In Section 5.1 Brin and Stuck speak about infinite $\varepsilon$-orbits of $E_m=mx \mod 1$ on $S^1$, $m>1$. I.e a sequence $(x_n)_{n=0}^\infty$ such that $d(x_{n+1},E_mx_n)<\varepsilon$ for all $n\in \mathbb N_0$. They then go onto discuss how for each finite $n$ there is a point $y_n$ such that $d(E_m^jy_n,x_j)<\varepsilon$ for all $0\leq j\leq n$. They then proceed to claim that $\lim_{n\to \infty}y_n$ exists (which I am actually still struggling to understand why, but that's not the point of this question.) After this they claim that if $f$ is $C^1$ close to $E_m$ then each infinite orbit $\varepsilon$-orbit of $f$ is shadowed by a unique real orbit of $f$.
What do they mean $f$ being $C^1$ close to $E_m$? I cannot recall having seen it in the book before, and I cannot find it in the index. My best guess coming from functional analysis, is that $\|E_m-f\|_\infty+\|E_m'-f'\|_\infty<\varepsilon$, but then this means that the definition of closeness depends on $\varepsilon$, and the way they phrase things makes me think this is unlikely. My only other guess is that they mean $\|E_m-f\|_\infty+\|E_m'-f'\|_\infty<\frac{1}{2m}$, as that is a significant figure in determining whether two points get expanded by a factor of $m$ by $E_m$. I would appreciate someone who knows the actual definition helping me out.