Entropy and flatness in local algebraic dynamics

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.

The second and third authors received funding from the NSF Grants DMS-0854746 and DMS-0739346.

Introduction
Let ϕ : X Ñ X be a self-morphism of an algebraic or analytic variety X and assume that P X is fixed by ϕ.If ϕ is finite, or if P is totally ramified, then the local ring O X,P inherits a local endomorphism ϕ U P whose closed fiber is an artinian ring.This situation arises, for example, at the vertex of the affine cone over a projective variety with a polarized self-morphism, or at infinity for a polynomial map of the projective line P 1 .There is a recent literature on constructing examples of finite endomorphisms ϕ U P : O X,P Ñ O X,P that are not automorphisms, with pX, P q a normal singularity.References include [6, 6.2-6.3] and [16, 2.3-2.5].Other authors, see e.g.[11], have addressed the question of whether the existence of a finite surjective self-morphism of degree ¡ 1 of a normal variety imposes restrictions on its local geometry or the nature of its singularities.
In [17] Favre and Jonsson studied a number of local invariants of endomorphisms of local rings of germs of analytic functions over the complex numbers (also see [20]).Their study of these local invariants explains the obstructions encountered in establishing an equidistribution theorem for p1, 1q-currents, that extends results of Brolin [10], Lyubich [32], the third author, Ullmo and Zhang [48].The work of Favre and Jonsson has also been a key to many subsequent works, including [49], [14], and [39].
This paper will primarily be concerned with dynamical systems generated by endomorphisms of finite length of noetherian local rings.Definition 1.A local homomorphism f : pR, mq Ñ pS, nq of noetherian local rings is of finite length, if one of the following equivalent conditions holds: a) f pmqS is n-primary.b) If p is a prime ideal of S such that f ¡1 ppq m, then p n. c) If q is any m-primary ideal of R, then f pqqS is n-primary.Note that for a homomorphism of noetherian local rings, finite ñ integral ñ finite length and finite ñ quasi-finite ñ finite length.Also the composition of two homomorphisms of finite length is again of finite length.
Definition 2. A local endomorphism f : pR, mq Ñ pR, mq of a noetherian local ring R is called contracting if for every x m the sequence tf n pxqu n¥1 converges to 0 in the m-adic topology of R.
ring pR, mq: h loc pϕq lim Our first theorem asserts: Theorem 1.Let pR, m, ϕq be a local algebraic dynamical system.Suppose R is of dimension d and embedding dimension δ.Then a) The invariants h loc pϕq, v h pϕq and w h pϕq are finite and non-negative.
We call the main invariant h loc pϕq local entropy.Roughly speaking, in the theory of dynamical systems, entropy is a notion that measures the rate of increase in dynamical complexity as the system evolves with time [51, p. 313].Various forms of entropy exist in the literature.For instance, Adler, Konheim, and McAndrew introduced the notion of topological entropy in [1] for continuous maps of compact topological spaces.Measure-theoretic entropy was introduced by Kolmogorov in [28] and later improved by Sinaȋ in [47] for measure-preserving morphisms of probability spaces, and in [4] Bellon and Viallet introduced a notion of algebraic entropy for dominant rational self-maps of projective space.We show that our notion of local entropy shares many standard properties of topological entropy.For instance, writing hpϕq for entropy of a self-map ϕ of a space X, both local and topological entropies satisfy: 1) hpϕ t q t ¤ hpϕq for all t N, where ϕ t ϕ ¥ ϕ ¥ ¤ ¤ ¤ ¥ ϕ (t copies).
2) If Y X is a closed ϕ-stable subspace, then hpϕae Y q ¤ hpϕq.
3) If f : X Ñ X I is an isomorphism, then hpϕq hpf ¥ ϕ ¥ f ¡1 q. 4) If X Y i , i 1, . . ., m, where the Y i are closed ϕ-stable subspaces, then hpϕq max The various invariants that measure the complexity of a given system are often related (see, e.g., [33]).We compare h loc pϕq to degree of ϕ when R is a domain and ϕ : R Ñ R is finite.Particular attention is given to the local ring of the vertex of the affine cone of an integral projective variety equipped with a polarized self-morphism, for which we prove (see Proposition 40) an analogue of a result of Gromov in [23], where he showed topological entropy of a self-morphism of P n pCq is equal to logarithm of its topological degree.
The invariant v h pϕq has been studied by Favre and Jonsson in [19] and [17] in a different guise.In [19, Theorem A] they prove the remarkable result that if k is an arbitrary field and ϕ is a k-endomorphism of the ring k X, Y , then v h pϕq is the logarithm of a quadratic algebraic integer.A priori it is assumed in [19, Theorem A] that the characteristic of the field is equal to 0, but the method relies on the technique of key polynomials of [18, Appendix E], which is valid in arbitrary characteristic.Bellon and Viallet have conjectured in [4] that their notion of algebraic entropy for dominant rational self-maps of projective space is also always the logarithm of an algebraic integer (see also [46,Conjecture 8]).This conjecture is proved for monomial self-maps in [24,Corollary 6.4].It is, therefore natural to ask a similar question about the invariants h loc pϕq, v h pϕq and w h pϕq: Question 6.Let pR, ϕq be a local algebraic dynamical system.Are the invariants h loc pϕq, v h pϕq and w h pϕq always logarithms of algebraic integers?
An endomorphism ϕ : R Ñ R induces a self-morphism a ϕ of Spec R. When ϕ is integral and Spec R V pker ϕq, we show that a ϕ permutes the irreducible components of Spec R. As a result, irreducible components of Spec R are stable under an iteration of ϕ.Even when Spec R $ V pker ϕq, there exist irreducible components that are stable under an iteration of ϕ.
In the second part of this paper we study the case of regular local rings.We use local entropy to extend numerical conditions of Kunz' regularity criterion to arbitrary contracting endomorphisms of finite length.This result of Kunz was established as a converse to a result of Peskine and the third author [40,Theorem 1.7], stating that the pullback of a free resolution of a module of finite projective dimension by the Frobenius endomorphism is a free resolution of the pullback of the module.This statement is still mysterious for a general endomorphism of a local ring, even for its iterates.One should note, however, that it is true for resolutions of modules of finite length.(Apply the acyclicity lemma [40,Lemma 1.8].) The second theorem of this paper asserts: Theorem 2. Let pR, m, ϕq be a local algebraic dynamical system and let d be the dimension of R. Let h loc pϕq be the local entropy of this system and define q ϕ : expph loc pϕq{dq.Consider the following conditions: d) lengthpR{ϕ n pmqRq q nd ϕ for some n N. Then a) ñ b) ñ c) ñ d).If in addition ϕ is contracting, all these conditions are equivalent.
Avramov, Iyengar and Miller have proved the equivalence of conditions a) and b) (and more results) in [3] using different methods.We use Herzog's proof from [25,Satz 3.1] that is based on a nontrivial argument in local cohomology, to prove the implication b)ña).He originally wrote it for the Frobenius endomorphism.This part of our proof has been previously used by Bruns and Gubeladze in [12,Lemma 3].Our proof of the implication d) ñ b) utilizes a flatness criterion that is due to Nagata.
We propose a characteristic-free interpretation of definition of the Hilbert-Kunz multiplicity, see [37], in terms of local entropy.From Theorem 2 it quickly follows that the Hilbert-Kunz multiplicity of a regular local ring with respect to any endomorphism of finite length is 1.This is a well-known fact in the case of the Frobenius endomorphism.The existence of the Hilbert-Kunz multiplicity attached to an arbitrary endomorphism of finite length, however, remains an open question.
Our third theorem in this paper is inspired by results of Fakhruddin [15, Corollary 2.2], and Bhatnagar and the third author [5, Theorem 2.1] on extending a polarized self-morphism of a projective variety over an infinite field to an ambient projective space.Recently in [41] Poonen gave a proof for the main result of [5] over finite fields.Here we consider a similar lifting problem for an endomorphism of finite length of an equicharacteristic complete noetherian local ring, and prove a Cohen-Fakhruddin type structure theorem.Note that it is easy to see that if a local ring has a nonzero contracting endomorphism, it must be of equal characteristic.In our local version of Fakhruddin's result we do not assume our fields to be infinite.
Our third theorem asserts: Theorem 3. Let pA, ϕq be a local algebraic dynamical system and assume that A is a homomorphic image π : R A of an equicharacteristic complete regular local ring R. Then ϕ can be lifted (non uniquely) to an endomorphism of finite length ψ of R such that π ¥ ψ ϕ ¥ π.Thus, π : pR, ψq Ñ pA, ϕq becomes a morphism of local algebraic dynamical systems.
The converse problem of characterizing homomorphic images of a regular local ring R to which a given endomorphism of finite length ψ of R descends, is an open question.Equivalently, this question is asking for characterization of ideals a R for which ψpaq a.If a is such an ideal then ψ descends to give an endomorphism of finite length of R{a.
The results and questions that we discussed in this introduction, as well as the equidistribution results for the measure of maximal entropy obtained by Brolin [10], Lyubich [32], the third author, Ullmo and Zhang [48], have motivated us to study local and global invariants of schemes with self-morphisms.Example 8.A power series ring R : k X 1 , . . ., X n over a field k has many endomorphisms of finite length.If elements f 1 , . . ., f n of R generate an ideal of height n in R, then we obtain an endomorphism of finite length by setting X i Þ Ñ f i for 1 ¤ i ¤ n.Conversely, in Theorem 3 we will show using Cohen's structure theorem that every endomorphism of finite length of a complete equicharacteristic local ring is induced by an endomorphism of a power series ring.Definition 9. Let ϕ : R Ñ R be an endomorphism of a noetherian local ring.An ideal a of R is called ϕ-stable, if ϕpaqR a.

Local entropy
Example 10.Let R : k X 1 , . . ., X n be a power series ring over a field k, and let ϕ be an endomorphism of finite length of R, e.g., as defined in Example 8. Let z $ 0 be an arbitrary element of the maximal ideal of R. Then the ideal a generated by z, ϕpzq, ϕ 2 pzq, . . .(orbit of z under ϕ) is ϕ-stable.Thus ϕ induces an endomorphism of finite length on R{a.Moreover, if ϕ is contracting, then so is the induced map.Macaulay 2 can be used to generate examples of this type.We mention three examples here.In these examples k is a field of characteristic zero, R and a are as above and µp ¤ q denotes the minimum number of generators of a finitely generated R-module.
Then µpaq 5 and dim R{a 4.
Example 11.Let R : k X 1 , . . ., X n be a power series ring over a field k, and let a be an ideal of R with generators that can be expressed in the form monomial monomial.Then the endomorphism of R given by X i Þ Ñ X r i for an integer r ¡ 1, induces a contracting endomorphism of finite length on R{a.In more geometric terms this example says any singularity of a toric variety admits a contracting self-morphism.
1.2.Existence and estimates for local entropy.In this section we prove Theorem 1.We will need a lemma that is often used in dynamical systems.For a proof we refer to [50,Theorem 4.9].
Lemma (Fekete).Let ta n u n¥1 be a sub-additive (super-additive) sequence of real numbers, that is, a n m ¤ a n a m (resp.a n m ¥ a n a m ) for all n, m N. Then lim nÑV a n {n exists and equals inf n ta n {nu (resp.sup n ta n {nu).(The limit could be ¡V (resp.V) but if a n ¥ c ¤ n (resp.a n ¤ c ¤ n) for a number c ¡ 0 and all n N, then the limit will be finite.) We will also need the proposition that follows: Proposition 12. Let f : pR, mq Ñ S be a homomorphism of finite length of noetherian local rings.Let M be an R-module of finite length.Then a) M R S is of finite length as an S-module.b) length S pM R Sq ¤ length S pS{fpmqSq ¤ length R pMq.c) If f is flat, then length S pM R Sq length S pS{fpmqSq¤length R pMq.Proof: By induction on length R pMq.Definition 13.Let f : pR, mq Ñ pS, nq be a local homomorphism of finite length of noetherian local rings.We define lengthpf q lengthpS{f pmqSq vpf q maxtr | f pmqS n r u wpf q mintr | n r f pmqSu.Remark 14.The definition of vpf q and wpf q was inspired by similar definitions of Samuel in [43, p. 11].Note that n wpf q f pmqS n vpf q .Thus: vpf q ¤ wpf q.
Lemma 15.Let f : pR, mq Ñ pS, nq and g : pS, nq Ñ pT, pq be two homomorphisms of finite length of noetherian local rings.Write lengthpf q, lengthpgq, lengthpg ¥ f q for the lengths of the closed fibers of f , g and g ¥ f , respectively.Then Proof: For a) and b) apply Proposition 12, using the canonical isomorphism of T -modules pS{fpmqSq S T !T {g pfpmqSq T (see, e.g., [7, Chapter II, §3.6, Corollaries 2 and 3, pp.253-254]).For c) and d) using the fact that for an ideal a of S and n N, gpa n qT pgpaqTq n , we can write ppg ¥ f qpmqqT gpf pmqSqT gpn vpf q qT pgpnqTq vpf q R p vpgqvpf q , and p wpgqwpf q pgpnqTq wpf q gpn wpf q qT gpf pmqSqT ppg ¥ f qpmqqT.By definition of vpg ¥fq and wpg ¥fq then we obtain vpg ¥fq ¥ vpgq ¤vpfq and wpg ¥ f q ¤ wpgq ¤ wpf q, respectively.Theorem 1.Let pR, m, ϕq be a local algebraic dynamical system.Suppose R is of dimension d and embedding dimension δ.Define The invariants h loc pϕq, v h pϕq and w h pϕq are finite and non-negative.In addition, We call the invariant h loc pϕq local entropy of ϕ.
Proof: a) Apply Fekete's Lemma, taking a n to be log lengthpϕ n q, log wpϕ n q and log vpϕ n q, respectively.The sub-additivity of tlog lengthpϕ n qu and tlog wpϕ n qu and super-additivity of tlog vpϕ n qu were established in Lemma 15.By Lemma 15 and Remark 14, for every n N 1 ¤ rvpϕqs n ¤ vpϕ n q ¤ wpϕ n q ¤ rwpϕqs n .
These inequalities give the finiteness and non-negativity of the limits.
b) From Definition 13 we get m wpϕ n q ϕ n pmqR m vpϕ n q .Thus length R pR{m vpϕ n q q ¤ lengthpR{ϕ n pmqRq ¤ length R pR{m wpϕ n q q.We consider two cases: vpϕ n q Ñ V and vpϕ n q Û V.In the first case by Remark 14 wpϕ n q Ñ V, as well.Then, for large n, length R pR{m vpϕ n q q and length R pR{m wpϕ n q q are polynomials of precise degree d in vpϕ n q and wpϕ n q, respectively, with highest degree terms epmqpvpϕ n qq d {d! and epmqpwpϕ n qq d {d!.Thus, for large which gives the result after applying logarithm, dividing by n and taking limits.In the second case, when vpϕ n q Û V, the sequence tvpϕ n qu is bounded.Hence, there is a number c such that 1 ¤ vpϕ n q ¤ c.Applying logarithm, dividing by n and taking limits, we get v h pϕq 0. Now, if wpϕ n q Ñ V, then starting with the inequality 1 ¤ lengthpR{ϕ n pmqRq ¤ length R pR{m wpϕ n q q and repeating the same argument as before, we arrive at the desired inequality , then the sequence twpϕ n qu is also bounded and there is a number c I such that 1 ¤ wpϕ n q ¤ c I .After applying logarithm, dividing by n and taking limits, get w h pϕq 0. Since v h pϕq 0 as well, the proof will be completed by showing h loc pϕ, Rq 0. This follows from inequalities 1 ¤ lengthpR{ϕ n pmqRq ¤ length R pR{m wpϕ n q q ¤ length R pR{m c 1 q.c) There is nothing to prove if δ 0, so assume δ ¡ 0. By Lemma 4 to say that ϕ is contracting is equivalent to saying ϕ δ pmqR m 2 , which by definition gives vpϕ δ q ¥ 2. Hence, using parts a) and b) we obtain d) If R is of characteristic p and ϕ is its Frobenius endomorphism, then by [29,Proposition 3.2] p nd ¤ lengthpR{ϕ n pmqRq ¤ min ty1,...,y d u rlength R pR{py 1 , . . ., y d qRqs ¤ p nd , where ty 1 , . . ., y d u runs over all systems of parameters of R. Apply logarithm, divide by n and take limits to get h loc pϕq d ¤ log p. Corollary 16.Let pR, m, ϕq be a local algebraic dynamical system.If dim R is zero, then h loc pϕq 0. Conversely, if ϕ is contracting and h loc pϕq 0, then dim R is zero.Proof: If dim R 0, then R is artinian and the result follows from inequalities 1 ¤ lengthpR{ϕ n pmRqq ¤ lengthpRq V.The converse statement follows from part c) of Theorem 1.
Local entropy can be computed using other methods.To show this we need a definition.Definition 17.Let R be a noetherian local ring, and let ϕ be an endomorphism of R. Let R-Mod be the category of R-modules.For every n N we define a functor Φ n : R-Mod Ñ R-Mod as follows: if M R-Mod, then Φ n pMq : M R ϕ n R, where the R-module structure of Φ n pMq is defined to be r ¤ x ¸mi r ¤ r i , if x ¸mi r i Φ n pMq and r R.
For the Frobenius endomorphism the functors defined in Definition 17 are known as Frobenius functors.They were first introduced in [40, Definition 1.2].Properties of Frobenius functors were established in [40] and [25].The same proofs can be re-written for the functors Φ n .Thus, these functors share similar properties with Frobenius functors.Proposition 18.Let pR, ϕq be a local algebraic dynamical system.If M is a nonzero module of finite length, then [40, p. 54] or [25, no. 2.6]).Thus, length R pΦ n pR{mqq length R pR{ϕ n pmqRq.
Since M is of finite length, there is a surjection M Ñ R{m Ñ 0. Apply the functor Φ n to obtain a surjection Φ n pMq Ñ Φ n pR{mq Ñ 0. Then using Proposition 12b) length R pΦ n pR{mqq¤length R pΦ n pMqq¤lengthpR{ϕ n pmqRq¤length R pMq.
The result follows after applying logarithm, dividing by n and letting n Ñ V.
1.3.Properties of local entropy.Many of the properties that we will establish for local entropy in this section are also shared by topological entropy.
Proposition 19.Let pR, m, ϕq be a local algebraic dynamical system and let r N. Then h loc pϕ r q r ¤ h loc pϕq.

Proof: By definition of local entropy
Proposition 20.Let f : pR, m, ϕq Ñ pS, n, ψq be a morphism between two local algebraic dynamical systems.Assume that f is of finite length.Then Apply logarithm, divide by n and take limits as n approaches infinity.
b) If f is flat, using Lemma 15a) and b) we obtain Apply logarithm, divide by n and take limits as n approaches infinity.For the inequality in the other direction we use part a).
With regard to Proposition 20, C. Huneke has asked us the following question: Question 21.Let f : pR, m, ϕq Ñ pS, n, ψq be a morphism, not necessarily of finite length, between two local algebraic dynamical systems.
The ideal f pmqS is easily seen to be ψ-stable and ψ induces an endomorphism ψ : S{f pmqS Ñ S{f pmqS that is of finite length.Corollary 23.Consider homomorphisms of finite length f : pR, mq Ñ pS, nq and g : pS, nq Ñ pR, mq of noetherian local rings.Then h loc pg ¥ f q h loc pf ¥ gq.Proof: f : pR, g ¥ f q Ñ pS, f ¥ gq and g : pS, f ¥ gq Ñ pR, g ¥ f q are morphisms between local algebraic dynamical systems.By Proposition 20 h loc pf ¥ gq ¤ h loc pg ¥ f q and h loc pg ¥ f q ¤ h loc pf ¥ gq.Corollary 24 (Invariance).Let pR, mq and pS, nq be noetherian local rings.Suppose f : R Ñ S is an isomorphism, and let ϕ be an endomorphism of of finite length of R. Then h loc pf ¥ ϕ ¥ f ¡1 q h loc pϕq.Proof: Apply Corollary 23 to homomorphisms f ¥ϕ: R ÑS and f ¡1 : S Ñ R.
Corollary 25.Let pR, ϕq be a local algebraic dynamical system and let a be a ϕ-stable ideal of R. Write ϕ and φ for endomorphisms induced by ϕ on R{a and R{ϕpaqR, respectively.Then h loc pϕq h loc p φq. Proof: Let f : R{a Ñ R{ϕpaqR and g : R{ϕpaqR Ñ R{a be homomorphisms induced by ϕ and the identity map of R. Apply Corollary 23.
The next two lemmas will be used in our proof of Proposition 28.Lemma 27.Let pR, m, ϕq be a local algebraic dynamical system.Let a 1 , . . ., a s be a collection of not necessarily distinct ϕ-stable ideals of R.
Let ϕ and ϕ i be the endomorphisms induced by ϕ on R{ ± i a i and R{a i , respectively.Then Proof: We proceed by induction on s, the number of ideals, counting possible repetitions.There is nothing to prove if s 1, so suppose s 2. We may assume a 1 a 2 0; if not, we can replace R with R{a 1 a 2 without loss of generality.Then a 2 is a finitely generated pR{a 1 q-module.
Hence for some integer r, we have an exact sequence Tensoring by R{ϕ n pmqR and taking lengths we get: lengthpϕ n q ¤ r ¤ lengthpϕ n 1 q lengthpϕ n 2 q.Now let n Ñ V and apply Lemma 26 to obtain h loc pϕq ¤ maxth loc pϕ 1 q, h loc pϕ 2 qu.On the other hand by Proposition 20a) we know that h loc pϕq ¥ h loc pϕ i q, for each i.This establishes the case s 2. The general case follows easily by induction.
Our next result shows that if all minimal prime ideals of a noetherian local ring R are stable under an endomorphism of the ring, then the local entropy is equal to the maximum local entropies of the endomorphisms induced on irreducible components of Spec R.
Proposition 28.Let pR, m, ϕq be a local algebraic dynamical system.Suppose all minimal prime ideal of R are ϕ-stable and for each p i Specmin R, let ψ i be the endomorphism induced by ϕ on R{p i .Then (1) h loc pϕq maxth loc pψ i q | p i Specmin Ru.Proof: Let Specmin R tp 1 , . . ., p s u and let a ± i p i .Then a is contained in the nilradical of R, hence a N p0q for some N , that is, R R{a N .Apply Lemma 27 to obtain h loc pϕq maxth loc pψ i q | p i Specmin Ru.
1.4.Reduction to equal characteristic.The main result in this section shows that computing local entropy in mixed characteristic can be reduced to the case of equal characteristic p ¡ 0. For a given local algebraic dynamical system pR, m, ϕq, we define (2) S : V n1 ϕ n pRq and n : V n1 ϕ n pmq.Lemma 30.Let pR, m, ϕq be a local algebraic dynamical system.Let S and n be as defined in equation (2), and let a be the ideal generated by n in R. Then a) S is a local subring of R with maximal ideal n. b) a is a ϕ-stable ideal of R.
c) If ϕ is in addition injective, then ϕpaqR a.
Proof: a) It is immediately clear that S is a subring of R and that n is an ideal of S. To show that n is the (only) maximal ideal of S, consider an element s Szn.Since s n, there is an n 0 such that s ϕ n0 pmq.In fact, since for n ¥ n 0 , ϕ n pmq ϕ n0 pmq, we see that s ϕ n pmq for all n ¥ n 0 .Hence, there are units y n Rzm such that s ϕ n py n q for all n ¥ n 0 .
Since s is clearly a unit in R, it has a unique multiplicative inverse s ¡1 in R. From uniqueness of multiplicative inverse it immediately follows that we must have s ¡1 ϕ n py ¡1 n q, for all n ¥ n 0 .Hence, s ¡1 S, that is, s is also a unit in S.
b) Note that by its definition, a has a set of generators x 1 , . . ., x g n.So ϕpaqR can be generated by ϕpx 1 q, . . ., ϕpx g q and it suffices to show that each ϕpx i q is in a.Since x i n, there is a sequence of element y i,n m such that x i ϕpy i,1 q ¤ ¤ ¤ ϕ n py i,n q ¤ ¤ ¤ .Thus, ϕpx i q ϕ 2 py i,1 q ¤ ¤ ¤ ϕ n 1 py i,n q ¤ ¤ ¤ , showing that ϕpx i q n a.
c) Now suppose ϕ is injective.To show ϕpaqR a it suffices to show that each x i is in ϕpaq.Since x i n, there is a sequence of element y i,n m such that x i ϕpy i,1 q ¤ ¤ ¤ ϕ n py i,n q ¤ ¤ ¤ .Since x i ϕpy i,1 q, we will be done by showing that y i,1 n.By injectivity of ϕ, y i,1 ϕpy i,2 q ¤ ¤ ¤ ϕ n¡1 py i,n q ¤ ¤ ¤ , which means y i,1 n.Remark 31.Let pR, m, ϕq be a local algebraic dynamical system and let n be as defined in equation ( 2).If n p0q, then by Lemma 30 the ring R contains a field and is of equal characteristic.As noted in [2, Remark 5.9, p. 10], this occurs, for example, if ϕ is a contracting endomorphism.
Proposition 32.Let pR, m, ϕq be a local algebraic dynamical system.
Let a be the ideal of R defined in Lemma 30, and let ϕ be the endomorphism induced by ϕ on R{a.Then Proof: a) Note that ϕ n pmqR a for all n ¥ 1. Hence ϕ n pmqR a ϕ n pmqR, showing that lengthpR{pϕ n pmqR aqq lengthpR{ϕ n pmqRq, giving the result.b) With reference to Lemma 30, the image of the subring S of R in R{a is a field, because it's maximal ideal n is contained in a and is mapped to 0. Hence R{a contains a field and must be a local ring of equal characteristic p ¡ 0, as its residue field is of characteristic p ¡ 0.
1.5.Local entropy and degree.The analogy between local and topological entropies also extends to their relation to degree.Misiurewicz and Przytycki showed in [36], that if f is a C 1 self-map of a smooth compact orientable manifold M , then For a holomorphic self-morphism f of P n pCq, Gromov showed in [23] h top pfq log |degpfq|.
Here degpf q is the topological degree of f .In this section we obtain similar formulas for finite endomorphisms of local domains, relating their local entropy to degree.For local Cohen-Macaulay domains we prove an analog of Gromov's formula.Definition 33.Let f : R Ñ S be a finite homomorphism of noetherian local rings.Assume that R is a domain.Then by degree of f , degpf q, we mean the rank of the R-module f S. Lemma 34.Let f : pR, mq Ñ pS, nq be a finite homomorphism of noetherian local rings with residue fields k R and k S , respectively.Assume that R is a domain.Let q be an m-primary ideal of R. Then χpϕ n px 1 q, . . ., ϕ n px d q; Rq q nd ϕ ¤ χpx 1 , . . ., x d ; Rq. c) Equation (4) holds for some system of parameters of R and some n N. Proof: Let tx 1 , . . ., x d u be a system of parameters of R and let q be the m-primary ideal that they generate.By [45, Chapter IV, Theorem 1] epqq χpx 1 , . . ., x d ; Rq.Since tϕ n px 1 q, . . ., ϕ n px d qu is also a system of parameters of R, the result quickly follows from equation (3) in Lemma 34.Remark 37. When f : pR, m, k R q Ñ pS, n, k S q is a finite homomorphism of local rings, then rf k S : k R s ¤ length S pS{mSq is the minimal number of generators of f S over R, by Nakayama lemma.Proof: a) By Theorem 1 we know v h pϕq sup n tplog vpϕ n qq{nu.Hence, to show the first inequality it suffices to show vpϕ n q d{n ¤ degpϕq{rϕ k : ks.Since ϕ n pmqR m vpϕ n q , with the aid of Lemma 34 we obtain epmqvpϕ n q d epm vpϕ n q q ¤ epϕ n pmqRq epmq ¤ degpϕ n q rϕ n k : ks .
For the second inequality, let µpϕ n Rq, or simply µ, be the minimum number of generators of the R-module ϕ n R. Localizing the surjec- tion R µ ϕ n R at p0q we see rank ϕ n R ¤ µpϕ n Rq.On the other hand, as mentioned in Remark 37, we have µpϕ n Rq rϕ k : ks n ¤ lengthpR{ϕ n pmqRq.Since by definition of degree, rank ϕ n R degpϕ n q pdegpϕqq n , we conclude pdegpϕqq n {rϕ k : ks n ¤ lengthpR{ϕ n pmqRq.Apply logarithm, divide by n and take limits as n Ñ V.
This inequality together with the inequality in a) give the desired equality.
1.6.Entropy at the vertex of the cone over a projective variety.
The following result, which will be used in this section was proved in [5, Theorem 2.1] by Bhatnagar and the third author over infinite fields and in [41,Theorem 1.4] by Poonen over finite fields.
Theorem.Let X be a projective variety defined over a field k, ı : X ã Ñ P N k a given embedding and L : ı O P N k p1q.Let ϕ : X Ñ X be a selfmorphism that is polarized by L, that is, ϕ pLq !L q for an integer q ¥ 2. Then there exists a positive integer r and a finite morphism Keeping the notation as above, let a be the largest homogeneous ideal in krX 0 , . . ., X N s defining ıpXq and let R : krX 0 , . . ., X N s{a.Also let R pqq : À n¥0 R nq be the qth Veronese subring of R. Using the above theorem, we can fix an r N such that ϕ r extends to a finite self- morphism P N k Ñ P N k .The proof of this theorem in [5], [41] shows that this self-morphism is given by N 1 forms F 0 , . . ., F N of degree q r that have no non-trivial common zeros in the algebraic closure of k.The assignment X i Þ Ñ F i defines a finite endomorphism ψ : R Ñ R that can be factored as R Ñ R pq r q ã Ñ R, where R pq r q ã Ñ R is inclusion and R Ñ R pq r q is a graded homomorphism that induces ϕ r on X Proj R. We want to calculate the local entropy of the local endomorphism induced by ψ at the vertex of the affine cone Spec R over X.
Remark 39.While the self-morphism P N k Ñ P N k obtained from the theorem is not unique, the endomorphism ψ : R Ñ R is unique up to a scalar multiple.
Proposition 40.Let X be an integral projective variety of dimension d over a field k with a given embedding ı : X ã Ñ P N k and set L : ı O P N k p1q.Let ϕ : X Ñ X be a self-morphism and assume that ϕ pLq !L q for an integer q ¥ 2. Let r, R and ψ be as defined in the previous paragraph and let h loc pψq be the local entropy of ψ at the vertex of the affine cone Spec R over X.Then (5) h loc pψq log degpψq pd 1q log q r .Proof: By Theorem 1b) and Proposition 38a) it suffices to show v h pψq ¥ log q r and w h pψq ¤ log q r .For the first inequality, note that as discussed prior to this proposition, ψ is induced by assignments X i Þ Ñ F i for N 1 homogeneous forms F 0 , . . ., F N of degree q r in the variables X i .Thus, ψ n is given by forms of degree q nr .This shows, with notations of Theorem 1, that vpψ n q ¥ q nr .Hence To prove the second inequality we use elimination theory: by [31, Corollary to Theorem 1, p. 169] we get xX 0 , . . ., X N y s xF 0 , . . ., F N y, where s q r pN 1q ¡ N .Thus, wpψ n q ¤ q nr pN 1q ¡ N and we obtain w h pψq lim nÑV 1 n log wpψ n q ¤ log q r .This concludes the proof.

Integral endomorphisms.
In this section we study local algebraic dynamical systems pR, ϕq generated by integral endomorphisms.We begin with a simple observation.Let pR, ϕq be a local algebraic dynamical system.Then for every n N, ϕ pker ϕ n q ker ϕ n¡1 ker ϕ n .
Hence ϕ induces a local endomorphism of R{ ker ϕ n .
Proposition 41.Let pR, m, ϕq be a local algebraic dynamical system.Let ϕ n be the local endomorphism induced by ϕ on R{ ker ϕ n , n N. Then Proof: a) Apply Corollary 25 to the endomorphism ϕ n of R, taking ker ϕ n as the ideal a in that corollary.Since ϕ n pker ϕ n qR p0q, by that corollary h loc pϕ n n q h loc pϕ n q.Now use Proposition 19.
b) This is clear (see [8, Chapter V, Proposition 2, p. 305]).c) R is noetherian, so the ascending chain ker ϕ ker ϕ 2 ker ϕ 3 ¤ ¤ ¤ is stationary.Let n 0 be such that ker ϕ n ker ϕ n 1 for n ¥ n 0 .We will show that if n ¥ n 0 , then ϕ n : R{ ker ϕ n Ñ R{ ker ϕ n is injective.Let x R{ ker ϕ n .Saying ϕ n pxq 0 is equivalent to saying ϕpxq ker ϕ n , which is equivalent to saying x ker ϕ n 1 .Since ker ϕ n 1 ker ϕ n , we see that x ker ϕ n , or x 0 in R{ ker ϕ n .Thus, ϕ n is injective.Proposition 42.Let pR, m, ϕq be a local algebraic dynamical system, where ϕ is integral.Let a be the ideal obtained as the stable limit of the ascending chain of ideals ker ϕ ker ϕ 2 ker ϕ 3 ¤ ¤ ¤ .Write a ϕ for the self-morphism of Spec R induced by ϕ.Then a) a ϕ permutes the minimal prime ideals of a. b) Every minimal prime ideal of a is in Specmin R.
c) Let ϕ : R{a Ñ R{a be the endomorphism induced by ϕ.Assume that the permutation in part a) is of order p.For a minimal prime ideal p i of a let ψ i be the endomorphism induced by ϕ p on R{p i .Then h loc pϕq 1 p ¤ max th loc pψ i q | p i is a minimal prime of au .
Proof: We recall that if a ring S is integral over a subring R, then over every prime ideal p of R there lies a prime ideal q of S.Moreover, if p is a minimal prime ideal, then so is q (see [35,Theorem 9.3]).
a) By Proposition 41, the endomorphism ϕ : R{a Ñ R{a is integral and injective.Apply the fact that we recalled above and note that a has only a finite number of minimal prime ideals.(Any surjective map from a finite set to itself is also injective.) b) Let n be large enough so that ker ϕ n a. Then ϕ n induces an integral injection φn : R{a ãÑ R and we have a commuting diagram If q is a minimal prime ideal of a, then by part a) (applied to ϕ n ) the ideal p : pϕ n q ¡1 pqq is a minimal prime of a. Thus, p φn q ¡1 pqq p{a.
We apply the fact that we recalled at the beginning of the proof to the ring injection φn : R{a ã Ñ R and conclude q Specmin R. c) Let n be large enough so that ker ϕ np a.By Proposition 41a) we know that h loc pϕ np q h loc pϕ np q.Since all minimal prime ideals of R{a are ϕ np -stable, by Proposition 28 we obtain h loc pϕ np q max th loc pψ n i q | p i is a minimal prime of au .

Now an application of Proposition 19 quickly concludes the proof.
Corollary 43.Let pR, ϕq be a local algebraic dynamical system.Assume that ϕ is integral and Spec R V pker ϕq.Then a) a ϕ permutes the minimal prime ideals of R.
b) If p Specmin R is a prime ideal of R, then ϕ ¡1 ppq Specmin R. c) An element x R belongs to a minimal prime ideal of R, if and only if ϕpxq belongs to a minimal prime ideal of R.
d) Assume that the permutation in part a) is of order p.For p i Specmin R let ψ i be the endomorphism induced by ϕ p on R{p i .Then h loc pϕq 1 p ¤ maxth loc pψ i q | p i Specmin Ru.Proof: a) Since ϕ is integral, imagep a ϕq V pker ϕq.Thus, from the hypothesis V pker ϕq Spec R we see that a ϕ is surjective.Hence, a ϕ n is also surjective for every n N, i.e., V pker ϕ n q imagep a ϕ n q Spec R.So for every n N, the set of minimal prime ideals of ker ϕ n is equal to Specmin R. Take n large enough and apply Proposition 42a).b) Suppose q : ϕ ¡1 ppq Specmin R. If p is not minimal, it contains a minimal prime ideal p I .Moreover, ker ϕ p I .By part a), ϕ ¡1 pp I q Specmin R. Since q ϕ ¡1 pp I q and q is minimal, we must have q ϕ ¡1 pp I q.This is a contradiction, because there can be no inclusion between prime ideals that lie over q in the integral ring inclusion ϕ : pR{ ker ϕq ã Ñ R.
c) Let x be an element of R. If ϕpxq p for some p Specmin R, then x ϕ ¡1 ppq.By part a), ϕ ¡1 ppq Specmin R. Conversely, suppose x q for some q Specmin R. Then by part a) there is a p Specmin R such that q ϕ ¡1 ppq.Hence ϕpxq p. d) Since all minimal prime ideals of R are ϕ p -stable, by Proposition 28, h loc pϕ p q maxth loc pψ i q | p i Specmin Ru.The result quickly follows from Proposition 19.

Regularity, flatness and entropy
In the second part of this paper we will present proofs of Theorems 2 and 3. Let pR, mq be a noetherian local ring of positive prime characteristic p and dimension d, and let ϕ be the Frobenius endomorphism of R. In [29] Kunz showed that the following conditions are equivalent: Later Rodicio showed in [42], that these conditions are also equivalent to At first glance, Kunz' conditions c) and d) may appear to be stated in terms of the characteristic p of the ring and one may not expect to be able to extend, or even state them in arbitrary characteristic.Nevertheless, local entropy can be used to make sense of Kunz' numerical conditions c) and d) for all endomorphisms of finite length in any characteristic.Theorem 2 states that with this new interpretation, all conditions in Kunz' result are equivalent.
We should note that in [3, Theorem 13.3] Avramov, Iyengar and Miller have extended the equivalence of conditions a) and b) of Kunz and e) of Rodicio to arbitrary contracting local endomorphisms of noetherian local rings.
We list two results here that we will need in our proof of Theorem 2.
Lemma 44 ([25, Lemma 3.2]).Let pR, mq be a noetherian local ring, and let M be a finitely generated R-module.Consider an ideal b m of R. Then there exists an integer µ 0 ¥ 0 such that depthpm, b µ M q ¡ 0 for all µ ¥ µ 0 .
Remark 45.In using Lemma 44 we must pay particular attention to the standard convention that the depth of the zero module is V (see, e.g., [26, p. 291]).Otherwise, if M is an R-module of finite length, then for µ 4 0 we have m µ M p0q, and this would have been a counterexample to Lemma 44.
Proposition 46.Let R be a noetherian ring and let a be an ideal of R.
Let 0 Ñ M I Ñ M Ñ M P Ñ 0 be an exact sequence of R-modules.If we define d I depthpa, M I q, d depthpa, M q, and d P depthpa, M P q, then one of the following mutually exclusive possibilities holds: 2.1.Kunz' regularity criterion via local entropy.Before we give the proof of Theorem 2 we need to establish two lemmas.We begin with a flatness criterion that is due to Nagata.A proof can be found in [38, Chapter II, Theorem 19.1].See also [35,Exercise 22.1,p. 178].
Theorem (Nagata).Let g : pR, mq Ñ pS, nq be an injective homomorphism of finite length of noetherian local rings.Then S is flat over R, if and only if for every m-primary ideal q of R, length R pR{qq ¤ length S pS{gpmqSq length S pS{gpqqSq.
We need a stronger version of Nagata's criterion that we state and prove here.Lemma 47.Let g : pR, mq Ñ pS, nq be a homomorphism of finite length of noetherian local rings.If equation (6) holds for a family of m-primary ideals tq α u αA that define the m-adic topology, then it holds for all m-primary ideals.
Proof: Let q be an m-primary ideal.We will show equation ( 6) holds for q.First, using Proposition 12 length S pS{gpqqSqlength S pS R R{qq¤ length S pS{gpmqSq¤length R pR{qq.
To show the reverse inequality, note that by assumption there is a q α q.The exact sequence 0 Ñ q{q α Ñ R{q α Ñ R{q Ñ 0 yields (7) length R pR{q α q length R pR{qq length R pq{q α q.
If we tensor the previous exact sequence with S, we obtain an exact sequence of S-modules pq{q α q R S Ñ S{gpq α qS Ñ S{gpqqS Ñ 0. Thus length S pS{gpq α qSq ¤ length S pS{gpqqSq length S ppq{q α q R Sq.
Since equation ( 6) holds for q α , and using Proposition 12 we quickly see length R pR{q α q¤length S pS{gpmqSq¤length S pS{gpqqSq length R pq{q α q¤length S pS{gpmqSq.
Now using equation ( 7) we obtain length S pS{gpmqSq ¤ length R pR{qq ¤ length S pS{gpqqSq.Lemma 48.Let pR, m, ϕq be a local algebraic dynamical system, and let a be a ϕ-stable ideal of R. Let ϕ be the endomorphism of R{a induced by ϕ.Set d : dim R and ρ : dim R{a and let q ϕ : expph loc pϕq{dq.Assume that lengthpR{ϕ n pmqRq q nd ϕ for an integer n N. Then i) lengthpR{ϕ nt pmqRq q ntd ϕ for all t N. ii) If in addition h loc pϕq h loc pϕq and ϕ is contracting, then a p0q.Proof: i) Fix t N. As the sequence tplog lengthpϕ nt qq{pntqu by Theorem 1 converges to its infimum, we have From this inequality we obtain q ntd ϕ ¤ lengthpR{ϕ nt pmqRq.By Lemma 15a) lengthpR{ϕ nt pmqRq ¤ plengthpR{ϕ n pmqRqq t .
Using our hypothesis and previous inequalities we obtain Hence, lengthpR{ϕ nt pmqRq q ntd ϕ for all t N.
ii) Similar to the previous part, we can write From our hypothesis in ii) we know q ρ ϕ q d ϕ .Thus, from equation ( 8) we can conclude (9) lengthpR{pϕ nt pmqR aqq lengthpR{ϕ nt pmqRq, d t N. The surjection R{ϕ nt pmqR Ñ R{pϕ nt pmqR aqq Ñ 0 and equation (9) then show Hence, where the last equality follows from Lemma 4 because ϕ is by assumption, contracting.
Theorem 2. Let pR, m, ϕq be a local algebraic dynamical system and let d be the dimension of R. Let h loc pϕq be the local entropy of this system and define q ϕ : expph loc pϕq{dq.Consider the following conditions: d) lengthpR{ϕ n pmqRq q nd ϕ for an integer n N. Gubeladze have also used this proof in [12,Lemma 3].We include it here for completeness.To show that R is regular, it suffices to show all finitely generated R-modules have finite projective dimension.So let M be a finitely generated R-module.Suppose M were of infinite projective dimension.Consider a minimal (infinite) free resolution of M L Ñ M Ñ 0. Let s : depthpm, Rq, and take an R-regular sequence of elements tx 1 , . . ., x s u in m.Write a for the ideal generated by this regular sequence.(If s 0, take a p0q.)Let Φ n be the functor defined in Definition 17.For every n N we set Using properties of Φ n , see [40] or [25], it is easy to see that C n i !L i {aL i and B n i ϕ n pmqC n i .Thus, for every i the module C n i is independent of n and is nonzero, finitely generated and of depth zero.Applying Lemma 44, let µ i0 be such that depthpm, m µi 0 C n i q ¡ 0. Since ϕ is contracting, using Lemma 4 if n is large enough then ϕ n pmqR m µi 0 , hence B n i ϕ n pmqC n i m µi 0 C n i .This shows that depthpm, B n i q ¡ 0 for large n.On the other hand, since ϕ is flat, Φ n pL q is exact.Thus, using properties of Φ n again, we see that Φ n pL q Ñ Φ n pMq Ñ 0 is a minimal (infinite) free resolution of Φ n pMq.Hence H i pC n q Tor R i pΦ n pMq, R{aq 0, for i ¡ s.This shows that if i ¡ s, then the sequences (10) 0 Ñ B n i 1 Ñ C n i 1 Ñ B n i Ñ 0 are exact for all n N. Take i s 1 in sequence (10), for instance.By the above argument, if we take n large enough, we will obtain depthpm, B n s 1 q ¡ 0 and depthpm, B n s 2 q ¡ 0, while depthpm, C n s 2 q 0. By Proposition 46 this is not possible.Hence, the projective dimension of M must be finite.d) ñ b) We will use Nagata's Flatness Theorem to show that ϕ n is flat.We first need to show that ϕ is injective.Clearly ker ϕ is ϕ-stable.
Let ϕ be the local endomorphism induced by ϕ on R{ ker ϕ.Then by Proposition 41, h loc pϕq h loc pϕq.By assumption, lengthpR{ϕ n pmqRq q nd ϕ for an integer n N. From Lemma 48 it follows that ker ϕ p0q.
Now since ϕ is contracting, using Lemma 4 we quickly see that the family tϕ nt pmqRu tN defines the m-adic topology of R. By Lemma 47 it suffices to verify equation (6) for this family of m-primary ideals.We need to show length R R{ϕ n pϕ nt pmqqR ¨ length R R{ϕ nt pmqR ¨¤length R R{ϕ n pmqR ¨.
Using Lemma 48, this equality holds, if and only if q npt 1qd ϕ q ntd ϕ ¤ q nd ϕ .Since this equality holds trivially, by Nagata's Flatness Theorem ϕ n is flat.The implication b) ñ a) applied to ϕ n then tells us that R is regular, and the implication a) ñ b) shows that ϕ is flat, as well.Remark 49.There exist normal singularities pX, P q such that O X,P admits a finite contracting endomorphism, see [6, or [16,.In this case by Theorem 2 the endomorphism is not flat.This gives examples of finite maps between normal varieties which are not flat.
2.2.Generalized Hilbert-Kunz multiplicity.Following ideas of Kunz, Monsky in [37] defined the Hilbert-Kunz multiplicity for the Frobenius endomorphism of noetherian local rings of positive prime characteristic.He then showed that in this case, Hilbert-Kunz multiplicity always exists.Since then, it has become evident through works of various authors, that the Hilbert-Kunz multiplicity provides a reasonable measure of the singularity of the local ring.Here, inspired by Theorem 1d), we propose a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity associated with an endomorphism of finite length.
The Hilbert-Kunz multiplicity of R with respect to ϕ is defined as (11) e ϕ HK pRq : lim nÑV lengthpR{ϕ n pmqRq q nd ϕ , provided that the limit exists.
Remark 51.We do not know whether the limit in equation ( 11) always exists or not.Nevertheless, the next corollary shows that in the case of a regular local ring the Hilbert-Kunz multiplicity is 1, as expected.
Corollary 52.Let ϕ be an endomorphism of finite length of a regular local ring R. Then e ϕ HK pRq 1.
Proof: This quickly follows from Theorem 2 and Lemma 15b).
We end this section with a note.Not every homological property of the Frobenius endomorphism can be immediately extended to arbitrary endomorphisms.For example, in [40, Theorem 1.7, p. 58] Peskine and the third author showed that in positive prime characteristic, a finite free resolution of a module remains exact after applying the Frobenius functor (see Definition 17).This property may fail in general, for an arbitrary endomorphism, even in the simple case of a Koszul complex with one element.The image of a non-zerodivisor under an integral endomorphism could be a zerodivisor, as the next example shows.
Example 53.Consider the polynomial ring krx, y, z, ws over a field k.
Define an endomorphism ϕ of krx, y, z, ws as Then a is ϕ-stable.Let ϕ be the endomorphism of A induced by ϕ.The A-module ϕ A is finitely generated.In fact, it is generated by 1 and x as an A-module.Now, y w is not a zerodivisor in A because it does not belong to any prime ideal in AsspAq.But ϕpy wq y z is a zerodivisor in A; it is killed by x, for example.On the other hand, y z is a zerodivisor but is mapped to y w, a non-zerodivisor.
Nonetheless, in the previous example ϕ 2 sends any A-regular sequence to an A-regular sequence.This motivates the following Question 54.Let pR, ϕq be a local algebraic dynamical system.Does there exist a positive integer n such that ϕ n will send any R-regular sequence to an R-regular sequence?

Endomorphisms of complete equicharacteristic local rings.
In this section we prove Theorem 3, which is inspired by results of Fakhruddin [15, Corollary 2.2], and Bhatnagar and the third author [5, Theorem 2.1] on extending a polarized self-morphism of a projective variety over an infinite field to an ambient projective space.Recently in [41] Poonen gave a proof for the main result of [5] over finite fields.
Consider a self-morphism ϕ of a projective variety X over an infinite field k and let L be an ample line bundle on X with ϕ pLq !L q for an integer q ¥ 1.In [15] Fakhruddin showed that there exists an embedding ı of X in P N k given by an appropriate tensor power of L and a self-morphism ψ of P N k such that ψ ¥ı ı ¥ϕ.In [5] Bhatnagar and the third author relaxed some of Fakhruddin's hypotheses and showed that (assuming L is very ample) one can keep the same embedding of X given by L and instead extend an appropriate iteration of ϕ to the ambient projective space.
Our Theorem 3 is an analogous result about lifting of endomorphisms of finite length of complete noetherian local rings of equal characteristic.In this local version of Fakhruddin's result we do not assume our fields to be infinite.We will begin with a few preparatory results that we will need in the proof of Theorem 3.
Definition 55 ( [44, p. 159]).In a noetherian local ring R of dimension d and of embedding dimension δ, a system of parameters tx 1 , . . ., x d u is called a strong system of parameters if it is part of a minimal set of generators tx 1 , . . ., x d , . . ., x δ u of the maximal ideal.Lemma 56.A noetherian local ring pR, mq has strong systems of parameters.
Proof: The proof is by induction on dim R. If dim R 0 then the statement is vacuous, since every system of parameters is empty.So assume dim R ¡ 0 and using the Prime Avoidance Lemma [34, p. 2], pick an element x m that is neither in any minimal prime ideal of R, nor in m 2 .Apply the induction hypothesis to R{ xxy.Lemma 57.Let pR, mq be a complete local ring of equal characteristic and assume that A is a homomorphic image π : R Ñ A of R. If K is a subfield of A, then there is a subfield L of R such that π| L : L Ñ K is an isomorphism.
Proof: Let B π ¡1 pKq.Then B is a local subring of R with maximal ideal q π ¡1 p0q.Note that q ker π as subsets of R. Since B{q !K, B is also of equal characteristic.In general B need not be noetherian.
We claim that B R is a closed subset in the m-adic topology of R.
To see this, let n be the maximal ideal of A and note that the topology induced from the n-adic topology of A on any subfield of A is the discrete topology.Therefore, any subfield of A is complete with respect to the topology induced from A, and hence is closed in A. Since π is a continuous map and B π ¡1 pKq, the claim follows.In particular, B is complete with respect to the topology induced from the m-adic topology of R.
Denote the q-adic completion of B by p B. Since B is a local subring of R and R is complete, we obtain a map p i : p B Ñ R, where i : B ãÑ R is the inclusion homomorphism.Furthermore, since B is complete with respect to the topology induced from the m-adic topology of R, we see that p ip p Proof: Let K be an arbitrary coefficient field of R. Then ϕ pπpKqq is a subfield of A, and can be lifted to a subfield L of R, by Lemma 57, in such a way that π| L : L Ñ ϕ pπpKqq is an isomorphism.We will use L at the end of our proof to construct an endomorphism ψ of R. Let d dim A and let δ be the embedding dimension of A. By Lemma 56 we can choose a strong system of parameters tx 1 , . . ., x d u of A which is part of a minimal set of generators tx 1 , . . ., x d , . . ., x δ u of n.Choose elements X 1 , . . ., X δ in m in such a way that π pX i q x i for each i.We claim that since the images of x 1 , . . ., x δ in n{n 2 are linearly independent over A{n, the images X 1 , . . ., X δ of X 1 , . . ., X δ in m{m 2 are also linearly independent over R{m.If not, there will be a dependence relation α 1 X 1 ¤ ¤ ¤ α δ X δ 0 with α i R{m not all zero.This means if we choose a i R such that they map to α i in R{m for 1 ¤ i ¤ δ, then a 1 X 1 ¤ ¤ ¤ a δ X δ m 2 .If we apply π to this relation, we obtain πpa 1 qx 1 ¤ ¤ ¤ πpa δ qx δ n 2 .But then the image in n{n 2 would provide a nontrivial dependence relation πpa 1 qx 1 ¤ ¤ ¤ πpa δ qx δ 0, contradicting the linear independence of x 1 , . . ., x δ in n{n 2 over A{n.Our claim follows.Hence, we can extend tX 1 , . . ., X δ u to a basis tX 1 , . . ., X δ , . . . ,X n u of m{m 2 over R{m, where n dim R. If we choose elements X i m such that they map to X i in m{m 2 for δ 1 ¤ i ¤ n, then by Nakayama's Lemma tX 1 , . . ., X n u is a minimal set of generators of m.Furthermore, it follows from the Cohen Structure Theorem that R K X 1 , . . ., X n .Now consider elements ϕ pπpX i qq in A and for 1 ¤ i ¤ d choose f i m such that πpf i q ϕ pπpX i qq.Next, we will choose elements f d 1 , . . ., f n m inductively, making sure at each step that πpf t q ϕpπpX t qq and that dim R{ xf 1 , . . ., f t y n ¡ t.Assume d ¤ t n and that f 1 , . . ., f t have been chosen with desired properties.To choose f t 1 we use the coset version of the Prime Avoidance Lemma due to E. Davis (see [27,Theorem 124] or [35,Exercise 16.8]), that can be stated as follows: let I be an ideal of a commutative ring R and x R be an element.Let p 1 , . . ., p s be prime ideals of R none of which contain I. Then x I s i1 p i .
Choose an element u m such that πpuq ϕ pπpX t 1 qq.If dim R{ xf 1 , . . ., f t , uy n ¡ t ¡ 1, then set f t 1 u.If not, let tp 1 , . . ., p s u be the set of minimal associated prime ideals of R{ xf 1 , . . ., f t y that satisfy dim R{p i dim R{ xf 1 , . . ., f t y .Since xf 1 , . . ., f t y ker π is an m-primary ideal in R, none of these p i 's can contain ker π.Therefore by the coset version of the Prime Avoidance Lemma there exists an element a ker π such that u a s i1 p i .
Setting f t 1 u a we see that dim R{ xf 1 , . . ., f t 1 y n ¡ t ¡ 1 and πpf t 1 q ϕ pπpX t 1 qq, as desired.After choosing tf 1 , . . .,f n u as described, we define an endomorphism ψ of R K X 1 , . . ., X n as follows.For each 1 ¤ i ¤ n, we define ψpX i q to be f i and for every element α of K we define ψpαq to be pπ| L q ¡1 pϕ pπpαqqq.Then we extend the definition of ψ to all elements of R by continuity.Since ψpmqR xf 1 , . . ., f n y is m-primary by construction of the f i 's, ψ is of finite length.Moreover, it is clear from the construction that ϕ¥π π ¥ψ, that is, π : pR, ψq Ñ pA, ϕq is a morphism of local algebraic dynamical systems.Question 59.Is it possible in Theorem 3 to take ψ to satisfy v h pψq v h pϕq, w h pψq w h pϕq and pdim Aq ¤ h loc pψq pdim Rq ¤ h loc pϕq?

1. 1 .
Examples of rings with endomorphisms.Example 7. If R is a noetherian local ring of positive prime characteristic p, then the Frobenius endomorphism x Þ Ñ x p is contracting and of finite length.

Lemma 26 .
Let ta n u and tb n u be two sequences of real numbers not less than 1 such that lim nÑV plog a n q{n α and lim nÑV plog b n q{n β exist.Then lim nÑV logpa n b n q{n maxtα, βu.Proof: See [1, p. 312].

( 3 )
e S pfpqqSq e R pqq ¤ degpf q rf k S : k R s .Proof: Let d dim R dim S. By definition of multiplicity and properties of length we quickly obtain e R pq, f Sq lim mÑV k S : k R s n ¤ e S pfpqqSq.On the other hand e R pq, f Sq e R pqq ¤ degpf q (see [35, Theorem 14.8]), and equation (3) follows.Corollary 35.Let pR, m, ϕq be a local algebraic dynamical system, where R is a domain and ϕ is finite, and let k be the residue field of R. Set d : dim R and define q ϕ : expph loc pϕ, Rq{dq.For elements x 1 , . . ., x d m let χpx 1 , . . ., x d ; Rq be the Euler-Poincaré characteristic of the Koszul complex on these elements.The following conditions are equivalent: a) log degpϕq logrϕ k : ks h loc pϕq.b) For any system of parameters tx 1 , . . ., x d u of R and for any n N (4)

Example 36 .
Let pR, mq be a noetherian local domain of prime characteristic p, and let ϕ be the Frobenius endomorphism of R. Then by [30, Proposition 2.3] condition a) of Corollary 35 holds.

Proposition 38 .
Let pR, m, ϕq be a local algebraic dynamical system, where ϕ is finite, R is a domain and dim R d.Let k be the residue field of R. Then a) d ¤ v h pϕq ¤ log pdegpϕq{rϕ k : ksq ¤ h loc pϕq.b) If in addition R is Cohen-Macaulay, log pdegpϕq{rϕ k : ksq h loc pϕq.

Theorem 3 .
Bq B. Let L I be a coefficient field of p B. (For the existence of coefficient fields in complete local rings that are not necessarily noetherian, see[38, Theorem 31.1],or[35, Theorem 28.3]  or[21, Corollary 2].)Let L : p ipL I q.Then L is subfield of B that is isomorphic to L I .Furthermore, the following diagram is commutative, and shows that π| L : L Ñ K is an isomorphism.Let pA, n, ϕq be a local algebraic dynamical system and assume that A is a homomorphic image π : R A of an equicharacteristic complete regular local ring pR, mq.Then ϕ can be lifted (non uniquely) to an endomorphism of finite length ψ of R such that π ¥ ψ ϕ ¥ π.Thus, π : pR, ψq Ñ pA, ϕq becomes a morphism of local algebraic dynamical systems.

Corollary 58 .
If ϕ in Theorem 3 is finite, then so is ψ.Proof: This follows from [13, Theorem 8]: a local homomorphism f : S Ñ T of complete noetherian local rings is finite if and only if f is of finite length, and rf k T : k S s is a finite (algebraic) field extension, where k S and k T are residue fields of S and T .
If f is flat, does the equality h loc pψq h loc pϕq h loc pψq hold?One can see quickly that the inequality h loc pψq ¤ h loc pϕq h loc pψq always holds even if f is not flat.
Corollary 22.Let pR, m, ϕq be a local algebraic dynamical system.If p R is the m-adic completion of R then h loc pϕq h loc pp ϕq.Proof: We have a flat morphism of finite length p ¤: pR, ϕq Ñ p p