well, if you right rotate instead, you can use chop.

sub rot_same
{
    my( $word )= @_;
    my %seen;

    for(1..length $word)
    {
        ## rotate right
        $word= chop($word) . $word;
        return 1 if exists $seen{$word};
        $seen{$word}++;
    }
}

print $_, ' => ', rot_same($_), $/ for qw(abcdef aaaaaa abcabc);

__END__
abcdef =>
aaaaaa => 1
abcabc => 1
[download]

and if you calculate the maximum number of same characters once, it will speed things up.

sub too_many_repetitions
{
    my( $word )= @_;
    my $same_char_percent= 0.5;

    my $max_same_char= $same_char_percent * length $word;

    my @chars= split // => $word;
    my %seen;

    $seen{$_}++ for @chars;
    return 1 if $max_same_char < pop @{[sort values %seen]};
}

print $_, ' => ', too_many_repetitions($_), $/ 
    for qw(abcdef aaaaaa abcabc);
__END__
abcdef =>
aaaaaa => 1
abcabc =>
[download]

~Particle *accelerates*