Let $F$ be a compact,
orientable surface with
negative Euler characteristic,
and let $x_1, €cdots , x_n$ be $n$ fixed
but arbitrarily chosen points on $€mathrm{int}F$;
each $x_i$ has a (small) diskal neighborhood $D_i €subset F$.
Denote by $€mathcal{S}_n(F)$
a subgroup of $€mathrm{Diff}(F)$ consisting of ``sliding" maps $f$
each of which satisfies
$(1)$ $f(€{x_1, €dots , x_n€}) = €{x_1, €dots , x_n€}$,
$f(D_1€cup €cdots €cup D_n)
= D_1 €cup €cdots €cup D_n$
and
$(2)$ $f$ is isotopic to the identity map on $F$.
Then by restricting such automorphisms to
$€hat{F} = F - €mathrm{int}(D_1 €cup €cdots €cup D_n)$,
we have automorphisms $€hat{f} : €hat{F} €to €hat{F}$,
which form a subgroup $€mathcal{S}_n(€hat{F})$
of $€mathrm{Diff}(€hat{F})$.
We give a Nielsen-Thurston classification of elements of
$€mathcal{S}_n(€hat{F})$ using braids in $F €times I$
which characterize the elements of
$€mathcal{S}_n(€hat{F})$.