NOWHERE-ANALYTIC SMOOTH CURVES WITH NON-TRIVIAL ANALYTIC ISOTROPY

Abstract

We study the smoothness properties of planar curves (Formula presented.), (Formula presented.), 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 (Formula presented.) 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 (Formula presented.). On the other hand, when these conditions are not verified we can construct examples of nowhere-analytic curves of class (Formula presented.), whose Taylor expansion is divergent at 0, which are invariant under non-trivial real-analytic local diffeomorphisms (either tangent to the identity or not). © 2020 The Authors. The publishing rights for this article are licensed to University College London under an exclusive licence.