Distortion of dimension by Sobolev and quasiconformal mappings J. Tyson 18 June 2014 I. Introduction and overview, quasiconformal maps of Rn and their effect on Hausdorff dimension II. Global quasiconformal dimension in Rn III. Conformal dimension of metric spaces IV. Sobolev dimension distortion in Rn and in metric spaces V. QC and Sobolev dimension distortion in the sub-Riemannian Heisenberg group 1 In this final lecture we will survey known results on distortion of dimension by Sobolev and quasiconformal mappings of the sub-Riemannian Heisenberg group Hn. n We equip H with the Carnot-Carath´eodory metric dcc and recall that the n Hausdorff dimension of (H ; dcc ) is Q = 2n + 2. A homeomorphism f :Ω ! Ω0 between domains in Hn is (metrically) K-quasiconformal if supfdcc (f (p); f (q)) : dcc (p; q) = rg lim sup 0 0 ≤ K 8 p 2 Ω: r!0 inffdcc (f (p); f (q )) : dcc (p; q ) = rg 2 A major difficulty, however, lies in the lack of effective methods for constructing Heisenberg quasiconformal maps. For instance, • naive versions of the radial stretch map x 7! jxja−1x fail to be quasiconformal (cf. recent work of Balogh{F¨assler{Platis), • PL techniques are not obviously available, • no theory of Heisenberg quasiuniform convexity exists to date. Many results of the Euclidean theory of quasiconformal maps carry over to the Heisenberg group (and even to more general metric measure spaces). 3 Many results of the Euclidean theory of quasiconformal maps carry over to the Heisenberg group (and even to more general metric measure spaces). A major difficulty, however, lies in the lack of effective methods for constructing Heisenberg quasiconformal maps. For instance, • naive versions of the radial stretch map x 7! jxja−1x fail to be quasiconformal (cf. recent work of Balogh{F¨assler{Platis), • PL techniques are not obviously available, • no theory of Heisenberg quasiuniform convexity exists to date. 4 1. For each 0 < s < t < Q there exists E ⊂ Hn compact and f : Hn ! Hn qc s.t. dim E = s and dim f (E) = t. f 2. Gehring{V¨ais¨al¨atheorem: for each 0 < s < Q and K ≥ 1 9 0 < β ≤ α < Q s.t. β ≤ dim f (E) ≤ α whenever E ⊂ Hn with dim E = s and f is K-qc. Distortion of dimension by quasiconformal maps in Hn Z. M. Balogh, \Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group", J. d'Analyse Math. (2001) 5 f 2. Gehring{V¨ais¨al¨atheorem: for each 0 < s < Q and K ≥ 1 9 0 < β ≤ α < Q s.t. β ≤ dim f (E) ≤ α whenever E ⊂ Hn with dim E = s and f is K-qc. Distortion of dimension by quasiconformal maps in Hn Z. M. Balogh, \Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group", J. d'Analyse Math. (2001) 1. For each 0 < s < t < Q there exists E ⊂ Hn compact and f : Hn ! Hn qc s.t. dim E = s and dim f (E) = t. 6 f Distortion of dimension by quasiconformal maps in Hn Z. M. Balogh, \Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group", J. d'Analyse Math. (2001) 1. For each 0 < s < t < Q there exists E ⊂ Hn compact and f : Hn ! Hn qc s.t. dim E = s and dim f (E) = t. 2. Gehring{V¨ais¨al¨atheorem: for each 0 < s < Q and K ≥ 1 9 0 < β ≤ α < Q s.t. β ≤ dim f (E) ≤ α whenever E ⊂ Hn with dim E = s and f is K-qc. 7 Proof sketch: For s ≤ 2n + 1, choose E to be the union of line segments along integral curves of the horiz vector field X1 emanating from a Cantor set in the z2 ··· znt-subspace. Criteria of the Bourdon{Pansu theorem hold. S For 2n + 1 ≤ s ≤ Q, let E = r2C @BH (o; r) be a Cantor set's worth of Kor´anyi n spheres @BH (o; r) = fp 2 H : dH (o; p) = rg. Apply a version of the Bourdon{Pansu criterion for such foliations. 4. (T, 2008) Kovalev's theorem holds for the Assouad dimension on Hn. n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA;H E = 0. Question: Kovalev's theorem in Hn for Hausdorff dimension? n n 3. For each s 2 [1; Q] 9 E ⊂ H compact s.t. GQCdimH E = dim E = s. 8 4. (T, 2008) Kovalev's theorem holds for the Assouad dimension on Hn. n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA;H E = 0. Question: Kovalev's theorem in Hn for Hausdorff dimension? n n 3. For each s 2 [1; Q] 9 E ⊂ H compact s.t. GQCdimH E = dim E = s. Proof sketch: For s ≤ 2n + 1, choose E to be the union of line segments along integral curves of the horiz vector field X1 emanating from a Cantor set in the z2 ··· znt-subspace. Criteria of the Bourdon{Pansu theorem hold. S For 2n + 1 ≤ s ≤ Q, let E = r2C @BH (o; r) be a Cantor set's worth of Kor´anyi n spheres @BH (o; r) = fp 2 H : dH (o; p) = rg. Apply a version of the Bourdon{Pansu criterion for such foliations. 9 Question: Kovalev's theorem in Hn for Hausdorff dimension? n n 3. For each s 2 [1; Q] 9 E ⊂ H compact s.t. GQCdimH E = dim E = s. Proof sketch: For s ≤ 2n + 1, choose E to be the union of line segments along integral curves of the horiz vector field X1 emanating from a Cantor set in the z2 ··· znt-subspace. Criteria of the Bourdon{Pansu theorem hold. S For 2n + 1 ≤ s ≤ Q, let E = r2C @BH (o; r) be a Cantor set's worth of Kor´anyi n spheres @BH (o; r) = fp 2 H : dH (o; p) = rg. Apply a version of the Bourdon{Pansu criterion for such foliations. 4. (T, 2008) Kovalev's theorem holds for the Assouad dimension on Hn. n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA;H E = 0. 10 n n 3. For each s 2 [1; Q] 9 E ⊂ H compact s.t. GQCdimH E = dim E = s. Proof sketch: For s ≤ 2n + 1, choose E to be the union of line segments along integral curves of the horiz vector field X1 emanating from a Cantor set in the z2 ··· znt-subspace. Criteria of the Bourdon{Pansu theorem hold. S For 2n + 1 ≤ s ≤ Q, let E = r2C @BH (o; r) be a Cantor set's worth of Kor´anyi n spheres @BH (o; r) = fp 2 H : dH (o; p) = rg. Apply a version of the Bourdon{Pansu criterion for such foliations. 4. (T, 2008) Kovalev's theorem holds for the Assouad dimension on Hn. n n In other words, if E ⊂ H , dimA E < 1 then GQCdimA;H E = 0. Question: Kovalev's theorem in Hn for Hausdorff dimension? 11 Examples: ' = 1, ' = xk , ' = yk , k = 1;:::; n generate left translation flows P ' = 2t V = 2tT + j (xj Xj + yj Yj ) generates dilation flow p0 7! δes (p0) Quasiconformal flows n Recall the standard left invariant vector fields X1; Y1;:::; Xn; Yn generating HH . @ @ @ @ Xj (p) = + 2yj ; Yj (p) = − 2xj @xj @t @yj @t Kor´anyiand Reimann (1995) give conditions on a vector field V which guarantee that the time s map p(0) 7! p(s) associated to the ODEp _ = V (p) is K(s)-qc. Basic structure is n 1 X V = ' T + (X ' Y − Y ' X ) for ' 2 C 1( n) 4 j j j j H j=1 with additional restrictions on the second horiz derivatives of '. 12 Quasiconformal flows n Recall the standard left invariant vector fields X1; Y1;:::; Xn; Yn generating HH . @ @ @ @ Xj (p) = + 2yj ; Yj (p) = − 2xj @xj @t @yj @t Kor´anyiand Reimann (1995) give conditions on a vector field V which guarantee that the time s map p(0) 7! p(s) associated to the ODEp _ = V (p) is K(s)-qc. Basic structure is n 1 X V = ' T + (X ' Y − Y ' X ) for ' 2 C 1( n) 4 j j j j H j=1 with additional restrictions on the second horiz derivatives of '. Examples: ' = 1, ' = xk , ' = yk , k = 1;:::; n generate left translation flows P ' = 2t V = 2tT + j (xj Xj + yj Yj ) generates dilation flow p0 7! δes (p0) 13 f Fix a ball B(p; R) and a family of disjoint balls Bj = B(pj ; rj ) ⊂ B(p; r). Choose potentials 'j generating contractive similarities with fixed points at the P pj , and define ' = j χj 'j for suitable smooth cutoff functions χj . For suitable s, the time s map F associated to ' is qc, fixes @B(p; r), and contracts each Bj . Moreover, F is conformal outside of B and inside each Bj . Iterate. As the flow method only works for smooth maps, use compactness of the space of normalized K-qc maps to pass to the limit. 14 In particular, it follows that f is a.e. diff'ble in horizontal directions, and that f is n n n a generalized contact map, dhfp = (Dfp)∗jHp H maps HpH to Hf (p)H a.e. Intuition: The CC ball at p of radius 0 < r 1 is comparable to the box [−r; r]2n × [−r 2; r 2], sheared so that the [−r; r]2n factor lies along the subspace n HpH . Metric quasiconformality ) at a point of differentiability, the size r and size r 2 directions cannot be interchanged by f .