This is related to this other question of mine about a paper of Colin and Honda.

I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the text so i ask for its meaning or a reference where it is used other than this. I quote the beginning of page 6 in the cited paper

Since $d(h(\gamma_0), \gamma_0) = N$, it follows that $d(h(\alpha), h(\gamma_0)) \approx N$ for every $\alpha$. (Here $\approx$ means

approximately equal to.)

For some context: $\gamma_0, \alpha$ are simple closed curves in a surface, $h$ is an automorphism of the surface, $d(\cdot, \cdot)$ is the distance in the complex of curves (which is a natural number) and $N$ is a natural number. Hence, we are saying that two natural numbers are *approximately equal*?

Since $N$ is a number that, in that context, is *arbitrarily large* one could think that it means that $\lim_{N\to \infty} d(h(\gamma_0), \gamma_0) - N = 0$? But later on the text a similar exppresion appears changing $N$ by $0$ (end of page $7$).

So I would appreciate someone explaining to me what it means.

oneof the functions is constant, but if they're both constants then they'd be approximately equal no matter how far apart those constants are. $\endgroup$ – Noah Snyder Jun 8 at 16:47