$x$ and $\psi(x)$ such that all $k$-th partial derivatives of $\varphi_2 \circ \psi \circ \varphi_1^{-1}$ are Hölder continuous with exponent $\alpha$. If the domain and codomain $M$ and $N$ are clear from context, we will write ${C_{\textnormal{loc}}}^{k,\alpha}$ instead of ${C_{\textnormal{loc}}}^{k,\alpha}(M,N)$.

Using the chain rule and the fact that compositions and products of locally $\alpha$-Hölder continuous functions are again locally $\alpha$-Hölder, it follows that the property of being ${C_{\textnormal{loc}}^{k,\alpha}}$ on a manifold does not depend on the choice of charts in Definition \[def:ckal-funcs\].

Given a differentiable map $F\colon M\to N$ between smooth manifolds, in the remainder of this paper we use the notation ${\mathsf{D}}_x F$ for the derivative of $F$ at the point $x\in M$. (Recall that the derivative ${\mathsf{D}}_x F\colon {\mathsf{T}}_x M \to {\mathsf{T}}_{F(x)}N$ is a linear map between tangent spaces [@lee2013smooth], which can be identified with the Jacobian of $F$ evaluated at $x$ in local coordinates.) In particular, given a dynamical system $\Phi\colon Q\times {\mathbb{T}}\to Q$ and fixed $t \in {\mathbb{T}}$, we write ${\mathsf{D}}_{x}\Phi^t\colon {\mathsf{T}}_{x}Q\to {\mathsf{T}}_{\Phi^t(x)}Q$ for the derivative of the time-$t$ map $\Phi^t\colon Q\to Q$ at the point $x\in Q$. Given $i \geq 2$, we similarly use the notation ${\mathsf{D}}_{x}^iF$ for the $i$-th derivative of $F$, which can 