Physica A 281 (2000) 151–172 www.elsevier.com/locate/physa Real Arnold complexity versus real topological entropy for a one-parameter-dependent two-dimensional birational transformation N. Abarenkova a; c , J.-Ch. Angl es d’Auriac c , S. Boukraa d; b , S. Hassani e , J.-M. Maillard b; ∗ a Steklov Mathematical Institute, 27 Fontanka, 191101 St. Petersburg, Russia b LPTHE, Tour 16, 1er  etage, 4 Place Jussieu, 75252 Paris Cedex, France c Centre de Recherches sur les Tr es Basses Temp eratures, B.P. 166, F-38042 Grenoble, France d Institut d’A eronautique, Universit e de Blida, BP 270, Blida, Algeria e CDTN, Boulevard F. Fanon, 16000 Alger, Algeria Abstract We consider a family of birational transformations of two variables, depending on one param- eter, for which simple rational expressions with integer coecients, 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 identication takes place for the birational mapping seen as a mapping bearing on two com- plex variables (acting in a complex projective space). We revisit this identication between these two quite “universal complexities” by considering now the mapping as a mapping bearing on two real variables. The denitions of the two previous “topological” complexities (Arnold com- plexity and topological entropy) are modied 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 identication 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. c  2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Arnold complexity; Topological entropy; Discrete dynamical systems of real variables; Birational mappings; Cremona transformations; Rational dynamical zeta functions; Complex mappings ∗ Corresponding author. E-mail address: maillard@lpthe.jussieu.fr (J.-M. Maillard) 0378-4371/00/$ - see front matter c  2000 Published by Elsevier Science B.V. All rights reserved. PII: S0378-4371(00)00020-0