X(t) is Brownian motion on the
axis −∞ < t < ∞, with paths in Rn, n ≧ 2. X(t) leads to composed mappings
f ∘ X, where f is a real-valued function of class Λα(Rn), whose gradient
never vanishes. To define the class Λα(Rn), when α > 1, take the integer p
in the interval α − 1 ≦ p < α and require that f have continuous partial
derivatives of orders 1,⋯,p and these fulfill a Lipschitz condition in exponent
α − p on each compact set; to specify further that gradf≠0 throughout
Rn, write Λ+α. Then a closed set T is a set of “Λα-multiplicity” if every
transform f(T) ⊆ R1(f ∈ Λ+α) is a set of strict multiplicity— an M0-set
(see below). Henceforth we define b = α−1 and take S to be a closed linear
set.
