Connectedness & Canonical Transformations

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This is a question I found no answer from books.

Is Connectedness and Simply-connectedness preserved by Canonical Transformations?

If an area in phase space is connected (simply connected), will it still connected (simply connected) in the new phase space of new canonical variables?
 
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Isn't a canonical transformation continuous?
 

Thread 'Off resonance driving of a MW/RF cavity'

Hello! Let's say I have a cavity resonant at 10 GHz with a Q factor of 1000. Given the Lorentzian shape of the cavity, I can also drive the cavity at, say 100 MHz. Of course the response will be very very weak, but non-zero given that the Loretzian shape never really reaches zero. I am trying to understand how are the magnetic and electric field distributions of the field at 100 MHz relative to the ones at 10 GHz? In particular, if inside the cavity I have some structure, such as 2 plates...
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