This looks more like you really want logfiles rather than redirecting STDERR

This is untested, but should work, if not it will give you a foundation on what to work with.

The ` backtick operator is explained in perlop

open(DUMP,">>./dump.txt") or die "could not open: $!\n";
open(ERR,">>./error.txt") or die "could not open: $!\n";

while ($DIR[$a]) {
    print "$directory\\$DIR[$a]\n";
    print DUMP "$directory\\$DIR[$a]";
    my $output = `identify -verbose $directory\\$DIR[$a]`;
        or print ERR "$directory\\$DIR[$a]\n";
    print DUMP "$output\n";
    $a=$a+1;
    }
[download]

You should take a look at you indenting style. You current style makes it difficult to follow your logic