Entropy for Fukaya Categories
Joint work with Hanwool Bae, Dongwook Choa, Wonbo Jeong, Jongmyeong Kim, Sangjin Lee
One can study a symplectic manifold by applying a symplectomorphism iteratively and studying the resulting dynamical system. Our broad goal is to study the induced auto-equivalences on the various types of Fukaya categories.
A related concept is entropy. There is “categorical entropy” defined by Dimitrov-Haiden-Katzarkov-Kontsevich which measures the logarithmic growth of the “complexity” of an object under the iterations of an auto-equivalence. There is also Floer-theoretic entropy (resp. Hochschild entropy) which measures the logarithmic growth of the fixed point Floer homology (resp. Hochschid (co)homology) of the iterations of an auto-equivalence. One of our goals is to relate all of them where categorical entropy here is for Fukaya categories.
My contribution to the project is as follows: Given a compactly supported symplectomorphism on a Weinstein manifold, the categorical entropy of the induced auto-equivalence on the wrapped Fukaya category and partial wrapped Fukaya categories match. Note that by choosing a large stop, one can ensure that the partial wrapped Fukaya category is smooth and proper. In that case, categorical entropy can be described by the logarithmic growth of Hom spaces, which is comparable with the other types of entropies mentioned above. See Fel’shtyn and Kikuta-Ouchi for some of the known results.
Another goal is to define a new type of entropy for the monodromy of the regular fibre of a Lefschetz fibration and relate it to the categorical entropy of a particular functor.
Draft of the current result is available upon request.