Kenneth Chang

What specifically was happening in the mid-20th century that Labourie was ignoring?

The specific question you're objecting to appears to be this: "…most mathematicians in the middle of the 20th century did not think much of probability, which was at the bottom in the hierarchy of mathematics, below number theory, algebra and differential geometry."

yes, probability was entirely mainstream in mid-century, embraced as an important method by prestigious folks in many places, including in Labourie's France. It was important in both pure and applied work (a distinction that was being actively re-negotiated in this period).

Yes, and Labourie said that most mathematicians looked down on it, just as physicists think of engineering as a lesser academic discipline because it's just "applied physics." Probability is and was of course of great use, but it wasn't for most mathematicians.

yes, I, a professional historian of 20th century mathematics, am telling you that Labourie's claim is wrong. He has no evidence that "most mathematicians looked down on it" because most did not. The comparison to physics/engineering is also flawed, but that's another story

Who used dynamical systems for a number theory or group theory proof?

Kolmogorov, among others

Ok, I get your point, but think Labourie has a valid point too even if you don't like how he phrased it. How would describe how Furstenberg's and Margulis's work differed in nature from what proceeded it.