Single-character substitution is usually easiest with tr:

my ($filename, $newfile);
foreach $filename (readdir(DIR))
{
    ($newfile = $filename) =~ tr/ /./;
    rename($filename, $newfile);
}
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As a bonus, this will replace every space with a period. (That is a bonus, right? ;-)

Also, you want to either chdir to the target directory, or prepend the directory name to the file names. When possible, I prefer chdir:

chdir $dir or die "Can't chdir to $dir: $!\n";
opendir(DIR, '.') or die "Can't open '.': $!\n";
[download]

HTH