We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such sets are `flat' on most scales and in most locations. Our result is quantitative, and when combined with work of I. Hahlomaa, gives a characterization of 1-Ahlfors regular subsets of 1-Ahlfors-regular curves in metric spaces. Our result is a generalization to the metric space setting of the Analyst's (Geometric) Traveling Salesman theorems of P. Jones, K. Okikiolu, and G. David and S. Semmes, and it can be stated in terms of average Menger curvature.
We characterize $Q$-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each $0 < α< Q$, an $α$-regular subspace. As an application, we give an alternative simple proof of the existence of $α$-regular subspaces of a $Q$-dimensional complete Ahlfors regular metric space $(X,ρ)$, which was proved in \cite{JJKRRS}.
Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions $\dim_{\mathrm{H}} A =: α$ and $\dim_{\mathrm{H}} B =: β$. I prove that $$\dim_{\mathrm{H}} [A + θB] \geq α+ β\cdot \tfrac{1 - α}{2 - α}$$ for all $θ\in \mathbb{R} \, \setminus \, E$, where $\dim_{\mathrm{H}} E = 0$.
We show that given a domain $Ω\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.
We introduce a definition of intrinsically quasi-symmetric sections in metric spaces and we prove the Ahlfors-David regularity for this class of sections. We follow a recent result by Le Donne and the author where we generalize the notion of intrinsically Lipschitz graphs in the sense of Franchi, Serapioni and Serra Cassano. We do this by focusing our attention on the graph property instead of the map one.
It is proved that for any $0<β<α$, any bounded Ahlfors $α$-regular space contains a $β$-regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most $O(α/(α-β))$. The bound on the distortion is asymptotically tight when $β\to α$. The main tool used in the proof is a regular form of the ultrametric skeleton theorem.
We show that a mapping $f\colon X\to Y$ satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class $N_{\textrm{loc}}^{1,1}(X;Y)$. Contrary to previous works, we only assume an asymptotic version of Ahlfors-regularity on $X,Y$. This allows many non-Ahlfors regular spaces, such as weighted spaces and Fred Gehring's bowtie, to be included in the theory. Unexpectedly, already in the classical setting of unweighted Euclidean spaces, our theory detects Sobolev mappings that are not recognized by previous results.
We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space. This approach is applicable even if the space does not support any Poincaré inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric measure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map.
Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$, we show that $f$ satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that $f$ belongs to the Sobolev class $N_{\rm{loc}}^{1,p}(X;Y)$, where $1< p\le Q$, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors $Q$-regularity, which particularly enables various weighted spaces to be included in the theory. Unexpectedly, we can apply this to obtain results that are new even in the cl…
In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space.
We prove that, up to topological conjugacy, every Smale space admits an Ahlfors regular Bowen measure. Bowen's construction of Markov partitions implies that Smale spaces are factors of topological Markov chains. The latter are equipped with Parry's measure which is Ahlfors regular. By extending Bowen's construction we create a tool for transferring the Ahlfors regularity of the Parry measure down to the Bowen measure of the Smale space. An essential part of our method uses a refined notion of approximation graphs over compact metric spaces. Moreover, we obtain new estimates for the Hausdorff,…
Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension $\dim_\mathrm{ARC}(X,d)$ of a metric space $(X,d)$ is the infimum over the Hausdorff dimensions of the Ahlfors regular images of the space by quasisymmetric transformations. For a given regular Dirichlet form with the heat kernel, the spectral dimension $d_s$ is an exponent which indicates the short-time asymptotic behavior of the on-diagonal part of the heat kernel. In this paper, we consider the Dirichlet form induced by a resistance form on a set $X$ and t…
The focus of this paper is on Ahlfors $Q$-regular compact sets $E\subset\mathbb{R}^n$ such that, for each $Q-2<α\le 0$, the weighted measure $μ_α$ given by integrating the density $ω(x)=\text{dist}(x, E)^α$ yields a Muckenhoupt $\mathcal{A}_p$-weight in a ball $B$ containing $E$. For such sets $E$ we show the existence of a bounded linear trace operator acting from $W^{1,p}(B,μ_α)$ to $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ when $0<θ<1-\tfrac{α+n-Q}{p}$, and the existence of a bounded linear extension operator from $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ to $W^{1,p}(B, μ_α)$ when $1-\tfrac{α+n-Q}{p}…
Successive divisions of compact metric spaces appear in many different areas of mathematics such as the construction of self-similar sets, Markov partitions associated with hyperbolic dynamical systems, dyadic cubes associated with a doubling metric space. The common feature in these is to divide a space into a finite number of subsets, then divide each subset into finite pieces and repeat this process again and again. In this paper we generalize such successive divisions and call them partitions. Given a partition, we consider the notion of a weight function assigning a "size" to each piece o…
We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \le…
We show that if an Ahlfors-David regular set $E$ of dimension $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by $BP(BP(LG))$), then $E\subset \tilde{E}$ where $\tilde{E}$ is Ahlfors-David regular of dimension $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by $BP(LG)$. Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors-David regular sets $\mathcal{F}$ replacing $LG$. A simple corollary is the stability of the BP operator after 2 iterations. This was previously only known in the Euclidean setting fo…
A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschi…
We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincaré inequality with respect to surface measure is uniformly rectifiable.
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $β$-numbers. These $β$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G.…
F. Gehring and W. Ziemer proved that the p-modulus of the family of paths connecting two continua is dual to the p^*-modulus of the corresponding family of separating hypersurfaces. In this paper we show that a similar result holds in complete Ahlfors-regular metric spaces that support a weak 1-Poincaré inequality. As an application we obtain a new characterization for quasiconformal mappings between such spaces.
We provide examples of infinitesimally Hilbertian, rectifiable, Ahlfors regular metric measure spaces having pmGH-tangents that are not infinitesimally Hilbertian.
The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy-Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extendin…
We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regular metric measure spaces supporting a $1$-Poincaré inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincaré inequality and $f:X\to Y$ is a quasiconformal mapping, then the $Q/(Q-1)$-modulus of the collection of measures $\mathcal{H}^{Q-1}\lfloor_{ΣE}$ corresponding to any collection of sets $E\subset X$ of finite perimeter is quasi-preserved by $f$. We also show that for $Q/(Q-1)$-modulus almost every $ΣE$, if the image surface $Σf(E)$ does not see the singular set of $f$ as…
We provide a new definition of a local walk dimension $β$ that depends only on the metric and not on the existence of a particular regular Dirichlet form or heat kernel asymptotics. Moreover, we study the local Hausdorff dimension $α$ and prove that any variable Ahlfors regular measure of variable dimension $Q$ is strongly equivalent to the local Haudorff measure with $Q=α,$ generalizing the constant dimensional case. Additionally, we provide constructions of several variable dimensional spaces, including a new example of a variable dimensional Sierpinski carpet. We show also that there exist …
We prove that the fractal dimension of a metric space equipped with an Ahlfors regular measure can be recovered from the persistent homology of random samples. Our main result is that if $x_1,\ldots, x_n$ are i.i.d. samples from a $d$-Ahlfors regular measure on a metric space, and $E^0_α\left(x_1,\ldots,x_n\right)$ denotes the $α$-weight of the minimum spanning tree on $x_1,\ldots,x_n:$ \[E_α^0\left(x_1,\ldots,x_n\right)=\sum_{e\in T\left(x_1,\ldots,x_n\right)} |e|^α\,,\] then there exist constants $0<C_1\leq C_2$ so that \[C_1\leq n^{-\frac{d-α}{d}} E^0_α\left(x_1,\ldots,x_n\right)\leq C_2\,\…
We give an example of an Ahlfors $3$-regular, linearly locally connected metric space homeomorphic to $\mathbb{R}^3$ containing a nondegenerate continuum $E$ with zero capacity, in the sense that the conformal modulus of the set of nontrivial curves intersecting $E$ is zero. We discuss this example in relation to the quasiconformal uniformization problem for metric spaces.
