# On the conformal dimension of product measures

@article{Bate2017OnTC, title={On the conformal dimension of product measures}, author={David Bate and Tuomas Orponen}, journal={Proceedings of the London Mathematical Society}, year={2017}, volume={117} }

Given a compact set E⊂Rd−1 , d⩾1 , write KE:=[0,1]×E⊂Rd . A theorem of Bishop and Tyson states that any set of the form KE is minimal for conformal dimension: If (X,d) is a metric space and f:KE→(X,d) is a quasisymmetric homeomorphism, then dimHf(KE)⩾dimHKE.We prove that the measure‐theoretic analogue of the result is not true. For any d⩾2 and 0⩽s0 , there exists a quasisymmetric embedding F:KE→Rd such that dimHF♯ν

## 3 Citations

### Every locally finite Borel measure on $\mathbb{R}$ has conformal dimension zero

- Mathematics
- 2017

A result of P. Tukia from 1989 says that Lebesgue measure on R has conformal dimension zero: for every ǫ > 0, there is a Borel set G ⊂ R of full Lebesgue measure, and a quasisymmetric homeomorphism f…

### Assouad dimension of planar self-affine sets

- MathematicsTransactions of the American Mathematical Society
- 2020

We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad…

### Singular quasisymmetric mappings in dimensions two and greater

- MathematicsAdvances in Mathematics
- 2019

## References

SHOWING 1-10 OF 29 REFERENCES

### Locally minimal sets for conformal dimension

- Mathematics
- 2001

We show that for each 1• fi < d and K <1 there is a subset X of R d such that dim(f(X))‚ fi = dim(X) for every K -quasiconformal map, but such that dim(g(X)) can be made as small as we wish for some…

### Every locally finite Borel measure on $\mathbb{R}$ has conformal dimension zero

- Mathematics
- 2017

A result of P. Tukia from 1989 says that Lebesgue measure on R has conformal dimension zero: for every ǫ > 0, there is a Borel set G ⊂ R of full Lebesgue measure, and a quasisymmetric homeomorphism f…

### On Hausdorff and packing dimension of product spaces

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1996

Abstract We show that for arbitrary metric spaces X and Y the following dimension inequalities hold: where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the…

### Lectures on Analysis on Metric Spaces

- Mathematics
- 2000

1. Covering Theorems.- 2. Maximal Functions.- 3. Sobolev Spaces.- 4. Poincare Inequality.- 5. Sobolev Spaces on Metric Spaces.- 6. Lipschitz Functions.- 7. Modulus of a Curve Family, Capacity, and…

### A doubling measure on R^d can charge a rectifiable curve

- Mathematics
- 2009

For d > 2, we construct a doubling measure v on ℝ d and a rectifiable curve Γ such that ν(Γ) > 0.

### Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability

- Mathematics
- 1995

Acknowledgements Basic notation Introduction 1. General measure theory 2. Covering and differentiation 3. Invariant measures 4. Hausdorff measures and dimension 5. Other measures and dimensions 6.…

### Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

- Mathematics
- 2012

We show that if $${f\colon X\to Y}$$f:X→Y is a quasisymmetric mapping between Ahlfors regular spaces, then $${dim_H f(E)\leq dim_H E}$$dimHf(E)≤dimHE for “almost every” bounded Ahlfors regular set…

### Techniques in fractal geometry

- Mathematics
- 1997

Mathematical Background. Review of Fractal Geometry. Some Techniques for Studying Dimension. Cookie-cutters and Bounded Distortion. The Thermodynamic Formalism. The Ergodic Theorem and Fractals. The…