Real Arnold complexity versus real topological entropy for a one-parameter-dependent two-dimensional birational transformation

Abstract

We consider a family of birational transformations of two variables, depending on one parameter, for which simple rational expressions with integer coefficients, for the exact expression of the dynamical zeta function, have been conjectured. Moreover, an equality between the (asymptotic of the) Arnold complexity and the (exponential of the) topological entropy has been conjectured. This identification takes place for the birational mapping seen as a mapping bearing on two complex variables (acting in a complex projective space). We revisit this identification between these two quite “universal complexities” by considering now the mapping as a mapping bearing on two real variables. The definitions of the two previous “topological” complexities (Arnold complexity and topological entropy) are modified according to this real-variables point of view. Most of the “universality” is lost. However, the results presented here are, again, in agreement with an identification between the (asymptotic of the) “real Arnold complexity” and the (exponential of the) “real topological entropy”. A detailed analysis of the “real Arnold complexity” as a function of the parameter of this family of birational transformations of two variables is given.