I opened it and it was from someone from Uruguay, saying that he had proved a series of theorems of the highest significance in this area of research, where all but one of them answered open problems. This was part of my work with Steve Smale, my advisor in Berkeley, California. This is how Ricardo Mañé presented himself, with an immense letter, not only enunciating these great theorems, but also giving sketches of the proofs. ... I was amazed that someone in Uruguay had been able to write a letter, possibly full of statements and incomplete or even wrong proofs, but with immense good taste about what the central problems of the area at that time would be, and also the audacity to propose even plans of proofs.

... as soon as he arrived in Rio, he moved to Leblon, where I also lived. I was always with him carrying his unfailing umbrella. He had the habit of carrying an umbrella even on sunny days. It didn't matter what the weather was like: he was always with his umbrella walking around Leblon. As he didn't have a car, I often took him to Maracanã, at the time when Maracanã was frequented. There we talked about all things, not only about mathematics and my insistence on the book he had to write, but about politics, not only in Brazil and Uruguay, but international politics. He was a very cultured person, very experienced, having lived in his youth more through the deep readings he did. He was a voracious reader. When he wasn't doing mathematics, he was always reading other things.

From the beginning he was very peculiar both as a person and as a student. He was exceptional, brilliant. He quickly completed the necessary credits for his doctorate. Soon, he took the initiative regarding his thesis. I did not choose the topic of his thesis ... which was already quite unusual at that time. It is true that I argued with him a lot. He came, presented proposals and we tried to get things right, I eliminated things that I thought were less possible or less interesting. Anyway, it was very much his own initiative. He completed his doctorate in an absolutely record period of one and a half years. In February 1973 he was defending his thesis receiving all the awards. Truly an exceptional figure who soon gained immense sympathy and respect from his colleagues.

It is a remarkable fact that each complex analytic endomorphism f of the Riemann sphere $\mathbb{C}$ exhibits highly non-trivial dynamical phenomena. In this paper we first describe classical and recent results which give the basic dynamical picture of these mappings. Then we make use of this picture to construct topological conjugacies or partial topological conjugates between certain nearby endomorphisms in analytic families of endomorphisms.

This book presents the theory of smooth ergodic theory, i.e. the part of modern ergodic theory which is connected with differential dynamical systems. P Walters' book [An introduction to ergodic theory, Springer, New York, 1982] is the most similar, but it only covers about two-thirds of the material. Much of this recent material is available in numerous research articles, but not a systematic textbook like the one under review. The first half of the book presents the classical theory, first through examples and then a more systematic presentation. The second half of the book presents the theory connected with entropy which was introduced by Kolmogorov .... Both measure-theoretic and topological entropy are introduced and connected through the variational principle establishing topological entropy as the supreme of measure-theoretic entropy. The last topics present the connection of entropy with Lyapunov exponents. ... The book is well written with problems at the end of each section. It is self-contained and includes a rapid review of the theorems needed from measure theory. The statements of the theorems are clear and easily found. This book provides an accessible introduction to an important and active area of current research.

This version differs from the Portuguese edition only in a few additions and many minor corrections. Naturally, this edition raised the question of whether to use the opportunity to introduce major additions. In a book like this, ending in the heart of a rich research field, there are always further topics that should arguably be included. Subjects like geodesic flows or the role of Hausdorff dimension in contemporary ergodic theory are two of the most tempting gaps to fill. However, I let it stand with practically the same boundaries as the original version, still believing these adequately fulfil its goal of presenting the basic knowledge required to approach the research area of Differentiable Ergodic Theory.

This book is the eagerly awaited translation into English of the Instituto de Matemática Pura e Aplicada monograph written in Portuguese and not easily available outside Brazil. The aim of the book is to describe the rudiments of measure-theoretic ergodic theory and apply these to the qualitative study of diffeomorphisms.

I like this book very much. It presents the right material in a coherent framework with just the right balance of general theory and concrete examples. The topics covered are well chosen and there is a wealth of material to be found here. ... It must be said that some of the topics covered by Mañé (the ergodic theory of Anosov diffeomorphisms for example) are technical and difficult. This is probably a difficulty with any book presenting a young and active field, however. The author in his introduction acknowledges that some proofs are "arid and demanding" and encourages the reader to concentrate on their complement in the text. This is good advice, and I found it quite feasible to do so and still learn a great deal.

The practice of awarding four researchers was not followed, perhaps due to the fact that a fourth name worthy of the laurel was not found. It could have been Ricardo Mañé, but, producing mathematics on the fringes of the Atlantic Forest, on the fringes of the great research centres and at a time when information circulation was so less than today, it would be surprising if anyone remembered him. "Brazil did not have enough articulation in the world of mathematics," explains Palis in his office at IMPA. Marcelo Viana, a researcher at the institution and a student at Palis, adds: "At the time, Brazil was a peripheral country and behaved like a peripheral country. We sat there hoping someone would come up with his name."

A specific text was required for each course. His course was Ergodic Theory. He wrote the text and asked me to proofread it. ... I had already read some of Ricardo's writings, but perhaps not as closely and not to the level where clarity of exposition is paramount, because this course was intended for an audience at the elementary level of ergodic theory. And I remember being deeply impressed by the way in which all the difficulties of exposing sophisticated topics to an audience that weren't supposed to be sophisticated had been resolved in a way that at some points was even brilliant. And at the same time, I followed the process relatively closely, and it was clear from the text and from direct contact with Ricardo, that he had a great joy in what he was doing. He was writing a text, where essentially everything was contained in his book, and yet he was finding new angles and having fun while writing the text.

This solves one of the most outstanding conjectures in the field of dynamical systems, posed by S Smale and J Palis in 1970.

Our subject will be minimising measures of Lagrangian dynamical systems.

Uniformly hyperbolic diffeomorphisms ... are stable: two $C^{^{*}}$ close uniformly hyperbolic diffeomorphisms are topologically conjugated on a neighbourhood of their respective chain recurrent sets. Actually, a deep theorem of Mañé (extended by Palis) states that the converse is also true: a diffeomorphism that is $C^{1}$ stable in this sense is uniformly hyperbolic.

... the doctors detected the beginning of a metastasis, due to lung cancer. While his health began to deteriorate rapidly, he dealt with great determination putting on paper all the mathematical ideas he planned to develop. I remember at that time going to his mother's house, where he received visits from family and friends every afternoon, taking half a dozen books that he asked me to borrow from our library. Many afternoons we would all find ourselves on the pavement outside his house, waiting for the regular interview that Ricardo had with a priest to finish. On 8 March 1995, Mañé was admitted to the Spanish sanatorium on Garibaldi street, where he continued to write his extensive notes. Finally, they gave rise to his famous posthumous publication "Lagrangian flows: the dynamics of globally minimizing orbits" in the Bulletin of the Brazilian Mathematical Society in the year 1997. Obviously, many details were left unfinished, but in the years that followed these omissions were corrected, his observations were expanded or corrected, and many of the proofs were finally established with the greater rigour that characterises mathematical work.

Mañé died prematurely at the age of 48 due to HIV, a disease that, during the 1980s and 1990s, hit mainly homosexuals who used their freedom. Ricardo found that freedom in Brazil. The same freedom with which he thought and created mathematics.

Ricardo Mañé was one of the most brilliant mathematicians ever in Latin America. In the words of Lennart Carleson, he had a unique approach to mathematics and will have no substitute. Certainly not. Yet, his powerful intellect will be with us for many years to come: He permanently influenced the area of dynamical systems as well as his friends, colleagues and students at his so much loved IMPA, to which he had always been adamantly faithful. We all remember him as a wonderful and most inspiring lecturer. Also as a funny and often ironic story teller, incredibly sharp, intelligent and knowledgeable (a compulsive reader), as well as a profound connoisseur of opera. In brief, a most unforgettable human being and mathematician.