start-ver=1.4 cd-journal=joma no-vol=305 cd-vols= no-issue= article-no= start-page=144 end-page=164 dt-received= dt-revised= dt-accepted= dt-pub-year=2017 dt-pub=20170110 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=The asymptotic behavior of Frobenius direct images of rings of invariants en-subtitle= kn-subtitle= en-abstract= kn-abstract= 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. en-copyright= kn-copyright= en-aut-name=HashimotoMitsuyasu en-aut-sei=Hashimoto en-aut-mei=Mitsuyasu kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=1 ORCID= en-aut-name=SymondsbPeter en-aut-sei=Symondsb en-aut-mei=Peter kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=2 ORCID= affil-num=1 en-affil=Department of Mathematics, Okayama University kn-affil= affil-num=2 en-affil=University of Manchester kn-affil= en-keyword=Frobenius direct image kn-keyword=Frobenius direct image en-keyword=Hilbert?Kunz multiplicity kn-keyword=Hilbert?Kunz multiplicity en-keyword=F-signature kn-keyword=F-signature en-keyword=Frobenius limit kn-keyword=Frobenius limit END