Shape of matchbox manifolds
In this work, we develop shape expansions of minimal matchbox manifolds without holonomy, in terms of branched manifolds formed from their leaves. Our approach is based on the method of coding the holonomy groups for the foliated spaces, to define leafwise regions which are transversely stable and are adapted to the foliation dynamics. Approximations are obtained by collapsing appropriately chosen neighborhoods onto these regions along a “transverse Cantor foliation”. The existence of the “transverse Cantor foliation” allows us to generalize standard techniques known for Euclidean and fibered cases to arbitrary matchbox manifolds with Riemannian leaf geometry and without holonomy. The transverse Cantor foliations used here are constructed by purely intrinsic and topological means, as we do not assume that our matchbox manifolds are embedded into a smooth foliated manifold, or a smooth manifold.
Key words and phrases:solenoids, matchbox manifold, laminations, Delaunay triangulations
In this work, we consider topological spaces which are continua; that is, compact, connected metric spaces. We assume that has the additional structure of a codimension-zero foliated space, so are matchbox manifolds. The path-connected components of form the leaves of a foliation of dimension . The precise definitions are given in Section 2 below. Matchbox manifolds arise naturally in the study of exceptional minimal sets for foliations of compact manifolds, and as the tiling spaces associated to repetitive, aperiodic tilings of Euclidean space which have finite local complexity. They also arise in some aspects of group representation theory and index theory for leafwise elliptic operators for foliations, as discussed in the books [CandelConlon2000, MS2006].
The class of Williams solenoids and the results about these spaces provide one motivation for this work. Recall that an expanding attractor for an Axiom A diffeomorphism of a compact manifold is a continuum; that is, a compact, connected metric space. Williams developed a structure theory for these spaces in his seminal works [Williams1967, Williams1970, Williams1974]. The hyperbolic splitting of the tangent bundle to along yields a foliation of the space by leaves of the expanding foliation for , and the contracting foliation for gives a transverse foliation on an open neighborhood . Williams used this additional structure on a neighborhood of to obtain a “presentation” of as an inverse limit of “branched -manifolds”, , where is the dimension of the expanding bundle for , and the map induces the map between the approximations. The notion of a -dimensional branched manifold is easiest to define, as the branches are required only to meet each other at disjoint vertices. In higher dimensions, the definition of branched manifolds becomes more subtle, and especially for the “transversality condition” imposed on the cell attachment maps. The spaces with this structure are called Williams solenoids in the literature. The topological properties of the approximating map is used to study the the dynamical system defined by , for example as discussed in the works [FJ1981, Jones1983, ShubSullivan1975, SullivanWilliams1976] and others.
The Riemann surface laminations introduced by Sullivan [Sullivan1988] are compact topological spaces locally homeomorphic to a complex disk times a Cantor set, and a similar notion is used by Lyubich and Minsky in [LM1997], and Ghys in [Ghys1999]. These are also well-known examples of matchbox manifolds.
Associated to the foliation of a matchbox manifold is a compactly generated pseudogroup acting on a “transverse” totally disconnected space , which determines the transverse dynamical properties of . The first two authors showed in [ClarkHurder2013] that if the action of on is equicontinuous, then is homeomorphic to a weak solenoid (in the sense of [McCord1965, Schori1966, FO2002]). Equivalently, the shape of is defined by a tower of proper covering maps between compact manifolds of dimension .
The purpose of this work is to study the shape properties of an arbitrary matchbox manifold , without the assumption that the dynamics of the associated action of is equicontinuous, though with the assumption that is minimal; that is, that every leaf of is dense. We also assume that is without holonomy. Our main result shows that all such spaces have an analogous structure as that of a William solenoid. An important difference between the case of Williams solenoids, and the general case we consider, is that the tower of approximations is not in general defined by a single map, but uses a sequence of maps between compact branched manifolds.
A presentation of a space is a collection of continuous maps , where each is a connected compact branched -manifold, and each is a proper surjective map of branched manifolds, as defined in Section 10. It is assumed that there is given a homeomorphism between and the inverse limit space defined by
A Vietoris solenoid [Vietoris1927] is a -dimensional solenoid, where each is a circle, and each is a covering map of degree greater than . More generally, if each is a compact manifold and each is a proper covering map, then we say that is a weak solenoid, as discussed in [McCord1965, Schori1966, FO2002].
Here is our main result.
Let be a minimal matchbox manifold of dimension . Assume that the foliation of is without holonomy. Then there exists a presentation where each is a triangulated branched -manifold and each is a proper simplicial map, so that is homeomorphic to the inverse limit of the system of maps they define,
The hypothesis that is without holonomy may possibly be removed, using a more general definition of the bonding maps allowed. The work by Benedetti and Gambaudo in [BG2003] gives a model for such a result, in the case of tiling spaces associated with an action of a connected Lie group . Also, the results of the first two authors in [ClarkHurder2013] applies to equicontinuous matchbox manifolds which may have non-trivial leafwise holonomy. On the other hand, a generalization of Theorem 1.1 to arbitrary minimal matchbox manifolds would yield solutions to several classical problems in the study of exceptional minimal sets for codimension-one foliations, for example as listed in [Hurder2002, Hurder2006], so an extension of Theorem 1.1 to this generality would be important, but is expected to involve additional subtleties.
In this work, no assumptions are made on the geometry and topology of the leaves in , beyond that they are complete Riemannian manifolds. In order to construct the branched manifold approximations to , we use a technique based on “dynamical codings” for the orbits of the pseudogroup associated to the foliation, a method which generalizes that used by Gambaudo and Martens [GambaudoMartens2006] for flows. This coding method can also be seen as extending a technique used by Thomas in his study of equicontinuous flows in [EThomas1973]. The first two authors used this coding approach in their study of equicontinuous actions of groupoids on Cantor sets in [ClarkHurder2013], and it also appears implicitly in the work by Forrest [Forrest2000] in his study of minimal actions of on Cantor sets. The coding method we use is more formal, so applies in complete generality. The other ingredient required is the existence of a transverse Cantor foliation in the generality we consider. We construct these foliations in this work, extending the results of the authors’ paper [CHL2013a]. The existence of a compatible transverse Cantor foliation on is fundamental for defining the branched manifold approximations.
As mentioned above, tiling spaces provide an important class of examples for which Theorem 1.1 applies. Let be a tiling of which is repetitive, aperiodic, and has finite local complexity. The tiling space is defined as the closure of the set of tilings obtained via the translation action of , in a suitable metric topology [AP1998, BBG2006, FHK2002, Sadun2008]. The assumptions imply that is locally homeomorphic to a disk in times a Cantor set, so that is a matchbox manifold. (See [FS2009] for a discussion of variants of this result.) The seminal result of Anderson and Putnam in [AP1998] showed that under these assumptions, and under the assumption that the tiling is defined by an expanding substitution, then is homeomorphic to the inverse limit of a tower of branched flat -manifolds. The approach of Anderson and Putnam, and the alternative approaches to generalizing their method and extending the conclusion to all tilings by Gähler and Sadun [Sadun2003], and in works [ALM2011, BBG2006, BG2003, GambaudoMartens2006, LR2013], used variations on a technique called “collaring” or a method called “inflation” of a Voronoi tessellation of , to obtain the branched manifold approximations for . The more recent work [BDHS2010] gives a version of this method closest to the approach we take in this paper. The work here is an important step in developing the general theory of point patterns and tilings for general, non-Euclidean spaces. As described in Senechal [Sen1995], the theory of point patterns and tilings is directly connected to the theory of quasicrystals, and so our work can be seen as a step towards understanding possible quasicrystalline structures in non-Euclidean spaces.
We discuss next the contents of the rest of this work, which culminates in the proof of Theorem 1.1. Section 2 introduces the basic definitions of a matchbox manifold, and Section 3 introduces the definitions and properties of the holonomy pseudogroups for matchbox manifolds, as developed in [ClarkHurder2013] and [CHL2013a]. Section 4 discusses the Voronoi tessellations of the leaves associated to transversal clopen sets, then Section 5 develops some fundamental properties of restricted pseudogroups.
A coding of the orbits of the action of the holonomy pseudogroup of the foliation on is developed in Section 6, and used to construct “dynamically defined” nested sequences of clopen coverings for of a Cantor transversal to , which are “centered” on the transverse orbit defined by a fixed leaf . The study of the dynamics of the induced action on a descending chain of clopen neighborhoods is a standard technique in almost all approaches to the study of the dynamics of minimal actions on a Cantor set. It corresponds to the initial steps in the construction of a Kakutani tower for the measurable theory of group actions.
In Section 7, we associate to each clopen covering of the transversal space a collection of open sets, called Reeb neighborhoods which form a covering of . The results of [CHL2013a] show that each such Reeb neighborhood projects along a transverse Cantor foliation to its compact base. Section 9 uses the results in [CHL2013a] to show that these bi-foliated structures on the Reeb neighborhoods can be chosen to be compatible on overlaps. In Section 10, the proof of Theorem 1.1 is completed, by showing that the Reeb neighborhoods collapsed along the Cantor transversal foliation, can be identified (or “glued”) each to the other on their overlaps, to obtain a branched -manifold. By iterating this process for a chain of clopen covers for , we obtain a tower of maps whose inverse limit is homeomorphic to .
Note that the works [ALM2011, LR2013] follow a similar approach to the above, but using an intuitive extension of the zooming technique for Voronoi cells as in [BBG2006, BG2003]. The geometry and topology of a generic leaf can be quite complicated, and not at all as intuitively straightforward as for the case of Euclidean spaces. Hence the geometry and topology of the Voronoi cells which decompose may be similarly non-intuitive. Some of the subtleties for Voronoi partitions at large scale in non-Euclidean manifolds are discussed, for example, in the work [LeibonLetscher2000]. The more formal approach using codings that we use in this work avoids the need to control the geometry of the Voronoi cells introduced. The reader will also find the care taken with the domains of holonomy maps makes the proofs of some results quite tedious at times, but a careful consideration of examples shows that these concerns are required.
This work is part of a program to generalize the results of the thesis of Fokkink [Fokkink1991], started during a visit by the authors to the University of Delft in August 2009. The papers [ClarkHurder2013, CHL2013a] are the initial results of this study, and this work is the preparation for the forthcoming paper [CHL2013c] which completes the program. The authors would like to thank Robbert Fokkink for the invitation to meet in Delft, and the University of Delft for its generous support for the visit. The authors’ stay in Delft was also supported by a travel grant No. 040.11.132 from the Nederlandse Wetenschappelijke Organisatie.
2. Foliated spaces and matchbox manifolds
In this section, we give the precise definitions and results for matchbox manifolds, as required for this work. More detailed discussions with examples can be found in the books of Candel and Conlon [CandelConlon2000, Chapter 11] and Moore and Schochet [MS2006, Chapter 2]. In the following, details of proofs are omitted whenever possible, as they are presented in detail in these two texts and the paper [ClarkHurder2013].
A foliated space of dimension is a continuum , such that there exists a compact metric space , and for each there is a compact subset , an open subset , and a homeomorphism defined on the closure of in , such that where . Moreover, it is assumed that each admits an extension to a foliated homeomorphism where is an open neighborhood.
The subspace of is called the local transverse model at .
Let denote the composition of with projection onto the second factor.
For the set is called a plaque for the coordinate chart . We adopt the notation, for , that , so that . Note that each plaque for is given the topology so that the restriction is a homeomorphism. Then .
Let . Note that if , then is an open subset of both and . The collection of sets
forms the basis for the fine topology of . The connected components of the fine topology are called leaves, and define the foliation of . For , let denote the leaf of containing .
Definition 2.1 does not impose any smoothness conditions on the leaves of , so they may just be topological manifolds. The next definition imposes a uniform smoothness condition on the leaves.
A smooth foliated space is a foliated space as above, such that there exists a choice of local charts such that for all with , there exists an open set such that and are connected open sets, and the composition
is a smooth map, where and . The leafwise transition maps are assumed to depend continuously on in the -topology on maps between subsets of .
A map is said to be smooth if for each flow box and the composition is a smooth function of , and depends continuously on in the -topology on maps of the plaque coordinates . As noted in [MS2006] and [CandelConlon2000, Chapter 11], this allows one to define leafwise smooth partitions of unity, vector bundles, and tensors for smooth foliated spaces. In particular, one can define leafwise smooth Riemannian metrics. We recall a standard result, whose proof for foliated spaces can be found in [CandelConlon2000, Theorem 11.4.3].
Let be a smooth foliated space. Then there exists a leafwise Riemannian metric for , such that for each , the leaf inherits the structure of a complete Riemannian manifold with bounded geometry, and the Riemannian geometry of depends continuously on .
Bounded geometry implies, for example, that for each , there is a leafwise exponential map which is a surjection, and the composition depends continuously on in the compact-open topology on maps.
A matchbox manifold is a continuum with the structure of a smooth foliated space , such that the transverse model space is totally disconnected, and for each , is a clopen (closed and open) subset. A matchbox manifold is minimal if every leaf of is dense.
Intuitively, a -dimensional matchbox manifold has local coordinate charts which are homeomorphic to a “box of matches”, which gave rise to this terminology in the works [AHO1991, AO1995, AM1988].
The maximal path-connected components of define the leaves of a foliation of . All matchbox manifolds are assumed to be smooth, with a leafwise Riemannian metric, and metric on .
2.1. Metric properties and regular covers
We introduce some local metric considerations for a matchbox manifold, and give the definition of a regular covering of .
For and , let be the closed -ball about in , and the open -ball about .
Similarly, for and , let be the closed -ball about in , and the open -ball about .
Each leaf has a complete path-length metric, induced from the leafwise Riemannian metric:
where denotes the path-length of the piecewise -curve . If are not on the same leaf, then set .
For each and , let be the closed leafwise ball.
For each , the Gauss Lemma implies that there exists such that is a strongly convex subset for the metric . That is, for any pair of points there is a unique shortest geodesic segment in joining and and contained in (cf. [doCarmo1992, Chapter 3, Proposition 4.2], or [Helgason1978, Theorem 9.9]). Then for all the disk is also strongly convex. As is compact and the leafwise metrics have uniformly bounded geometry, we obtain:
There exists such that for all , is strongly convex.
The following proposition summarizes results in [ClarkHurder2013, Sections 2.1 - 2.2].
For a smooth foliated space , given , there exist constants and , and a covering of by foliation charts with the following properties: For each , let be the projection, then
Locality: for , where .
For , the plaque of the chart through is denoted by .
Convexity: the plaques of are strongly convex subsets for the leafwise metric.
Uniformity: for let , then
The projection is a clopen subset for all .
A regular foliated covering of is one that satisfies these conditions.
We assume in the following that a regular foliated covering of as in Proposition 2.6 has been chosen. Let denote the corresponding open covering of . We can assume without loss of generality, that the spaces form a disjoint clopen covering of , so that .
Let be a Lebesgue number for . That is, given any there exists some index such that the open metric ball . Also, introduce a form of “leafwise Lebesgue number”, defined by
Thus, for all , . Note that for all and , the triangle inequality implies that . Note that for , we have
which imply that , and hence . This last inequality will be used in Section 5.
For , let be the projection, so that for each the restriction is a smooth coordinate system on the plaque .
For each , the set is a compact transversal to . Without loss of generality, we can assume that the transversals are pairwise disjoint in , so there exists a constant such that
In particular, this implies that the centers of disjoint plaques on the same leaf are separated by distance at least . Also, define sections
Then is the image of and we let denote their disjoint union, and is defined by the union of the maps .
Define the metric on via the restriction of to , and use the map to pull it back to .
2.2. Local estimates
The local projections and sections are continuous maps of compact spaces, so admit uniform metric estimates as shown in [ClarkHurder2013].
There exists a modulus of continuity function which is continuous and increasing, such that:
Set . ∎
There exists a modulus of continuity function which is continuous and increasing, such that:
Set . ∎
Next, for each , consider the projection map , and define
The assumption that implies that has distance at least from the exterior of . Then the projection of the closed ball to the transversal contains an open neighborhood of by the continuity of projections, and is the distance from this center to the exterior of the projected ball. Then introduce given by
Finally, we give a plaque-wise estimate on the Riemannian metrics. The assumption that the leafwise Riemannian metric on is continuous, means that for each coordinate chart , the push-forwards of the Riemannian metric to the slices vary continuously with . Let denote the norm defined on the tangent bundle , and denote a tangent vector by , for and . Let denote the Euclidean norm on . Then by the compactness of and the continuity of the metric, for , there exists such that the following holds, for each :
2.3. Foliated maps
A map between foliated spaces is said to be a foliated map if the image of each leaf of is contained in a leaf of . If is a matchbox manifold, then each leaf of is path connected, so its image is path connected, hence must be contained in a leaf of .
A leafwise path is a continuous map such that there is a leaf of for which for all . If is a matchbox manifold and is continuous, then is a leafwise path. This yields:
Let and be matchbox manifolds, and a continuous map. Then maps the leaves of to leaves of . In particular, any homeomorphism of a matchbox manifold is a foliated map.
3. Holonomy of foliated spaces
The holonomy pseudogroup of a smooth foliated manifold generalizes the induced discrete dynamical system associated to a section of a flow. A standard construction in foliation theory (see [CN1985], [CandelConlon2000, Chapter 2] or [Haefliger1984]) associates to a leafwise path a holonomy map . The collection of all such maps with initial and ending points on a transversal to defines the holonomy pseudogroup for a matchbox manifold . We recall below the ideas and notations, as required in the proofs of our main theorems, especially the delicate issues of domains which must be considered. See [ClarkHurder2013, CHL2013a] for a complete discussion of the ideas of this section and related technical results.
3.1. Holonomy pseudogroup
Let be a regular foliated covering of as in Proposition 2.6. A pair of indices , for , is said to be admissible if .
For admissible, set . The regular foliated covering assumption implies that plaques in admissible charts are either disjoint, or have connected intersection. This implies that there is a well-defined transverse change of coordinates homeomorphism with domain and range . By definition they satisfy , , and if then on their common domain of definition. Note that the domain and range of are clopen subsets of by Proposition 2.6.(5).
Recall that for each , denotes the transverse section for the chart , and denotes their disjoint union. Then is the coordinate projection restricted to , which is a homeomorphism, and denotes its inverse.
The holonomy pseudogroup of is the topological pseudogroup modeled on generated by the elements of . A sequence is admissible if each pair is admissible for , and the composition
has non-empty domain , which is defined to be the maximal clopen subset of for which the compositions are defined. Given any open subset , we obtain a new element by restriction. Introduce
The range of is the open set . Note that each map admits a continuous extension as is a clopen set for each .
Let be an admissible sequence. For each , set , and let denote the corresponding holonomy map. For , let . Note that and .
Given , let . For each , set and . Recall that , where each is a strongly convex subset of the leaf in the leafwise metric . Introduce the plaque-chain
Adopt the notation . Intuitively, a plaque-chain is a sequence of successively overlapping convex “tiles” in starting at , ending at , where each is “centered” on the point .
3.3. Leafwise paths to plaque-chains
Let be a path. Set , and . Let be an admissible sequence with . We say that covers , if there is a partition such that satisfies
For a path , we construct an admissible sequence with so that covers , and has “uniform domains”. Inductively choose a partition of the interval , say , such that for each ,
As a notational convenience, we have let , so that . Choose to be the largest value of such that for all , then .
For each , choose an index so that . Note that, for all , , so that . It follows that is an admissible sequence. Set and note that .
The construction of the admissible sequence above has the important property, that is the composition of generators of which each have a uniform lower bound estimate on the radii of the metric balls centered at the orbit and which are contained in their domains. To see this, let , and note that implies that for some , we have that . Hence,
Then for all , the uniform estimate defining in (8) implies that
Let be a path starting at and ending at . Then a good plaque-chain covering of is the plaque-chain, starting at , associated to an admissible sequence