For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to the infinite-dimensional setting, as discussed for example in the answer to this post.

Let $G=\operatorname{Diff}_c(M)$ the space of compactly supported diffeomorphisms on $M$, say $M$ is Riemannian manifold diffeomorphic to $\mathbb{R}^k$, and let $\mathfrak{g}=C_c^{\infty}(M,M)$ be the $C^{\infty}$-vector fields on $M$. Again making use of comments in the answer to this post the exponential map taking a vector field $V$ in $\mathfrak{g}$ to its flow $\Phi^V:x\to x_1^x$ where $t\to x_t$ is well-defined by $$ \partial x_t^x = V(x_t^x), \, x_0^x=x . $$

• Is this map continuous?
• Moreover, when is it locally-Lipschitz, in the sense that, for every $\emptyset \subset K \subset M$ compact there exists some $L_K>0$ such that for every $U,V \in C^{\infty}_c(M,M)$ $$ \sup_{x \in K} d_M\left( \Phi^V(x),\Phi^U(x) \right)\leq L_K \sup_{x \in X} d_M\left( V(x),U(x) \right) $$ where $d_M$ is the metric induced by the Riemannian metric on $M$?