When you define a style for the DropDownListBox, you can specify whether you want to display the arrow at the left of the DropDownListBox automatically. 

To display the arrow:

Click the Use as Border option in the DropDownListBox Edit Style window.  X displays in the check box.

To stop display the arrow:

Click the Use as Border option in the DropDownListBox Edit Style window.  The X no longer displays in the check box.