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Conformal dimension of a metric space $X$, denoted by $\dim_C X$, is theinfimum of the Hausdorff dimension among all its quasisymmetric images. Ifconformal dimension of $X$ is equal to its Hausdorff dimension, $X$ is said tobe minimal for conformal dimension. In this paper we show that the graph of theone dimensional Brownian motion is almost surely minimal for conformaldimension. We also give many other examples of minimal sets for conformaldimension, which we call Bedford-McMullen type sets. In particular we show thatBedford-McMullen self-affine sets with uniform fibers are minimal for conformaldimension. The main technique in the proofs is the construction of ``richfamilies of minimal sets of conformal dimension one''. The latter concept isquantified using Fuglede's modulus of measures.

Conformal Dimension of the Brownian Graph