NOWHERE‐ANALYTIC SMOOTH CURVES WITH NON‐TRIVIAL ANALYTIC ISOTROPY

We study the smoothness properties of planar curves γ ⊂ R 2 , 0 ∈ γ , which are invariant under a local real‐analytic diffeomorphism ψ fixing the origin. Under certain conditions, depending on the first‐order jet (if the eigenvalues of d ψ ( 0 ) are not both of modulus one) or on a higher order jet (if ψ is tangent to the identity) of ψ and γ, we show that γ must be real analytic as soon as it is smooth enough — in particular, if it is of class C ∞ . On the other hand, when these conditions are not verified we can construct examples of nowhere‐analytic curves of class C ∞ , whose Taylor expansion is divergent at 0, which are invariant under non‐trivial real‐analytic local diffeomorphisms (either tangent to the identity or not).