Open Access
2013 Entropy and Flatness in Local Algebraic Dynamic
Mahdi Majidi-Zolbanin, Nikita Miasnikov, Lucien Szpiro
Publ. Mat. 57(2): 509-544 (2013).


For a local endomorphism of a noetherian local ring we introduce a notion of entropy, along with two other asymptotic invariants. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the affine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.


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Mahdi Majidi-Zolbanin. Nikita Miasnikov. Lucien Szpiro. "Entropy and Flatness in Local Algebraic Dynamic." Publ. Mat. 57 (2) 509 - 544, 2013.


Published: 2013
First available in Project Euclid: 12 December 2013

zbMATH: 1302.37059
MathSciNet: MR3114781

Primary: 13D40 , 14B25 , 37P05 , 37P55 , 37P99

Keywords: endomorphism of finite length , generalized Hilbert-Kunz multiplicity , Kunz' regularity criterion , Local algebraic dynamics , local entropy

Rights: Copyright © 2013 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.57 • No. 2 • 2013
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