Entropy and flatness in local algebraic dynamics

Abstract

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.

Equidistribution of preimages over nonarchimedean fields for maps of good reduction

“…In general, the interaction between flatness and dynamics is not entirely understood. This is the subject of the recent work [MZMS11], where it is proved that f is flat at a superattracting periodic point x ∈ X if and only if X is smooth that x.…”

Entropy in the category of perfect complexes with cohomology of finite length

“…That is, if f : pR, mq Ñ S is a flat homomorphism of commutative Noetherian local rings, then If dim R " dim S, then Question 1 has an affirmative answer. This is proved in [14,Corollary 16 and Proposition 20]. Question 1 has also an affirmative answer when φ and ψ, respectively, are the Frobenius endomorphisms of two local rings R and S of characteristic p ą 0.…”

“…Two types of entropies can be associated to an endomorphism of finite length φ : R Ñ R (see Definition 1.1). On one hand there is the local entropy of φ, denoted h loc pφq, defined in [14,Theorem 1]. On the other hand, there is a category-theoretical entropy defined in [5, Definition 2.1] for exact endofunctors of a triangulated category with generator.…”

On the growth of local intersection multiplicities in holomorphic dynamics: a conjecture of Arnold

“…It follows that e(f n * a i ) ≤ e(m s n r ) = (s n r) d e(m) = r d s dn , which grows subexponentially, completing the proof. We mention this because quantities such as that on the right hand side have been recently studied by Majidi-Zolbanin, Miasnikov, and Szpiro [MZMS13]. In their notation, the right hand side of this inequality is exactly w h (f ).…”

“…We mention this because quantities such as that on the right hand side have been recently studied by Majidi-Zolbanin, Miasnikov, and Szpiro [MZMS13]. In their notation, the right hand side of this inequality is exactly w h (f ).…”