Consider a compact set $K \subset\mathbb{R}^d$ and a $\mathscr{C}^1$-diffeomorphism $\varphi:\mathbb{R}^d\rightarrow\mathbb{R}^d$.

Fix $\varepsilon>0$. Since $K$ is compact, we may define \begin{align*} N(\varepsilon,K) := \min\left\{n\in\mathbb{N}\,:\, \exists x_1,\dots,x_n \in K\text{ s.t. } K\subseteq\bigcup_i B(x_i,\varepsilon)\right\}. \end{align*}

Can something be said about $N(\varepsilon,\varphi(K))$ generally ? Or we have to impose some conditions on $K$ ?

More precisely if $(\varphi_t)_{t\in[0,T]}$ is a family of such diffeomorphisms, is there a simple condition to insure the boundedness of $[0,T] \ni t\mapsto N(\varepsilon,\varphi_t(K))$ ? Denoting $\Theta:(t,x) \mapsto \varphi_t(x)$, I expect $\Theta\in\mathscr{C}^0([0,T];\mathscr{C}^1(\mathbb{R}^d))$ to be sufficient but I am not sure and not really familiarized with that kind of issues !

Thanks in advance for any piece of information.

Ayman