Sorry, Tweetlogix does not offer its own push notification service. However, you can configure Boxcar to receive Twitter push notifications and open Tweetlogix. Checkout Boxcar in the App Store.

If you're not familiar with Boxcar, here's how to setup Boxcar to open Tweetlogix:

1. In the Boxcar app, tap Add a Service.
2. Scroll down and select Twitter Account.
3. Tap the Sign into Twitter button and login.
4. Select your desired message settings. For example, if you want to receive notifications just for mentions, turn on only the Mentions.
5. Scroll down to Advanced Settings and tap the Opens item. Select Tweetlogix from the list of clients.

When notified by Boxcar, Tweetlogix will automatically switch accounts/timelines and invoke a refresh.