[Paper Review] Dynamics and entropy in local algebra

Mahdi Majidi-Zolbanin, Nikita Miasnikov+1 more|arXiv (Cornell University)|Sep 29, 2011

Mathematical Dynamics and Fractals11 references3 citations

TL;DR

This paper introduces algebraic entropy for finite-length self-maps of Noetherian local rings, establishing properties analogous to topological entropy. It provides a characteristic-free interpretation of Hilbert-Kunz multiplicity and proves that algebraic entropy is zero when the map's index sequence is bounded, linking it to the ring's dimension and growth of ideals.

ABSTRACT

We introduce and study a notion of algebraic entropy for self-maps of finite length of Noetherian local rings, and develop its properties. We show that it shares the standard properties of topological entropy. For finite self-maps we explore the connection between the degree of the map and its algebraic entropy, when the ring is a Cohen-Macaulay domain. As an application of algebraic entropy, we give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity.

Motivation & Objective

• To define and study algebraic entropy for self-maps of finite length on Noetherian local rings.
• To establish properties of algebraic entropy analogous to those of topological entropy, such as scaling under iteration and invariance under isomorphism.
• To provide a characteristic-free interpretation of the Hilbert-Kunz multiplicity using algebraic entropy.
• To explore the relationship between algebraic entropy and the degree of finite self-maps in Cohen-Macaulay domains.
• To derive upper and lower bounds for algebraic entropy based on the growth of ideals under iteration.

Proposed method

• Define algebraic entropy as the logarithmic growth rate of the length of the ring modulo powers of the maximal ideal under iterated maps.
• Use the notion of $ v(\varphi^n) $ and $ w(\varphi^n) $, representing the smallest and largest integers such that $ \varphi^n(\mathfrak{m})^k \subseteq \mathfrak{m} $ for $ k \geq v(\varphi^n) $ and $ \mathfrak{m}^{w(\varphi^n)} \subseteq \varphi^n(\mathfrak{m}) $.
• Apply the Hilbert-Samuel polynomial to estimate lengths $ \ell_R(R/\mathfrak{m}^{v(\varphi^n)}) $ and $ \ell_R(R/\mathfrak{m}^{w(\varphi^n)}) $, showing they grow as polynomials of degree $ d $, the dimension of the ring.
• Establish bounds on algebraic entropy via logarithmic growth rates $ v_h(\varphi) $ and $ w_h(\varphi) $, leading to $ 0 \leq d \cdot v_h(\varphi) \leq h_{\mathrm{alg}}(\varphi,R) \leq d \cdot w_h(\varphi) < \infty $.
• Prove that if $ w(\varphi^n) $ is bounded, then $ h_{\mathrm{alg}}(\varphi,R) = 0 $, using the finiteness of $ \ell_R(R/\mathfrak{m}^{c'}) $ for bounded $ c' $.
• Use the structure of minimal prime ideals and induced maps on quotients to show that the entropy of the ring equals the maximum entropy over its quotients modulo minimal primes.

Experimental results

Research questions

• RQ1 How can algebraic entropy be defined for self-maps of finite length on Noetherian local rings?
• RQ2 Does algebraic entropy satisfy properties analogous to topological entropy, such as scaling under iteration and invariance under isomorphism?
• RQ3 Can algebraic entropy provide a characteristic-free interpretation of the Hilbert-Kunz multiplicity?
• RQ4 What is the relationship between algebraic entropy and the degree of finite self-maps in Cohen-Macaulay domains?
• RQ5 What are the upper and lower bounds for algebraic entropy in terms of the growth of ideals under iteration?

Key findings

• Algebraic entropy satisfies the scaling property: $ h_{\mathrm{alg}}(\varphi^k, R) = k \cdot h_{\mathrm{alg}}(\varphi, R) $ for all $ k \in \mathbb{N} $.
• If $ \mathfrak{a} $ is a $ \varphi $-invariant ideal, then $ h_{\mathrm{alg}}(\overline{\varphi}, R/\mathfrak{a}) \leq h_{\mathrm{alg}}(\varphi, R) $, analogous to topological entropy on invariant subspaces.
• Algebraic entropy is invariant under ring isomorphism: $ h_{\mathrm{alg}}(f \circ \varphi \circ f^{-1}, R') = h_{\mathrm{alg}}(\varphi, R) $.
• When all minimal primes are $ \varphi $-invariant, the entropy of the ring equals the maximum entropy over the quotients modulo minimal primes: $ h_{\mathrm{alg}}(\varphi, R) = \max_i h_{\mathrm{alg}}(\overline{\varphi}_i, R/\mathfrak{p}_i) $.
• If $ w(\varphi^n) $ is bounded, then $ h_{\mathrm{alg}}(\varphi, R) = 0 $, and in general $ 0 \leq d \cdot v_h(\varphi) \leq h_{\mathrm{alg}}(\varphi, R) \leq d \cdot w_h(\varphi) < \infty $, where $ d $ is the dimension of $ R $.
• The algebraic entropy of the Frobenius endomorphism in positive characteristic is $ d \cdot \log p $, while its topological entropy is zero, showing a key distinction between the two notions.