Real Arnold complexity versus real topological entropy for birational transformations

Abstract

We consider a family of birational measure-preserving 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, together with an equality between the (multiplicative rate of growth of the) Arnold complexity and the (exponential of the) topological entropy. 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 (multiplicative rate of growth of some) `real Arnold complexity' and the (exponential of some) `real topological entropy'. A detailed analysis of this `real Arnold complexity' as a function of the parameter of this family of birational transformations of two variables is given. One can also slightly modify the definition of the dynamical zeta function, introducing a `real dynamical zeta function' associated with the counting of the real cycles only. Similarly, one can also introduce some `real Arnold complexity' generating functions. We show that several of these two `real' generating functions seem to have the same singularities. Furthermore, we actually conjecture several simple rational expressions for them, yielding again algebraic values for the (exponential of the) `real topological entropy'. In particular, when the parameter of our family of birational transformations becomes large, we obtain two interesting compatible nontrivial rational expressions. These rational results for real mappings cannot be understood in terms of any obvious Markov's partition, or symbolic dynamics hyperbolic systems interpretation: the birational transformation is far from being hyperbolic, it is measure preserving. It would be useful to know whether this relation between the Arnold complexity and the topological entropy, as well as the rationality of the degree generating functions and dynamical zeta functions, are a consequence of the measure-preserving property of the mapping.