Several ways:

1. Simple closure:

   {
      my %state;
      sub recursive {
          my $arg = shift;
          ...
          if( ( $arg = $state{ $arg } ) == ... ) {
               return recursive( $arg );
          else {
               return 0
          }
      }
   }
   [download]

2. Closure via helper function:

   sub rec_helper {
       my %state;
       my $recursive;
       $recursive = sub {
           my( $arg ) = shift;
           ...
           if( ( $arg = $state{ $arg ) ) == ... ) {
                return $recursive->( $arg );
           }
           else {
                return 0;
           }
       };
   }
   [download]

3. State passed through args (often via helper function ):

   sub recursive {
       my( $state, $arg ) = @_;
       ...
       if( ( $arg = $state->{ $arg } ) == ... ) {
            return recursive( $state, $arg );
       }
       else {
            return 0;
       }
   }
   
   sub rec_helper{
       my $arg = shift;
       my %state;
       return recursive( \%state, $arg );
   }
   [download]

And many variations on those themes.