... informed Palis that he had attended Elon Lima's course at the recent Brazilian Mathematical Colloquium in Poços de Caldas. He added that he would very much like to study Dynamical Systems to obtain a Ph.D. in this area.

... he was informed that de Melo was an excellent student and very determined to learn new relevant topics in mathematics. With all the positive reactions Palis had received, he asked Welington to present one of the important Seminar topics. It was a success! With that, Jacob formalised his request to the Director of IMPA to accept de Melo as a Ph.D. student, under his guidance. A couple of weeks after that, Welington came to Jacob's office to say that he had discovered a serious gap in one of his mathematical papers. Happily, it turned out it was not a serious gap.

What I learned about mathematics without taking any courses, simply talking to people in the corridors, was great.

This paper is based in my doctoral thesis at IMPA under the guidance of J Palis. We are grateful to E Lima, P Mendes, S Newhouse, G L dos Reis, C Robinson, R Williams, and especially to J Palis for many helpful conversations.

Welington was very much influenced by Steve Smale who in the early 1970s was quite interested in economics. Inspired by Steve's work, Welington studied the problem of simultaneously optimising several functions. Forever the pure mathematician, he wrote that this 'is a natural question to consider, even in some basic models in economics'.

De Melo is a man of systematic habits. When he is in Angra, he wakes up early, works in the morning and goes sailing in the afternoon. Although he doesn't take paper and pencil to do mathematics on board, the activity helps to sharpen his reasoning. "It is the time when I organise my ideas," he explains.

The author discusses the connection between the density of structurally stable maps in the space of unimodal maps of the interval, the finiteness of attractors and the nonexistence of wandering intervals. It is proved that in the space of unimodal maps having an eventually periodic flat critical point, there is a residual subset whose maps have infinitely many sinks and there are also maps having a wandering interval.

... is my first contribution to my main field of research: one-dimensional dynamics. In it I discuss the problem of existence of wandering intervals and finiteness of attractors and relate these problems to the density of structurally stable interval endomorphisms.

[*] is my most important mathematical contribution: a complete description is given of the topological behaviour of 1-dimensional real dynamical systems. It turns out that the Julia-Fatou-Sullivan theory for conformal mappings of the Riemann sphere is also valid for smooth endomorphisms of the interval (or the circle).

Julia gave the following description of the topological structure of 1-dimensional complex dynamics: every rational map of the Riemann sphere has periodic domains attracting an open set of points, outside this open set there is a closed invariant set, called the Julia set, and maybe some open sets wandering around and being attracted to this Julia set.

The main problems left over were

1) Can a map have infinitely many periodic domains?

2) Do these wandering domains exist?

Fatou showed that there can be only finitely many periodic domains and Sullivan showed that these wandering domains do not exist.

The first result about the topological structure of 1-dimensional real dynamics was obtained by Denjoy and Poincaré. Poincaré gave a description of the topological structure of the dynamics of circle diffeomorphisms. As in the theory of complex dynamics the existence of wandering domains played an important role. Denjoy showed that they cannot exist for $C^{1+b\nu}$ diffeomorphisms of the circle. The topological work of Poincaré could not be generalised to endomorphisms of the interval (or circle). The topological description of the dynamics of interval dynamics was done by Milnor and Thurston. The description was exactly like the one given by Julia and again the only thing left over was to answer the questions above.

A sequence of authors achieved partial results on the non-existence of wandering intervals (Denjoy, Guckenheimer, van Strien and de Melo, Lyubich and Blokh). The general non-existence of wandering intervals was proved in [*]. Furthermore [*] also gives the finiteness of periodic attractors. As a corollary of the theory developed in [*] we got the classical Denjoy-Schwarz theorem under much milder smoothness conditions.

Whenever he takes the sailboat out, de Melo likes to set himself a little mathematical constraint. "Each time I try to make a path that is homotopically non-trivial," he explained. Faced with the stunned silence of the interlocutor, he tried to expand: it is equivalent to circumventing at least one island. He was proud to have already sailed those waters in the company of five Fields medallists: René Thom (1958), Stephen Smale (1966), Sergei Novikov (1970), Jean-Christophe Yoccoz (1994) and Stanislav Smirnov (2010). Artur Avila - now sixth on the list - has a photo on his IMPA page in his swimming trunks aboard the Doisdu, but he is not exactly a sailor. "He is very much from the city," lamented his advisor de Melo.

Remembering Welington is remembering someone who was fundamental for our institution [IMPA] and for many people. He showed, in addition to exceptional mathematical talent, an enormous commitment to IMPA. He wanted IMPA to succeed. His contribution was not just mathematics. ... He was an extremely rigorous mathematician, committed to the highest quality of the work carried out at IMPA. He wanted the work to be at the highest possible level, but, because of this rigour, his reputation scared students a little. I found it a bit risky to enrol, I think in 1997, in a course taught by Welington at IMPA. With great caution, I did not enrol in the course, but went to attend classes. He turned out to be, unlike his reputation, very nice to me. At the end of the course, he said that I had done well. ... He asked me to go on his boat in Angra. He provided the food, the beer. But the students had the task of getting the barnacles out of the hull. He even provided the appropriate instruments.

If two smooth unimodal maps or real analytic critical circle maps have the same bounded combinatorial type then there exists a $C^{1+\alpha}$ diffeomorphism conjugating the two maps along the corresponding critical orbits for some α > 0. The proof is based on a detailed understanding of the orbit structure of an infinite dimensional dynamical system: the renormalisation operator.

Just before welcoming the Brazilians to Stony Brook, Lyubich had written a series of articles in which he proved his most important results. "Very few people really understood what it was about", he commented, "and Welington was a notable exception. It was his proposal that Artur explore this line of research." ... The three spent a month tossing ideas back and forth, in a style of doing mathematics that only requires a blackboard, chalk, and space to walk back and forth. The daily conversations took place in the Institute's rooms, at Lyubich's house, in restaurants or during walks through the woods around the university. ... "Occasionally, amazed, I realised that Lyubich and I were a little behind Artur," recalled de Melo, "he was so young. .. I would forget about it but then be scared." ... After a month of intense discussions, the trio had a clear strategy for resolving the problem that consumed them, but the proof was still out of reach. There was an obstacle that refused to budge. Lyubich and de Melo decided to leave it in the boy's hands. "That was in March", remembers Artur. "I kept the problem in my head and a few months later, in September or October, I had a weird idea."

Welington met his wife Gilza while he was still quite young. They were inseparable, truly adored each other and even felt comfortable squabbling in front of others. No doubt, Gilza was instrumental in Welington's success. Usually Welington and Gilza lived relatively modestly. However, occasionally Welington would love to eat out in famous restaurants and talked about these memorable meals for years after. Welington loved to be near the sea and in particular to sail and the sense of adventure and freedom.

The occasion was particularly gratifying for us because of the active participation of a number of experts in this field, many of Welington de Melo's co-authors and all his former doctoral students. Collecting the articles for this volume was initially intended as an opportunity to celebrate the success of the meeting and Welington de Melo's joy at the excellent scientific level of the lectures and the friendly atmosphere that week. Unfortunately, Welington de Melo passed away a month after the meeting. He is sorely missed by the Brazilian mathematical community, and the publication of this volume became more a tribute to Welington de Melo. His role in the development of mathematics is indisputable, especially in the area of low-level dynamics, and his legacy includes, in addition to numerous articles with fundamental contributions, books that are mandatory references for beginners in this area. Welington had only seven formal Ph.D. students, and we were both honoured of being two of them. It is worth mentioning that Artur Avila, the 2014 Fields Medal winner, was also Welington's Ph.D. student.