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I would really like to see live update of the split screen preview while writing tex code, like in the whizzy-tex plugin for emacs. It is especially handy while working with complex diagrams or formulas.
Maybe it would be possible to trigger recompilation of the PDF upon changing (or only upon saving perhaps) the .tex file? Then the "live updating" would be possible in the built-in preview pane as well as in Evince, if you have the PDF open there...