For a simple electric axe, I would actually recommend a standard drill motor, with a ~5:1 reduction. For years I've been planning on using big fancy motors or complex arrangements but never got anything built. Yet when I just used a drill motor, I was able to build something in a few hours. Here's a picture of the basic setup (click to enlarge):

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It is a 5:1 ratio using 8mm chain and sprockets from Technobots. 8-tooth sprocket on the drill, 40-tooth sprocket on the axe. Here's a video showing how effective it can be:
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http://www.youtube.com/watch?v=WOSJBug1Y7I (copy and paste, didn't want to clog up the thread with a big embed)

If you stick to a simple setup like this, you should be able to get an axe mechanism made up reasonably easily