We study the smoothness properties of planar curves $\gamma \subset {\mathbb{R}}^2$, $0\in \gamma$, 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 $d\psi (0)$ 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 $C^{\infty }$. On the other hand, when these conditions are not verified we can construct examples of nowhere‐analytic curves of class $C^{\infty }$, whose Taylor expansion is divergent at 0, which are invariant under non‐trivial real‐analytic local diffeomorphisms (either tangent to the identity or not).

中文翻译：

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具有非平凡分析各向同性的任何分析平滑度曲线

我们研究平面曲线的平滑度特性 $\gamma \subset {\mathbb{[R}}^2$， $0\in \gamma$，在固定原点的局部实解析微分ψ不变的情况下不变。在某些情况下，取决于一阶射流（如果$d\psi （0）$ 既不是模数为1的），也不是在ψ和γ的高次射流（如果ψ与恒等式相切）上，我们证明γ必须足够平滑，因此必须立即进行实解析，尤其是类 $C^{\infty }$。另一方面，当不验证这些条件时，我们可以构造无位置分析曲线的示例$C^{\infty }$，其泰勒展开式在0处发散，在非平凡的实解析局部微分形（与正切线不相切）下是不变的。