Reading your notes I’ve just remembered the following paper by Bochi and Berger, with a similar taste: https://arxiv.org/pdf/1901.03300.pdf

They develop a definition quantifying how large the set of ergodic measures is as a subset of the set of invariant measures.

Best,

Lucas. ]]>

Indeed, measurability of sections is not enough to guarantee measurability of the set itself: if is a nonmeasurable set and are distinct, then is nonmeasurable in but all of its cross-sections in vertical fibers are singletons, hence measurable.

In the case described in the blog post, you can prove measurability of directly, in roughly the same way you would prove it for the cross-sections. Define a measurable map by , let , and let ; observe that is a measurable function. Then , which can be rewritten as , hence is measurable.

]]>I’m not enough of an expert on cone techniques to give a definitive answer about condition 4, and I don’t know how it enters into the results mentioned here (if at all) beyond the superficial observation that it imposes some kind of topological restriction. The earliest use of cone techniques in dynamics that I know of is Liverani’s 1995 Annals paper (https://mathscinet.ams.org/mathscinet-getitem?mr=1343323), which includes a topological condition very similar to this and which lists Birkhoff’s 1940 book “Lattice Theory” (https://mathscinet.ams.org/mathscinet-getitem?mr=1959) and a 1988 paper in Memoirs of the AMS by Nussbaum (https://mathscinet.ams.org/mathscinet-getitem?mr=961211) as places to find further details. I have not looked at either of those references to see what role this condition plays.

]]>This just means that if you hold all the other variables fixed and then treat your function as a function of a single variable (the one that you do not fix), then it is continuous / smooth.

]]>Let me ask you: In (6), where you rewrite the correlation term with the transfer operator (using its definition), everything works fine integrating against Lebesgue as long as it is an invariant measure. However, it Lebesgue is not invariant, the (relevant) correlation should be expressed as an integral against an invariant measure, while the definition of the transfer operator will still be that integrating against Lebesgue — in which case the rephrasing in (6) may not work anymore.

Is this really it and the writing trick in (6)should be generalized in a different manner when Lebesgue is not invariant?

Best! ]]>