Nuclear operators/spaces and transfer operators

I believe your question is: What is the importance of being nuclear of order zero for the transfer (=Perron-Frobenius) operator in Ergodic theory?

The importance of being nuclear is a theorem by Grothendieck: If $L$ is nuclear of order less or equal than $2/3,$ then $det(I-zL)$ is entire in $z$ and we have that $det(I-zL)=\prod_n(1-z\lambda_n)$ where $\lambda_n$ are the eigenvalues of $L.$ From this and the trace $tr(L)=\sum_n\lambda_n$ we can write $\log det(I-zL)=-\sum_n \frac{z^n}{n}Tr(L^n).$ This is quite helpfull because of many reasons. The main reason is that Ruelle succeeded to prove the nuclearity (of order zero) of the Perron-Frobenius operator associated to Axiom A diffeomorphisms in the seventies, so by Grothendieck's theorem he related the eigenvalues of the Perron-Frobenius operator with the inverses of the zeros of $\exp(-\sum_n \frac{z^n}{n}Tr(L^n)).$ This motivates many accurate approximation results in Ergodic theory by Pollicott, as eigenvalues of the Perron-Frobenius operator are known to be related to mixing properties of the associated dynamical system, and on the other hand, the inverses of the zeros of $\exp(-\sum_n \frac{z^n}{n}Tr(L^n))$ can be approximated using the Taylor expansion of holomorphic functions. A few examples of these mixing properties are the following:

1. Invariant densities are fixed points of the Perron-Frobenius operator.
2. If there exist an ergodic invariant density, then the first eigenvalue of the Perron-Frobenius operator is equal to 1 and it is simple.
3. If there exist a mixing invariant density, then the first eigenvalue of the Perron-Frobenius operator is equal to 1 and moreover, every other eigenvalue has modulus strictly less than 1.

I suggest you to read Pollicott, Baladi and Ruelle, in this order. They have some lecture notes available on the web. On the other hand, I do not think you will find anything useful directly from Grothendieck's 1956 notes, Ruelle did it!

Another reason of its importance (of $L$ being nuclear) it is that isolated eigenvalues are stable under perturbations (in Kato's sense). Therefore, as a consequence, one can usually use that $L$ is nuclear in order to prove stability results. For example, of the invariant density.

This is intended more as a long comment than an answer. My main experience of nuclearity in this area is that it is a key ingredient in the proofs of correctness (or the proof of the convergence rate) for some fast computational algorithms based on transfer operators, notably Mark Pollicott's algorithm for the calculation of Lyapunov exponents. Some papers which use nuclearity of transfer operators in this way include:

• Pollicott, Maximal Lyapunov exponents for random matrix products, Inventiones Mathematicae 181 (2010) 209 - 226
• Jenkinson and Pollicott, Calculating Hausdorff dimension of Julia sets and Kleinian limit sets, American Journal of Mathematics, 124 (2002) 495-545
• Jenkinson and Pollicott, Computing Invariant Densities and Metric Entropy, Comm. Math. Phys. 211 (2000), 687-703.
• Pollicott and Vytnova, Estimating singularity dimension, Math. Proc. Cambridge Philos. Soc. 158 (2015), 223–238.

I understand that nuclearity is critical to these arguments, but unfortunately I can't claim to understand these papers well enough to indicate exactly why it is necessary.

Another application of nuclearity is, I think, that it leads to quick proofs that certain dynamical zeta functions are meromorphic on the whole plane, a property which is related to some conjectures of Stephen Smale.