In the center of this well-documented book is the ‘Grand Prix des sciences mathématiques’ which in 1915 – with some allusion to work by the then recently deceased H. Poincaré – was advertised by the French Academy of Sciences, to be given for the solution of the problem of the iteration of complex rational functions (fractions rationnelles). This problem has historical origins in the method by Newton-Raphson and is today a central topic within complex dynamics. It has been widely discussed since the 1980s within chaos theory with Julia sets and Fatou sets being central notions. Audin gives some but not too detailed analysis of the respective publications by Gaston Julia and Pierre Fatou between 1917 and 1919, which both used Montel’s notion of normal families of analytic functions. Based on unknown archival material, mostly from the French Academy of Sciences, Audin discusses reasons for Fatou’s non-participation in the competition which finally was won by Julia. She sees them primarily in the different personalities of the two mathematicians and in the political conditions after the First World War. Audin opposes the view that Julia’s work was more modern than Fatou’s. On the contrary she sees Julia in this particular work more in the tradition of classical complex function theory and Fatou in the tradition of modern general topology. The entire chapter V (pp. 125-175) is devoted to providing new biographical material on Fatou. The mathematician earned a living as an astronomer, and he is today mainly known for his thesis of 1907 on trigonometric series, a central event in the modern theory of functions of real variables. Audin does not hide her predilection for the personal character traits of Fatou. She discusses further aggressive behavior of Julia’s in his successful competition against the much older Paul Montel during the election to the French Academy in 1934. In the end, Audin quotes the American John Milnor (2006) approvingly: ‘The most fundamental and incisive contributions were those of Fatou himself. However, Julia was a determined competitor and tended to get more credit because of his status as a wounded war hero.’ Audin gives a partial explanation for the temporary decline of the problems occupying Julia and Fatou around 1920 by pointing to communication problems and in particular the late reception of Hausdorff dimension in France. Not aiming at a full history of Complex Dynamics and referring in this respect to an earlier book by {it D. S. Alexander} [« A history of complex dynamics. From Schröder to Fatou and Julia (1994; Zbl 0788.30001)], Audin does not discuss reasons for the renaissance of complex dynamics in the 1980s either. The report in chapter VI on a curious polemics around 1965 about priority for Julia sets between the aged Frenchmen Julia and Montel seems a bit too detailed. The documents in the appendices (which are not listed in the table of contents) related to the work of Fatou, Julia and Montel, could have received a more detailed annotation. Audin’s book, written by a mathematician but arguing mainly from the standpoint of the social and political history of mathematics, is a valuable contribution to the historiography of mathematics in France, including the time of Nazi occupation. The impending English edition will hopefully make the book known to a broader readership. (ZDM/Mathdi)