Elsevier Acta Medica Okayama 0001-8708 305 2017 The asymptotic behavior of Frobenius direct images of rings of invariants 144 164 EN Mitsuyasu Hashimoto Department of Mathematics, Okayama University Peter Symondsb University of Manchester We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; we also calculate the Frobenius limit for the restriction of these to the ring of invariants. As an application, we generalize the description of the generalized F-signature of a ring of invariants by the second author and Nakajima to the modular case. No potential conflict of interest relevant to this article was reported.