and there is at least one leaf at level k 1. If L is dense in M then L is at level k. 3 M is at infinite level if it is not at finite level. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

27 Structure of a space of graph matchbox manifolds Theory of levels for graph matchbox manifolds Proposition 1 says that the set of compact leaves is dense in M n, n > 1. Proposition 1 says that M n contains a leaf L which is dense in M n. Therefore, by definition such a leaf L is at infinite level. How many infinite levels? Does there exists a hierarchy of infinite levels in M n? Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

28 Structure of a space of graph matchbox manifolds Theory of levels for graph matchbox manifolds Theorem 2 (OL 2011) Let M n, n > 1, be a space of graph matchbox manifolds. Then 1 There exists an infinite increasing chain M 0 M 1 M 2 of graph matchbox manifolds such that M = i Mi is at infinite level and M is a proper subset of M n. Such an M is said to be at infinite level of Type 1. 2 Let M be a graph matchbox manifold at infinite level and suppose M is a proper subset of M n. Then there exists a graph matchbox manifold M such that M M M n are proper inclusions. The space M is at infinite level of Type 2. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

29 Structure of a space of graph matchbox manifolds Theory of levels for graph matchbox manifolds Scheme of the proof of Theorem 2(1): 1 Choose a generator g F n, and start with a simple 2-ended graph containing vertices g n, n Z, and form the corresponding compact graph matchbox manifold M 0. 2 Apply fusion to create an infinite chain M 0 M 1 M 2 making sure that decorations are attached to the axes at distances larger than, say, 4. 3 Prove that all M i are distinct by using the depth characteristic. 4 It is easy to find a clopen neighborhood in X n which does not intersect the transversal to M i. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

30 Structure of a space of graph matchbox manifolds Theory of levels for graph matchbox manifolds Scheme of the proof of Theorem 2(2): 1 Since M M n is a proper subset, there is an open set U X n such that α(x n ) M =, where α : X n M n is an embedding. 2 Since the set of compact leaves is dense in M n, there exists (T, e) X n such that the corresponding leaf L T is compact. 3 Apply fusion to create a graph (S, e) X n such that L S L T M. 4 To prove that L S is a proper subset of M n, it is enough to show that L S does not accumulate on itself. 5 This follows from the choice of S, and the properties of the fusion construction. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

31 Structure of a space of graph matchbox manifolds Dynamics of graph matchbox manifolds Question Can any matchbox manifold be obtained by the construction of Kenyon and Ghys? The answer is negative. Proposition 3 (OL 2011) Let (T, e) X and let L T M n be a leaf. Then the following holds. Then for all ɛ < e 2 the restriction of the pseudogroup G to the closure of the orbit of T in X is ɛ-expansive. In particular, M n does not contain weak solenoids, which are minimal matchbox manifolds with equicontinuous dynamics. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

32 Structure of a space of graph matchbox manifolds Dynamics of graph matchbox manifolds Question Can any matchbox manifold be obtained by the construction of Kenyon and Ghys? The answer is negative. Proposition 3 (OL 2011) Let (T, e) X and let L T M n be a leaf. Then the following holds. Then for all ɛ < e 2 the restriction of the pseudogroup G to the closure of the orbit of T in X is ɛ-expansive. In particular, M n does not contain weak solenoids, which are minimal matchbox manifolds with equicontinuous dynamics. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

33 Structure of a space of graph matchbox manifolds Theory of levels for graph matchbox manifolds It is likely that there is no relation between the level of a leaf and its growth properties. Theorem 4 (OL 2011) There exists a totally proper graph matchbox manifold M L M n at level 1 where the transitive leaf L has linear growth. There exists a totally proper graph matchbox manifold M L at level 1 where the transitive leaf L has exponential growth. However, graph matchbox manifolds provide an interesting class of examples with various dynamical properties. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

34 Structure of a space of graph matchbox manifolds Theory of levels for graph matchbox manifolds It is likely that there is no relation between the level of a leaf and its growth properties. Theorem 4 (OL 2011) There exists a totally proper graph matchbox manifold M L M n at level 1 where the transitive leaf L has linear growth. There exists a totally proper graph matchbox manifold M L at level 1 where the transitive leaf L has exponential growth. However, graph matchbox manifolds provide an interesting class of examples with various dynamical properties. Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

35 Structure of a space of graph matchbox manifolds Non-minimal example Properties of the matchbox manifold M: Contains a compact leaf Contains a recurrent leaf Is at level 2 Transversal contains a clopen neighborhood with equicontinuous restricted dynamics Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31

36 Structure of a space of graph matchbox manifolds Thank you for your attention! Olga Lukina (University of Leicester, UK) Graph matchbox manifolds July 2th, / 31