Cosmological deflation? Wikipedia tells us:


  In physical cosmology, cosmic inflation, cosmological inflation, or just inflation is the exponential expansion of space in the early universe. The inflationary epoch lasted from 10-36 seconds after the Big Bang to sometime between 10-33 and 10-32 seconds. Following the inflationary period, the Universe continues to expand, but at a less accelerated rate.


I've no idea about astronomy, I just wondered whether somebody ever published a theory about cosmological deflation, which would make the universe shrink in a fraction of a second. 

In a SFI novel you could travel by locally causing a deflation of space in the direction of your spaceship; the unshrunken space of a lightyear would (only locally in front of the ship) measure 1000 km for the fraction of the second. Or maybe a technically very advanced civilization could keep its solar system inflated by the time of a deflation of the whole universe and thus survive, which leads to the interesting question, whether this solar system could reenter the space of the next universe after its new cosmolocical inflation and if yes, does somebody look out for stellar systems at places in space where you wouldn't expect them?

However: cosmological deflation?
There are a number of flaws with your idea (despite the relativity breaking transatlantic journey times that would be possible).

Inflation is a popular theoretical model put forward to solve three main issues with the Big Bang cosmology: the horizon problem, the flatness problem &amp; the magnetic monopole problem (see the wiki for details). It consists of a scalar field called the inflaton, a physical field that pervades all space. During inflation it undergoes a phase transition to a lower energy state, releasing huge amounts of energy that drive the expansion. To force the field back to a higher energy state would require inconceivable amounts of energy, and have to be applied across the entire universe. Not only is this impossible, but also annoyingly prevents its use for local galactic travel.


  Or maybe a technically very advanced civilization could keep its solar system inflated by the time of a deflation of the whole universe 


I assume you mean contraction, as opposed to the current expansion that we see today due to dark energy. Again this would be unfeasible due to the reasons outlined above, and if contraction was to proceed to it's conclusion, i.e. a singularity like the Big Bang, then no information from this universe could be carried through that singularity, even if there was another universe on the other side.