A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Eq... more A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Equivalently, the geodesic flow on the unit tangent bundle of X (equipped with the hyperbolic metric) is ergodic. Given a Riemann surface X of arbitrary topological type and a hyperbolic pants decomposition of X we obtain sufficient conditions for parabolicity of X in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as for half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, and Abelian covers of compact surfaces are also studied.
Thurston boundary of the universal Teichmüller space T (D) is the space P M L bdd (D) of projecti... more Thurston boundary of the universal Teichmüller space T (D) is the space P M L bdd (D) of projective bounded measured laminations of D. A geodesic ray in T (D) is of generalized Teichmüller type if it shrinks the vertical foliation of a holomorphic quadratic differential. We provide the first examples of generalized Teichmüller rays which diverge near Thurston boundary P LM bdd (D). Moreover, for every k ≥ 1 we construct examples of rays with limit sets homeomorphic to k-dimensional cubes. For the latter result we utilize the classical Kronecker approximation theorem from number theory which states that if θ 1 ,. .. , θ k are rationally independent reals then the sequence ({θ 1 n},. .. , {θ k n}) is dense in the k-torus T k .
Proceedings of The London Mathematical Society, Jun 10, 2022
A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Eq... more A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Equivalently, the geodesic flow on the unit tangent bundle of X (equipped with the hyperbolic metric) is ergodic. Given a Riemann surface X of arbitrary topological type and a hyperbolic pants decomposition of X we obtain sufficient conditions for parabolicity of X in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as for half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, and Abelian covers of compact surfaces are also studied.
A Riemann surface X is said to be of parabolic type if it supports a Green's function. Equivalent... more A Riemann surface X is said to be of parabolic type if it supports a Green's function. Equivalently, the geodesic flow on the unit tangent of X is ergodic. Given a Riemann surface X of arbitrary topological type and a hyperbolic pants decomposition of X we obtain sufficient conditions for parabolicity of X in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, Abelian covers of compact surfaces are also studied.
A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into its... more A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into itself is onto. We construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional Sierpiński spaces, which are quasisymmetrically co-Hopfian. We also show that the collection of quasisymmetric equivalence classes of spaces homeomorphic to the Menger curve is uncountable. These results answer a problem and generalize results of Merenkov from [Mer10].
We show that if f:X→ Y is a quasisymmetric mapping between Ahlfors regular spaces, then _H f(E)≤_... more We show that if f:X→ Y is a quasisymmetric mapping between Ahlfors regular spaces, then _H f(E)≤_H E for "almost every" bounded Ahlfors regular set E⊆ X. If additionally, X and Y are Loewner spaces then _H f(E)=_H E for "almost every" Ahlfors regular set E⊂ X. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if f is a quasiconformal map of R^N, N≥ 2, then for Lebesgue a.e. y∈R^N we have _H f(y+E) = _H E. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if E ⊂R is Ahlfors d-regular, d<1, then some component of f(E ×R) has dimension at most 2/(d+1), and we construct examples to show this bound is sharp. In addition, we show there is a 1-dimensional set S⊆ R and planar quasiconformal map f such that f(R× S) contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson T...
A Thurston boundary of the universal Teichmuller space T(ⅅ) is the space PMLbdd(ⅅ) of projective ... more A Thurston boundary of the universal Teichmuller space T(ⅅ) is the space PMLbdd(ⅅ) of projective bounded measured laminations of ⅅ. A geodesic ray in T(ⅅ) is of generalized Teichmuller-type if it shrinks the vertical foliation of a holomorphic quadratic differential. We consider the limit points in PMLbdd(ⅅ) of certain geodesic rays induced by non-integrable holomorphic quadratic differentials. In particular, we show that there is a geodesic ray in T(ⅅ), such that the limiting projective measured lamination is supported on a geodesic lamination, whose leaves are not homotopic to leaves of either vertical or horizontal foliation of the corresponding holomorphic quadratic differential.
We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T (... more We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T (H). Our main result states that the limits of the Teichmüller geodesics in the Thurston's boundary of T (H) may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.
A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into its... more A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into itself is surjective. We construct the first example of a metric space homeomorphic to the universal Menger curve, which is quasisymmetrically co-Hopfian. This answers a problem of Merenkov from arXiv:1305.4161. We also show that for every $n\geq 1$ there is a metric space which is homeomorphic to the Sierpi\'nski space of topological dimension $n$ and is quasisymmetrically co-Hopfian. More generally we show that doubles of porous slit Sierpi\'nski spaces are quasisymmetrically co-Hopfian and that there are also many non-porous examples. These results generalize the results from the same paper of Merenkov arXiv:1305.4161.
We show that middle interval Cantor sets of Hausdorff dimension 1 are minimal for quasisymmetric ... more We show that middle interval Cantor sets of Hausdorff dimension 1 are minimal for quasisymmetric maps of a line. Combining this with a theorem of Wu we conclude that there are “rigid” subsets of a line whose every quasisymmetric image has zero length and Hausdorff dimension 1.
Proceedings of the London Mathematical Society, 2018
Thurston's boundary to the universal Teichmüller space T (H) is the set of asymptotic rays to the... more Thurston's boundary to the universal Teichmüller space T (H) is the set of asymptotic rays to the embedding of T (H) in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations P M L bdd (H) of H. We prove that each Teichmüller geodesic ray in T (H) has a unique limit point in Thurston's boundary to T (H) unlike in the case of closed surfaces.
We show that if f : X → Y is a quasisymmetric mapping between Ahlfors regular spaces, then dimH f... more We show that if f : X → Y is a quasisymmetric mapping between Ahlfors regular spaces, then dimH f (E) ≤ dimH E for "almost every" bounded Ahlfors regular set E ⊆ X. If additionally, X and Y are Loewner spaces then dimHf (E) = dimH E for "almost every" Ahlfors regular set E ⊂ X. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if f is a quasiconformal map of R N , N ≥ 2, then for Lebesgue a.e. y ∈ R N we have dimH f (y + E) = dimHE. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if E ⊂ R is Ahlfors d-regular, d < 1, then some component of f (E × R) has dimension at most 2/(d + 1), and we construct examples to show this bound is sharp. In addition, we show there is a 1-dimensional set S ⊆ R and planar quasiconformal map f such that f (R × S) contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson [6] and answer questions posed in [6] and [10].
Proceedings of the American Mathematical Society, 2015
We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T (... more We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T ( H ) T(\mathbb {H}) . Our main result states that the limits of the Teichmüller geodesics in the Thurston’s boundary of T ( H ) T(\mathbb {H}) may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.
Proceedings of the American Mathematical Society, 2008
The central set of a domain D is the set of centers of maximal discs in D. Fremlin proved that th... more The central set of a domain D is the set of centers of maximal discs in D. Fremlin proved that the central set of a planar domain has zero area and asked whether it can have Hausdorff dimension strictly larger than 1. We construct a planar domain with central set of Hausdorff dimension 2.
In this paper we give several conditions for a space to be minimal for conformal dimension. We sh... more In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede [7]. It implies in particular that if E ⊂ R is minimal for conformal dimension and supports a measure λ such that for every ε > 0 ther is a constant 0 < C < ∞ such that C −1 r 1+ε ≤ λ(E ∩ B(x, r)) ≤ Cr 1−ε then X × Y is minimal for conformal dimension for every compact Y .
In this paper we give several conditions for a space to be minimal for conformal dimension. We sh... more In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede [5]. r 1+ε ≤ λ(E ∩ B(x, r)) (0.1) It implies in particular that there are many sets E ⊂ R of zero length such that X × Y is minimal for conformal dimension for every compact Y .
We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet... more We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a Transboundary Loewner Property (TLP) which is a transboundary analogue of the Loewner property of Heinonen and Koskela [Acta Math. 181 (1998), pp. 1–61]. We show that a dyadic slit carpet can be quasisymmetrically embedded into the plane if and only if it is TLP. Moreover, every dyadic slit carpet X X can be associated to a “pillowcase sphere” X ^ \widehat {X} which is a metric space homeomorphic to the sphere S 2 \mathbb {S}^2 . We show that X X quasisymmetrically embeds into the plane if and only if X ^ \widehat {X} is quasisymmetric to S 2 \mathbb {S}^2 if and only if X ^ \widehat {X} is Ahlfors 2 2 -regular.
We study the distortion of Hausdorff dimension of families of Ahlfors regular sets under quasisym... more We study the distortion of Hausdorff dimension of families of Ahlfors regular sets under quasisymmetric map $f$ between metric spaces. We show that $f$ cannot increase the dimension of "most" $d$-regular sets and we estimate the number of exceptional sets whose images have dimension $\geq d' > d$; the precise statements of both results involve modulus estimates for families of measures. For planar quasiconformal maps, the general estimates imply that if $E \subset \reals$ is $d$-regular, then some component of $f(E \times \reals)$ has dimension $\leq 2/(d+1)$, and we construct examples to show this bound is sharp. In addition, we construct 1-dimensional sets $E$ so that $f(E \times \reals)$ contains no rectifiable sub-arc. These results generalize work of Balogh, Monti and Tyson \cite{Tyson:frequency} and answer questions posed in \cite{Tyson:frequency}, \cite{AimPL}.
A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Eq... more A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Equivalently, the geodesic flow on the unit tangent bundle of X (equipped with the hyperbolic metric) is ergodic. Given a Riemann surface X of arbitrary topological type and a hyperbolic pants decomposition of X we obtain sufficient conditions for parabolicity of X in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as for half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, and Abelian covers of compact surfaces are also studied.
Thurston boundary of the universal Teichmüller space T (D) is the space P M L bdd (D) of projecti... more Thurston boundary of the universal Teichmüller space T (D) is the space P M L bdd (D) of projective bounded measured laminations of D. A geodesic ray in T (D) is of generalized Teichmüller type if it shrinks the vertical foliation of a holomorphic quadratic differential. We provide the first examples of generalized Teichmüller rays which diverge near Thurston boundary P LM bdd (D). Moreover, for every k ≥ 1 we construct examples of rays with limit sets homeomorphic to k-dimensional cubes. For the latter result we utilize the classical Kronecker approximation theorem from number theory which states that if θ 1 ,. .. , θ k are rationally independent reals then the sequence ({θ 1 n},. .. , {θ k n}) is dense in the k-torus T k .
Proceedings of The London Mathematical Society, Jun 10, 2022
A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Eq... more A Riemann surface X is said to be of parabolic type if it does not support a Green's function. Equivalently, the geodesic flow on the unit tangent bundle of X (equipped with the hyperbolic metric) is ergodic. Given a Riemann surface X of arbitrary topological type and a hyperbolic pants decomposition of X we obtain sufficient conditions for parabolicity of X in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as for half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, and Abelian covers of compact surfaces are also studied.
A Riemann surface X is said to be of parabolic type if it supports a Green's function. Equivalent... more A Riemann surface X is said to be of parabolic type if it supports a Green's function. Equivalently, the geodesic flow on the unit tangent of X is ergodic. Given a Riemann surface X of arbitrary topological type and a hyperbolic pants decomposition of X we obtain sufficient conditions for parabolicity of X in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, Abelian covers of compact surfaces are also studied.
A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into its... more A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into itself is onto. We construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional Sierpiński spaces, which are quasisymmetrically co-Hopfian. We also show that the collection of quasisymmetric equivalence classes of spaces homeomorphic to the Menger curve is uncountable. These results answer a problem and generalize results of Merenkov from [Mer10].
We show that if f:X→ Y is a quasisymmetric mapping between Ahlfors regular spaces, then _H f(E)≤_... more We show that if f:X→ Y is a quasisymmetric mapping between Ahlfors regular spaces, then _H f(E)≤_H E for "almost every" bounded Ahlfors regular set E⊆ X. If additionally, X and Y are Loewner spaces then _H f(E)=_H E for "almost every" Ahlfors regular set E⊂ X. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if f is a quasiconformal map of R^N, N≥ 2, then for Lebesgue a.e. y∈R^N we have _H f(y+E) = _H E. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if E ⊂R is Ahlfors d-regular, d<1, then some component of f(E ×R) has dimension at most 2/(d+1), and we construct examples to show this bound is sharp. In addition, we show there is a 1-dimensional set S⊆ R and planar quasiconformal map f such that f(R× S) contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson T...
A Thurston boundary of the universal Teichmuller space T(ⅅ) is the space PMLbdd(ⅅ) of projective ... more A Thurston boundary of the universal Teichmuller space T(ⅅ) is the space PMLbdd(ⅅ) of projective bounded measured laminations of ⅅ. A geodesic ray in T(ⅅ) is of generalized Teichmuller-type if it shrinks the vertical foliation of a holomorphic quadratic differential. We consider the limit points in PMLbdd(ⅅ) of certain geodesic rays induced by non-integrable holomorphic quadratic differentials. In particular, we show that there is a geodesic ray in T(ⅅ), such that the limiting projective measured lamination is supported on a geodesic lamination, whose leaves are not homotopic to leaves of either vertical or horizontal foliation of the corresponding holomorphic quadratic differential.
We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T (... more We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T (H). Our main result states that the limits of the Teichmüller geodesics in the Thurston's boundary of T (H) may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.
A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into its... more A metric space X is quasisymmetrically co-Hopfian if every quasisymmetric embedding of X into itself is surjective. We construct the first example of a metric space homeomorphic to the universal Menger curve, which is quasisymmetrically co-Hopfian. This answers a problem of Merenkov from arXiv:1305.4161. We also show that for every $n\geq 1$ there is a metric space which is homeomorphic to the Sierpi\'nski space of topological dimension $n$ and is quasisymmetrically co-Hopfian. More generally we show that doubles of porous slit Sierpi\'nski spaces are quasisymmetrically co-Hopfian and that there are also many non-porous examples. These results generalize the results from the same paper of Merenkov arXiv:1305.4161.
We show that middle interval Cantor sets of Hausdorff dimension 1 are minimal for quasisymmetric ... more We show that middle interval Cantor sets of Hausdorff dimension 1 are minimal for quasisymmetric maps of a line. Combining this with a theorem of Wu we conclude that there are “rigid” subsets of a line whose every quasisymmetric image has zero length and Hausdorff dimension 1.
Proceedings of the London Mathematical Society, 2018
Thurston's boundary to the universal Teichmüller space T (H) is the set of asymptotic rays to the... more Thurston's boundary to the universal Teichmüller space T (H) is the set of asymptotic rays to the embedding of T (H) in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations P M L bdd (H) of H. We prove that each Teichmüller geodesic ray in T (H) has a unique limit point in Thurston's boundary to T (H) unlike in the case of closed surfaces.
We show that if f : X → Y is a quasisymmetric mapping between Ahlfors regular spaces, then dimH f... more We show that if f : X → Y is a quasisymmetric mapping between Ahlfors regular spaces, then dimH f (E) ≤ dimH E for "almost every" bounded Ahlfors regular set E ⊆ X. If additionally, X and Y are Loewner spaces then dimHf (E) = dimH E for "almost every" Ahlfors regular set E ⊂ X. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if f is a quasiconformal map of R N , N ≥ 2, then for Lebesgue a.e. y ∈ R N we have dimH f (y + E) = dimHE. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if E ⊂ R is Ahlfors d-regular, d < 1, then some component of f (E × R) has dimension at most 2/(d + 1), and we construct examples to show this bound is sharp. In addition, we show there is a 1-dimensional set S ⊆ R and planar quasiconformal map f such that f (R × S) contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson [6] and answer questions posed in [6] and [10].
Proceedings of the American Mathematical Society, 2015
We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T (... more We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space T ( H ) T(\mathbb {H}) . Our main result states that the limits of the Teichmüller geodesics in the Thurston’s boundary of T ( H ) T(\mathbb {H}) may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.
Proceedings of the American Mathematical Society, 2008
The central set of a domain D is the set of centers of maximal discs in D. Fremlin proved that th... more The central set of a domain D is the set of centers of maximal discs in D. Fremlin proved that the central set of a planar domain has zero area and asked whether it can have Hausdorff dimension strictly larger than 1. We construct a planar domain with central set of Hausdorff dimension 2.
In this paper we give several conditions for a space to be minimal for conformal dimension. We sh... more In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede [7]. It implies in particular that if E ⊂ R is minimal for conformal dimension and supports a measure λ such that for every ε > 0 ther is a constant 0 < C < ∞ such that C −1 r 1+ε ≤ λ(E ∩ B(x, r)) ≤ Cr 1−ε then X × Y is minimal for conformal dimension for every compact Y .
In this paper we give several conditions for a space to be minimal for conformal dimension. We sh... more In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede [5]. r 1+ε ≤ λ(E ∩ B(x, r)) (0.1) It implies in particular that there are many sets E ⊂ R of zero length such that X × Y is minimal for conformal dimension for every compact Y .
We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet... more We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a Transboundary Loewner Property (TLP) which is a transboundary analogue of the Loewner property of Heinonen and Koskela [Acta Math. 181 (1998), pp. 1–61]. We show that a dyadic slit carpet can be quasisymmetrically embedded into the plane if and only if it is TLP. Moreover, every dyadic slit carpet X X can be associated to a “pillowcase sphere” X ^ \widehat {X} which is a metric space homeomorphic to the sphere S 2 \mathbb {S}^2 . We show that X X quasisymmetrically embeds into the plane if and only if X ^ \widehat {X} is quasisymmetric to S 2 \mathbb {S}^2 if and only if X ^ \widehat {X} is Ahlfors 2 2 -regular.
We study the distortion of Hausdorff dimension of families of Ahlfors regular sets under quasisym... more We study the distortion of Hausdorff dimension of families of Ahlfors regular sets under quasisymmetric map $f$ between metric spaces. We show that $f$ cannot increase the dimension of "most" $d$-regular sets and we estimate the number of exceptional sets whose images have dimension $\geq d' > d$; the precise statements of both results involve modulus estimates for families of measures. For planar quasiconformal maps, the general estimates imply that if $E \subset \reals$ is $d$-regular, then some component of $f(E \times \reals)$ has dimension $\leq 2/(d+1)$, and we construct examples to show this bound is sharp. In addition, we construct 1-dimensional sets $E$ so that $f(E \times \reals)$ contains no rectifiable sub-arc. These results generalize work of Balogh, Monti and Tyson \cite{Tyson:frequency} and answer questions posed in \cite{Tyson:frequency}, \cite{AimPL}.
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