A note on topological dimension, Hausdorff measure, and rectifiability

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ A note on topological dimension, Hausdorff measure, and rectifiability © 2020 American Mathematical Society Accepted version (Final draft) David, Guy C.; Le Donne, Enrico David, G. C., & Le Donne, E. (2020). A note on topological dimension, Hausdorff measure, and rectifiability. Proceedings of the American Mathematical Society, 148(10), 4299-4304. https://doi.org/10.1090/proc/15051 2020 A NOTE ON TOPOLOGICAL DIMENSION, HAUSDORFF MEASURE, AND RECTIFIABILITY GUY C. DAVID AND ENRICO LE DONNE Abstract. We give a sufficient condition for a general compact metric space to admit an n-rectifiable piece, as a consequence of a recent result of David Bate. Let Xbe a compact metric space of topological dimension n. Suppose that the n-dimensional Hausdorff measure of X,Hn(X), is finite. Suppose further that the lower n-density of the measure Hnis positive, Hn-almost everywhere in X. Then Xcontains an n-rectifiable subset of positive Hn-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Cs¨ornyei-Jones. 1. Introduction The purpose of this note is to record a consequence, for general metric spaces, of a recent result of Bate [2]. We prove the following fact: Theorem 1.1. Let Xbe a compact metric space of topological dimension n. Suppose that the n-dimensional Hausdorff measure of X,Hn(X), is finite. Suppose further that (1) lim inf r→0 Hn(B(x, r)) rn>0for Hn-a.e. x∈X. Then Xcontains an n-rectifiable subset of positive Hn-measure. Moreover, assumption (1) is unnecessary if one uses recently announced results of Cs¨ornyei-Jones. The use in Theorem 1.1 of the results of Cs¨ornyei-Jones arises purely through our use of the work of Bate [2] (Theorem 2.1 below), and does not directly appear in any of the proofs here. See Bate’s discussion just below [2, Theorem 1.1] for details concerning the announcement of Cs¨ornyei-Jones and the dependence of Theorem 2.1 on them. When Xis a subset of some Euclidean space, Theorem 1.1, without assuming (1) or the results of Cs¨ornyei-Jones, appears to already be known Date: July 17, 2018. 2010 Mathematics Subject Classification. Primary: 28A75. Secondary: 28A78, 30L99. G.C.D was supported by the National Science Foundation under Grant no. NSF DMS1758709. E.L.D. was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). 1 2 GUY C. DAVID AND ENRICO LE DONNE (see [16, p. 880]), as a consequence of the Besicovitch-Federer projection theorem. For general metric spaces, the Besicovitch-Federer theorem is unavailable [3], but Bate’s work [2] serves as our replacement. As a general rule, sufficient conditions for finding rectifiability in an abstract metric space are much rarer than for subsets of Euclidean space, where tools such as projection and density theorems are available. When n= 1, Theorem 1.1 (without assuming (1) or relying on the results of Cs¨ornyei-Jones) is a consequence of the fact that continua of finite H1measure are Lipschitz images of [0,1] (see, e.g., [15, Lemma 3.7]), but this particular fact does not extend to n > 1. We now recall some background: For compact metric spaces, the commonly used notions of topological dimension (Lebesgue covering dimension, large/strong inductive dimension, and small/weak inductive dimension) agree. We refer the reader to [14, Sections I.4 and II.5] for this fact and the relevant definitions. For Hausdorff measure and dimension, we refer the reader to [9, Chapter 8]. An Hn-measurable subset Eof a metric space Xis called n-rectifiable if Hn(E\ ∞ [ i=1 fi(Fi)) = 0 where Fiare measurable subsets of Rnand fi:Fi→Xare Lipschitz maps. By a theorem of Kirchheim [12, Lemma 4], one can equivalently take fito be bi-Lipschitz mappings. A subset Eof a metric space Xis called purely n-unrectifiable if it contains no n-rectifiable subsets of positive Hn-measure. If a compact metric space Xhas topological dimension n, then it is a well-known fact (see, e.g., [9, Theorem 8.15]) that Hn(X)>0, although certainly Xmay have infinite n-dimensional Hausdorff measure or even Hausdorff dimension strictly larger than n, as is the case for classical fractals. Thus, Theorem 1.1 says that in the extremal situation, one must see some Euclidean structure in the space. Related results, in which a combination of n-dimensional topological behavior and n-dimensional measure theoretic behavior implies some type of rectifiability, can be found, for example, in [5,7,8,11,16]. These results typically employ more quantitative assumptions to obtain more quantitative conclusions than our Theorem 1.1. It is easy to see that the assumptions of Theorem 1.1 (including (1)) do not imply n-rectifiability of the whole space X. For example, Xmay be the disjoint union of the unit ball in Rnwith a metric space that is a purely n-unrectifiable Cantor set of positive n-dimensional Hausdorff measure. More surprising is that the assumptions of Theorem 1.1 do not imply n-rectifiability even if one assumes that Xis a compact n-dimensional topological manifold. In the appendix to [19], Schul and Wenger construct a compact topological n-sphere with Hn(X)<∞that contains a purely nunrectifiable subset of positive measure. TOPOLOGICAL DIMENSION, HAUSDORFF MEASURE, AND RECTIFIABILITY 3 On the other hand, Theorem 1.1 implies that every open ball in a compact n-manifold with finite Hn-measure contains an n-rectifiable subset of positive Hn-measure. Note that there exist such manifolds with no bi-Lipschitz embedding into any Euclidean space [13,18]. As a final remark, we point out two well-known, purely unrectifiable examples that contrast with Theorem 1.1. For one, consider the closed unit ball Bin the Heisenberg group, which is a compact metric space of topological dimension 3 and Hausdorff dimension 4. This metric space Bis purely 4-unrectifiable, as one can show with a standard “blowup” argument. In fact, Bis also purely 2and 3-unrectifiable, but this is more difficult to establish (see [1, Theorem 7.2]). For a second example, consider any compact metric space (X, d) of topological dimension mand Hausdorff dimension n≥m, and let Y= (X, dp) for some p∈(0,1). Then Yis a compact metric space of topological dimension mand Hausdorff dimension n/p > m that is purely k-unrectifiable for each k∈N. Indeed, if E⊆Rkis compact and f:E→f(E)⊂Yis a bi-Lipschitz map, then blowing up f:E→f(E), in the Gromov-Hausdorff sense, at a point of density of Eyields a bi-Lipschitz embedding of Rkinto a metric space of the form (Z, dp). This is impossible, as such a space can contain no rectifiable curves. For more on such blowup arguments, we refer the reader to [6, Chapters 8-9]. Acknowledgments. The authors are grateful for comments from Giovanni Alberti, Luigi Ambrosio, Matthew Badger, David Preiss, and Raanan Schul. 2. Proof of Theorem 1.1 Given a metric space Xand m∈N, let Lip1(X, m) denote the space of bounded, 1-Lipschitz functions f:X→Rm, equipped with the supremum distance, which we denote dist. This is a complete metric space, and hence residual subsets (in the sense of Baire category) are dense. The proof of Theorem 1.1 is based on the following recent result. Theorem 2.1 (Bate [2, Theorem 1.1]).Let Xbe a complete, purely nunrectifiable metric space with Hn(X)<∞. Suppose further that (1) holds. Then the set of all f∈Lip1(X, m)with Hn(f(X)) = 0 is residual. Moreover, assumption (1) is unnecessary if one uses recently announced results of Cs¨ornyei-Jones. This has the following easy consequence. Corollary 2.2. Let Xbe a compact, purely n-unrectifiable metric space with Hn(X)<∞and satisfying (1). Let g:X→[0,1]nbe continuous. Then there is a sequence of Lipschitz functions fi:X→[0,1]nthat converge to gin the supremum distance and satisfy Hn(fi(X)) = 0 for all i∈N. Moreover, assumption (1) is unnecessary if one uses recently announced results of Cs¨ornyei-Jones. 4 GUY C. DAVID AND ENRICO LE DONNE Proof. Let hibe a sequence of Li-Lipschitz functions converging to gin the supremum distance. (The existence of such a sequence is a consequence of the Stone-Weierstrass theorem, or see [17, Lemma 2.4] for a simple direct proof.) Thus L−1 ihi∈Lip1(X, n). By Theorem 2.1, we can find, for each i∈N, a Lipschitz function gi∈ Lip1(X, n) satisfying dist(L−1 ihi, gi)< L−1 i2−iand Hn(gi(X)) = 0. Consider the 1-Lipschitz retraction r:Rn→[0,1]ngiven by (2) r(x1, x2, . . . , xn) = (ψ(x1), ψ(x2), . . . , ψ(xn)), where ψ(t) =      0t < 0 t0≤t≤1 1t > 1. Lastly, set fi=r◦(Ligi). Since ris Lipschitz and Hn(gi(X)) = 0, we have Hn(fi(X)) = 0 for all i∈N. Furthermore, dist(fi, g) = dist(r◦(Ligi), r ◦g)≤dist(Ligi, g)<2−i+ dist(hi, g)→0.  To prove Theorem 1.1, we will also need some topological information. Definition 2.3. Let f:X→Ybe a continuous map between metric spaces. A point y∈Yis called a stable value of fif there is  > 0 such that y∈g(X) for every continuous g:X→Ywith dist(g, f)< . Some basic and well-known facts about stable values of mappings to [0,1]n are collected in the following lemma. Lemma 2.4. Let Xbe a metric space and let ybe a stable value of a continuous map f:X→[0,1]n. Then (i) y /∈∂([0,1]n), (ii) yis a stable value of gfor each continuous g:X→[0,1]nwith dist(g, f)sufficiently small, and (iii) f(X)contains an open neighborhood of yin [0,1]n. Proof. Part (i) is simple and explained in [10, Example VI 4]. Part (ii) is an immediate consequence of the definition of stable value. For part (iii), recall the 1-Lipschitz retraction r:Rn→[0,1]ndefined in (2). Note that rmaps Rn\[0,1]nonto the boundary of [0,1]n. Let ybe a stable value of f:X→[0,1]n, with parameter  > 0. Then, by part (i), y∈(0,1)n. We claim that f(X) contains B(y, )∩[0,1]n. Consider any y0∈B(y, )∩[0,1]n. The formula h(x) = r(x+y−y0) TOPOLOGICAL DIMENSION, HAUSDORFF MEASURE, AND RECTIFIABILITY 5 defines a continuous map from [0,1]nto itself such that h(y0) = yand |h(x)−x|<  for all x∈[0,1]n. Consider the map g:X→[0,1]ndefined by g=h◦f. Then dist(f, g)< , so g(x) = yfor some x∈X. Therefore, r(f(x) + y−y0) = y. Since yis not on the boundary of [0,1]n, we must have f(x) + y−y0=y, i.e., f(x) = y0. The following theorem is the second main ingredient in the proof of Theorem 1.1. Theorem 2.5 (Theorem III.1 of [14]).Let Xbe a compact metric space of topological dimension n. Then there is a continuous map g:X→[0,1]n with a stable value. The technique of using stable values to find some rectifiable structure in a metric space was used by David and Semmes [6, Section 12.3] and Bonk and Kleiner [4] in similar contexts. Proof of Theorem 1.1. Let Xbe a compact metric space of topological dimension nand Hn(X)<∞. We claim that Xcontains an n-rectifiable subset of positive measure. Suppose, to the contrary, that Xis purely nunrectifiable. Then, by Theorem 2.5, there is a continuous map g:X→[0,1]nwith a stable value y. By Corollary 2.2, there is a sequence fiof Lipschitz maps from Xto [0,1]nthat converge to gin the supremum distance and satisfy (3) Hn(fi(X)) = 0 for all i∈N. On the other hand, by Lemma 2.4(ii), when i∈Nis sufficiently large, the map fimust also have yas a stable value. In that case, fi(X) contains an open subset of [0,1]n, by Lemma 2.4(iii). This contradicts (3).  References [1] L. Ambrosio and B. Kirchheim. Rectifiable sets in metric and Banach spaces. Math. Ann., 318(3):527–555, 2000. [2] D. Bate. Purely unrectifiable metric spaces and perturbations of Lipschitz functions. Preprint, 2017. to appear, Acta Math. arXiv:1712.07139. [3] D. Bate, M. Cs¨ornyei, and B. Wilson. The Besicovitch-Federer projection theorem is false in every infinite-dimensional Banach space. Israel J. Math., 220(1):175–188, 2017. [4] M. Bonk and B. Kleiner. 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Checkerboards, Lipschitz functions and uniform rectifiability. Rev. Mat. Iberoamericana, 13(1):189–210, 1997. [12] B. Kirchheim. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc., 121(1):113–123, 1994. [13] T. Laakso. Plane with A∞-weighted metric not bi-Lipschitz embeddable to RN.Bull. London Math. Soc., 34(6):667–676, 2002. [14] J Nagata. Modern dimension theory, volume 2 of Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin, revised edition, 1983. [15] R. Schul. Subsets of rectifiable curves in Hilbert space—the analyst’s TSP. J. Anal. Math., 103:331–375, 2007. [16] S.. Semmes. Finding structure in sets with little smoothness. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨urich, 1994), pages 875–885. Birkh¨auser, Basel, 1995. [17] S. Semmes. Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincar´e inequalities. Selecta Math. (N.S.), 2(2):155– 295, 1996. [18] S. Semmes. On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A∞-weights. Rev. Mat. Iberoamericana, 12(2):337–410, 1996. [19] C. Sormani and S. Wenger. Weak convergence of currents and cancellation. Calc. Var. Partial Differential Equations, 38(1-2):183–206, 2010. With an appendix by Raanan Schul and Wenger. (David) Department of Mathematical Sciences, Ball State University, Muncie, IN 47306 E-mail address:[email protected] (Le Donne) Dipartimento di Matematica, Universit` a di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy, &, University of Jyv¨ askyl¨ a, Department of Mathematics and Statistics, P.O. Box (MaD), FI-40014, Finland E-mail address:[email protected] E-mail address:[email protected]