Hi Aurélien,
To elaborate on Michael's reply: There isn't an out of box synchronization between the dates you need. The challenge in developing it on your own (if you are to undertake this), is how to design the data synchronization. The APIs can be of use but they are not the main issue here.
The main challenge in my opinion is to decide between an 'online' or 'offline' updates. What I mean by an 'online' update is behavior that is immediately reactive to updates on the user stories and will sync the Feature dates immediately. This is what Michael meant when said: "you would have to have something running constantly checking for updates.. .. ". There are a number of drawbacks to this approach, the most important one is the elevated performance of this solution, you may also need to consider that many/most of the stories are updated very frequently but probably the dates aren't updated as frequently. This approach will mean you will need to follow on each update, find out if it was the dates , if so then sync your Feature. It's much work and performance may take a hit. Technically speaking, going about this approach you may consider creating a webhook to fire on stories updates, then in your target page use the WSAPI to get the info of the story and decide if an update to the Feature is needed.
The 'offline' approach means you will develop an external component/program and schedule it to periodically find which stories had their dates updated which justifies an update to the Features. What you're compromising in this approach is these dates will not be in sync in-between the cycles of your sync app. With this approach you can schedule nightly scans/updates which perhaps will less interfere with users activity and performance/stability will be better preserved. It's a less intrusive approach and you won't need to create a webhook or initiate anything out of Rally itself. But, again, the drawback is your info is not always in sync. Technically speaking, if you went about this approach then using the Lookback API to find the stories that had their dates updates per the last period is what you'll want to use to figure out all you will need to act on (update the Features) since this ran last.
There is effort in either approach in terms of how to go about it.
I just wanted to point it all out in case you are going to consider it.
Sagi