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The authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one-dimensional mappings. They show that the probability density of the asymmetric component satisfies a scaling law. The exponent in this scaling law varies continuously with the distance from the bifurcation point, and is determined by the spectrum of local Lyapunov exponents of the uncoupled map. Finally they show that the topology of the invariant set is rather unusual: though the attractor for supercritical coupling is a line, it is surrounded by a strange invariant set which is dense in a two-dimensional neighbourhood of the attractor.