Algebraic entropy for algebraic maps

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations.

💡 Research Summary

The paper extends the notion of algebraic entropy, originally introduced by Bellon and Viallet for rational (birational) maps, to a class of multivalued algebraic maps (or correspondences). The authors begin by recalling that algebraic entropy measures the growth rate of the degree of iterates of a map: exponential growth signals chaotic behaviour, while polynomial growth is associated with integrability. For a rational map φ: ℙ^m → ℙ^m written in homogeneous coordinates, the degree of the n‑th iterate is simply d_n = d^n unless cancellations occur; the entropy is defined as E = lim_{n→∞} log d_n / n.

To treat algebraic maps, the authors consider a field extension C(x)