I believe you have two questions: 1) How to turn your rankings into a useful time series, and 2) How to analyze the resulting time series.
For #1, it seems to me that you could simply look at the Top 20 apps in each category and each week calculate how far each of the Top 20 apps has moved since the last week. Count how many slots they move, whether up or down, so that the moves are always a positive number, then sum up the total for each category.
Or calculate how much each app went up or down (positive or negative) and sum the squares of the values. Use the Top 25 or Top 50 if you want.
For #2, you could start with exploratory methods. Plot all the categories' turbulence each week for a year and see if some are consistently higher than most, or if there are jumps at key times (Christmas, the release of new Macs, MacWorld, etc.). Plot density plots to see how each category's turbulence is distributed and to compare them.
The particular kind of data you have is called "panel data", so you'll want to google that. Also, the tricky thing about time series is "autocorrelation": the tendency of weekly results to be related across weeks. And also "seasonal effects", which are repeating patterns (like stores selling more around Christmas).