is there a driver for the apple genuine dvi to composite adapter to make it work on Maverick OS?

Hi there djshotgunn,

There is not a specific driver that you would need to download outside of Software Update. I would recommend taking a look at the troubleshooting steps in the article below if you are experiencing video issues with an external monitor.

Apple computers: Troubleshooting issues with video on internal or external displays

http://support.apple.com/kb/ht1573

Many common display issues can be resolved easily by making sure the software on your Apple computer and the firmware on your video adapter, display and Thunderbolt cabling is up to date. Check for updates for Apple products by using Software Update, the Mac App Store, or visiting Apple downloads. Check with the manufacturer for any updates that may be available for displays, extenders, or adapters not made by Apple.

-Griff W.

There is a known issue with 10.9.3 and 10.9.4 and display running through the Thunderbolt port.. for example, see here 10.9.4 - Thunderbolt display for 6630M (Mini mid-2011) still broken and here Second display problem with 10.9.4