Algebraic Entropy

Abstract

For any discrete time dynamical system with a rational evolution, we define an entropy, which is a global index of complexity for the evolution map. We analyze its basic properties and its relations to the singularities and the irreversibility of the map. We indicate how it can be exactly calculated. Postal address: Laboratoire de Physique Théorique et des Hautes EnergiesUniversité Pierre et Marie Curie, boîte postale 126.

On solutions to evolution equations defined by lattice operators

“…They are often classified into some different groups by their characteristics; periodic or non-periodic, integrable or nonintegrable, and so on. To judge the integrability of the equations, several entropy-based complexity measures have been proposed and employed [3,4], which are relevant to the Lyapunov exponent used in the chaos theory [5].…”

Entropy and flatness in local algebraic dynamics

“…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.…”

“…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].…”

Рекурсионные Операторы, Законы Сохранения И Условия Интегрируемости Для Разностных Уравнений

“…Еще одно характеристическое свойство интегрируемого уравнения -это обра-щение в нуль его алгебраической энтропии [20]. Этим свойством воспользовался Виале [21], чтобы выделить уравнение …”