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Selected Works. Edited by Welington de Melo. (English) Zbl 1306.37004

Cham: Springer (ISBN 978-3-319-06967-8/hbk). xix, 711 p. (2014).
This book is a compilation of \(24\) papers published by Jacob Palis along its career. It contains papers from his Ph.D. thesis and papers on what is now known as Palis Conjecture. The book also contains a short summary of the scientific life of Jacob Palis written by Welington de Melo and Marcelo Viana, who are both former (and distinguished) students of Jacob Palis. Indeed, at the end of the book one can find a list of publications of Jacob Palis from 1968 to 2013 and a list of \(41\) Ph.D. students of Jacob Palis at IMPA from 1972 to 2013.
Throughout the review, \(M\) denotes a compact \(\mathcal{C}^\infty\) manifold without boundary and \(f\) denotes a diffeomorphism on it, in general of class \(\mathcal{C}^r\), with \(r \geq 1\).
The first paper of the book is entitled “On Morse-Smale dynamical system” [(1969; Zbl 0189.23902)], and in it is proved that set of Morse-Smale diffeomorphisms on \(M\), with \(\dim M \leq 3\) is open and each of its elements is structurally stable. The second paper of the book is a generalization of this property for manifolds of any dimension [(1970; Zbl 0214.50702)].
The third paper is entitled “A note on \(\Omega\)-stability” [(1970; Zbl 0214.50801)]. The result in this note is motivated by Smale’s result that if a \(f\) satisfies Axiom A and \(\Omega(f)\) has the no-cycle property then \(f\) is \(\Omega\)-stable. Here it is proved that if \(f\) satisfies Axiom A and is \(\Omega\)-stable, the \(\Omega(f)\) has the no cycle property.
The fourth paper is “Neighborhoods of hyperbolic sets” [(1970; Zbl 0191.21701)] and in it a study of the asymptotic properties of points near a compact hyperbolic set of \(f\) is performed.
The fifth paper, “Hyperbolic nonwandering sets of two-dimensional manifolds” [(1973; Zbl 0279.58010)], is a study of diffeomorphisms \(f\) whose nonwandering sets are hyperbolic. If \(\Omega\) is the nonwandering set of \(f\), it is proved that the periodic orbits of \(f\) are dense in \(\Omega\) and that \(f\) can be approximated by an \(\Omega\)-stable diffeomorphism.
The title of the sixth paper is “The topology of holomorphic flows with singularity” [(1978; Zbl 0411.58018)]. In this paper the authors consider differential equations defined by a holomorphic vector field on a complex manifold and study the topology of the foliation it defines in a neighborhood of a singularity.
The seventh paper contains a work in collaboration with Floris Takens entitled “Topological equivalence of normally hyperbolic dynamical systems” [(1977; Zbl 0391.58015)]. The authors investigate the concept of topological equivalence, that is, the existence of a homeomorphism of the ambient space carrying orbits of one vector field onto orbits of the other.
The eighth paper is the contribution of Jacob Palis to the International Congress of mathematics in 1980 and it is entitled “Moduli of stability and bifurcation theory” [(1980; Zbl 0467.34032)]. An important role in bifurcation theory of dynamical systems is performed by certain differentiable invariants of topological equivalence. This fact is exhibited in this paper.
The ninth paper is entitled “Characterizing diffeomorphisms with modulus of stability one” [(1981; Zbl 0482.58022)]. This paper contains the characterization of a large class of diffeomorphisms with modulus of stability one.
The tenth paper is entitled “Bifurcations and stability of families of diffeomorphisms” [(1983; Zbl 0518.58031)]. It contains a long survey around the collaborative work of Jacob Palis and Floris Takens, together with S. Newhouse. Structural properties of diffeomorphisms and arcs of diffeomorphisms are compiled in it.
The eleventh paper contains one of the most important papers authored by Palis and Takens [(1983; Zbl 0533.58018)]. One-parameter families of gradient vector fields, endowed with the Whitney topology, are considered and it is shown that the subset of the stable ones is open and dense.
The twelfth paper is “A Note on the inclination lemma (\(\lambda\)-Lemma) and Feigenbaum’s rate of approach” [(1983; Zbl 0528.58005)]. Quoting the author, this note provides an elementary discussion of the dynamical behavior of a transformation in a Banach space, near a hyperbolic fixed point having its stable manifold of finite codimension.
The thirteenth and fourteenth papers are collaborative works by Palis and Takens [(1985; Zbl 0579.58005); (1987; Zbl 0641.58029)]. In the first one the authors provide a large class of one-parameter families of two-dimensional diffeomorphisms which are stable when the parameter is negative, exhibit a cycle when the parameter is \(0\) and have a bifurcation set of positive but arbitrarily small relative measure for the parameter in small (positive) intervals. In the second one the authors keep on considering one-parameter families of diffeomorphisms on surfaces which are hyperbolic when the parameter is negative, have a homoclinic tangency when the parameter is \(0\) and show that most of these diffeomorphisms are also hyperbolic for small positive values of the parameter.
The fifteenth paper is entitled “On the \(\mathcal{C}^1\) \(\Omega\)-stability conjecture” [(1988; Zbl 0648.58019)] which is motivated by the proof of the \(\mathcal{C}^1\) stability conjecture done by Mañé in [(1988; Zbl 0678.58022)].
The sixteenth paper starts a series of various collaborative works by J. Palis and J.-C. Yoccoz. This one is entitled “Centralizers of Anosov diffeomorphisms on tori” [(1989; Zbl 0675.58029)]. Recall that an Anosov diffeomorphism is a diffeomorphism on a manifold with rather clearly marked local directions of ‘expansion’ and ‘contraction’.
The seventeenth paper is a contribution in determining which phenomena are more common, in terms of the Lebesgue measure in the parameter space, when unfolding the diffeomorphisms which exhibit a homoclinic tangency through \(k\)-parameter families [(1994; Zbl 0801.58035)].
The eighteenth paper, authored by Palis and Marcelo Viana [(1994; Zbl 0817.58004)] generalizes to higher dimensions the result of Newhouse in two dimensions that many smooth diffeomorphisms near one with a homoclinic tangency have infinitely many coexisting sinks.
The nineteenth paper is entitled “On the arithmetic sum of regular Cantor sets” and is authored by Palis and Yoccoz [(1997; Zbl 0895.58020)].
In the twentieth paper, Jacob Palis poses a conjecture on the denseness of finitude of attractors for dynamical systems [(2000; Zbl 1044.37014)].
The twenty-first work is a note by Palis and Yoccoz, in French, in which they study the diffeomorphisms on surfaces which are near a diffeomorphism with a horseshoe [(2001; Zbl 1015.37024)].
The twenty-second paper is entitled “Homoclinic tangencies and fractal invariants in arbitrary dimension” and studies generic families of diffeomorphisms unfolding a homoclinic tangency associated to a hyperbolic basic set [(2001; Zbl 1192.37032)].
The twenty-third paper is a survey entitled “A global perspective for non-conservative dynamics” [(2005; Zbl 1143.37016)]. This paper contains most contributions of the author on what is known as Palis conjecture.
The last paper is a joint work by Palis and Yoccoz entitled “Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles” [(2009; Zbl 1181.37024)]. In this paper the authors provide a long and detailed review of what is known on the topic of bifurcations of heteroclinic cycles for smooth parameterized families of surface diffeomorphisms.

MSC:

37-03 History of dynamical systems and ergodic theory
01A75 Collected or selected works; reprintings or translations of classics
01A70 Biographies, obituaries, personalia, bibliographies
01A60 History of mathematics in the 20th century
37Dxx Dynamical systems with hyperbolic behavior
37Gxx Local and nonlocal bifurcation theory for dynamical systems
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