Research articles

Books and chapters of books


Notes of courses

Slides (talks)


1) Robust existence of nonhyperbolic ergodic measures with positive entropy and full support (with Ch. Bonatti and D. Kwietniak) arXiv:1810.02384

Research papers.

62)  Non-hyperbolic Iterated Function Systems: semifractals and the chaos game (with E. Matias) arXiv:1808.10283 Fundamenta Math. (to appear).

61)  The structure of the space of ergodic measures of transitive partially hyperbolic sets (with K. Gelfert, T. Marcarini, and M. Rams) arXiv:1805.07308v1 Monatshefte für Mathematik(to appear).

60) Weak* and entropy approximation of nonhyperbolic measures: a geometrical approach (with K. Gelfert and B. Santiago) Math. Proc. Cambridge Phil. Soc.  (to appear).

59) Hénon-like families and blender-horseshoes at non-transverse heterodimensional cycles, (with S. Pérez)  Internat. J. Bifur. Chaos Appl. Sci. Engrg. 29 (2019), no. 3, 1930006, 22 pp. 37D25 (37E20)

58) Nonhyperbolic step skew-products: Entropy spectrum of Lyapunov exponents (with K. Gelfert and M. Rams) aXiv:1610.07167v2 Comm. in Math. Phys. 367 (2019) 351-416.

57) Hyperbolic graphs: critical regularity and box dimension, (with K. Gelfert, M. Gröger, T. Jäger,), Trans. Amer. Math. Soc. 371 (2019).

56) Attracting graphs of skew products with non-contracting fiber maps,  (with E. Matias), Math. Z. 291 (2019), no. 3-4, 1543–1568.

55) Non-hyperbolic iterated function systems: Attractors and stationary measures, (with E. Matias), Nonlinearity 31 (2018) 1782.

54) A criterion for zero averages and full support of ergodic measures (with Ch. Bonatti, Jairo Bochi), Moscow Math. J. 18 (2018) 15-61. 

53) Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew-products, (K. Gelfert, M. Rams), Proc. Steklov Math. Inst. 297 (2017) 98-115.

What is a... blender? (with Ch. Bonatti, S. Crovisier, A. Wilkinson). Notices Amer. Math. Soc. 63 (2016) 1175-1178.

51) Nonhyperbolic step skew-products: Ergodic approximations, (with K. Gelfert and M. Rams). Publ. Inst. Henri Poincaré Ann. Nonlineare 34 (2017) 1561-1598.

50) Robust criterion for the existence of nonhyperbolic ergodic measures, (with J. Bochi and Ch. Bonatti). Comm. in Math. Phys. 344 (2016) 751-795.

49) Skew product cycles with rich dynamics: From totally non-hyperbolic dynamics to fully prevalent hyperbolicity, (with S. Esteves and J. Rocha). Dynamical Systems 31 (2016) 1-40.

48) Generation of spines in porcupine-like horseshoes, (with T. Marcarini) Nonlinearity 28 (2015) 4249-4279.

47) Abundant rich phase transition is step skew-products, (with K. Gelfert and M. Rahms) Nonlinearity 27 (2014) 2255–2280.

46) Blender in center-unstable Henon-like families: with an application to heterodimensional bifurcations, (with K. Shinohara, S. Kiriki), Nonlinearity 27 (2014) 353-378.

45) Robust vanishing of all Lyapunov exponents for iterated function systems, (with J. Bochi and Ch. Bonatti) Mathematische Zeitschrift 276 (2014) 469-503;

44) Almost complete Lyapunov spectrum in step skew-products, (with K. Gelfert and M. Rahms). Dynamical Systems 28 (2013), 76--110.

43) Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication, (with Ch. Bonatti, S. Crovisier, and N. Gourmelon), Ergodic Theory and Dynamical Systems, 33 (2013), 739--776.

42) Stabilization of heterodimensional cycles (with Ch. Bonatti and Shin Kiriki), Nonlinearity 25, 931-960, (2012).

41) Entropy-expansiveness for partially hyperbolic diffeomorphisms, (with T. Fisher, M. J. Pacifico, and J. L. Vieitez), Discrete and Continuous Dynamical Systems, 32, 4195-4207, (2012).

40) Abundance of C1 robust homoclinic tangencies, (with Ch. Bonatti), Trans. Amer. Math. Soc. 364, 5111-5148, (2012).

39) Fragile cycles (with CDh. Bonatti), Jour. of Differential Equations, 252, 4176-4199, (2012).

38) Porcupine-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions, (with K. Gelfert), Fundamenta Mathematicae, 216, 55-100, (2012).

37) Rich phase transitions in step skew products, (with K. Gelfert and M. Rahms), Nonlinearity, 24, 3391-3412, (2011).

36) Symbolic extensions for partially hyperbolic diffeomorphisms, (with T. Fisher), Discrete and Continuous Dynamical Systems 29 (4), 1419-1441, (2011).

35) Non-hyperbolic ergodic measures with large support, (with Ch. Bonatti and A. Gorodetsky), Nonlinearity, 23, 687-705, (2010).

34) Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes, (with A. Gorodetsky), Ergodic Theory and Dynamical Systems, 29, 1479-1513, (2009).

33) Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes, (with V. Horita, I. Rios and M. Sambarino), Ergodic Theory and Dynamical Systems, 29, 433-474, (2009).

32) Robust heterodimensional cycles and C1-generic dynamics, (with C. Bonatti), Journal de l'Institut de Mathematiques de Jussieu, 7(3), 469-523, (2008).

31) Super-exponential growth of the number of periodic orbits inside homoclinic classes, (with C. Bonatti and T. Fisher), Discrete and Continuous Dynamical Systems, 20(3), 589-604, (2008).

30) How to put a stop to domination once and for all (and what comes afterwards, Partially hyperbolic dynamics, laminations, and Teichmuller flow, Fields Institute Communications-- American Mathematical Society, 51, 3-34, (2007).

29) Heterodimensional tangencies, (with A. Nogueira and E. R. Pujals), Nonlinearity, 19, 2543-2566, 2006.

28) Periodic points and homoclinic classes, (with F. Abdenur, C. Bonatti, S. Crovisier, and L. Wen). Ergodic Theory and Dynamical Systems, 27(1), 1-22, (2007).

27) How do homoclinic classes collide at heterodimensional cycles? (with J. Rocha). Discrete and Continuous Dynamical Systems, 17(3), 589-627, (2007).

26) Pseudo-orbit shadowing in the C1 topology, (with F. Abdenur), Discrete and Continuous Dynamical Systems, 17(2), 223-245, (2007).

25) Generic diffeomorphisms of compact surfaces, (with F. Abdenur, C. Bonatti, and S. Crovisier), Fundamenta Mathematicae, 187(2), 127-159 (2005).

24) Collision, explosion and collapse of homoclinic classes, (with B. Santoro), Nonlinearity, 17, 1001-1032, (2004).

23) Non-wandering set with non-empty interior, (with F. Abdenur and Ch. Bonatti), Nonlinearity, 17(1), 175-903, (2004).

22) A C1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, (with Ch. Bonatti and E.R. Pujals), Annals of Mathematics, 158(2), 355-418, (2003).

21) Robustly transitive sets and heterodimensional cycles, (with Ch. Bonatti, E.R. Pujals, and J. Rocha), Astérisque, 286, 187-222, (2003).1

20) Critical saddle-node horseshoes: bifurcations and entropy, (with I. Rios), Nonlinearity, 16(3) 897-929, (2003).

19) On maximal transitive sets of generic diffeomorphisms, (with Ch. Bonatti), Publications Mathématiques I.H.E.S., 96, 171-197, (2002).

18) Minimality of the strong foliations for partially hyperbolic diffeomorphisms, (with Ch. Bonatti and R. Ures), Journal de l'Institut de Mathématiques de Jussiieu, 1(4), 513-541, (2002).

17) Heterodimensional cycles, partial hyperbolicity and limit dynamics, (with J. Rocha), Fundamenta Mathematicae, 184, 127-186, (2002).

16) Critical saddle-node cycles: Persistence of tangencies and Haussdorf dimension, (with R. Ures), Ergodic Theory and Dynamical Systems, 22, 1117-1140, (2002).

15) Partially hyperbolic and transitive dynamics generated by heteroclinic cycles, (with J. Rocha), {\it Ergodic Theory and Dynamical Systems, 21, 25-76, (2001).

14) Pas de Shadowing Lemma pour les dynamiques partiellment hyperboliques, (with Ch. Bonatti and G. Turcat), Comptes Rendus de Academie de Sciencies de Paris, t 330, 587-592, (2000).

13) Partial hyperbolicity and robust transitivity, (with E.R. Pujals and R. Ures), Acta Mathematica, 183, 1-43, (1999).

12) Connexions hétérocliniques et généricite d'une infinité de puits ou de sources, (with Ch. Bonatti), Annales Scientifiques de l'École Normal Supérieure de Paris, 32 (4), 135-150, (1999)

11) Cubic tangencies and hyperbolic diffeomorphisms, (with C. Bonatti and F. Villeumin), Boletim Sociedade Matemática Brasileira, 29(1), 99-144, (1998).

10) Large measure of hyperbolic dynamics when unfolding heteroclinic cycles (with J. Rocha), Nonlinearity, 10, 857 - 884, (1997).

9) Noncritical saddle-node cycles and robust nonhyperbolic dynamics (with J. Rocha), Journal of Dynamics and Stability of Systems, 12 (2), 109-135, (1997).

8) Saddle-node cycles and strange attractors: prevalence and globality (with J.Rocha and M.Viana), Inventiones Math., 125 (1), 37-74, (1996).

7) Persistent Transitive Diffeomorphisms, (with C. Bonatti), Annals of Mathematics, 143(2), 357-396, (1996).

6) Persistence of cycles and nonhyperbolic dynamics at the unfolding of heteroclinic bifurcations, Nonlinearity, 8, 693-715, (1995). 5

5) Discountinuity of Hausdorff dimension of hyperbolic sets (with Ch. Bonatti and M. Viana), Comptes Rendus de Academie de Sciencies de Paris, t320-I, 713-718, (1995).

4) Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory and Dynamical Systems, 15, 291-315, (1995).

3) Persistent homoclinic tangencies and the unfolding of cycles, (with R. Ures), Publ. Inst. Henri Poincaré Ann. Nonlineare, 6, 643-689, (1994).

2) Non-connected heterodimensional cycles: bifurcation and stability, (with J. Rocha), Nonlinearity, 5, 1315-1341, (1992).

1) Discountinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphism, (with M. Viana), Ergodic Theory and Dynamical Systems, 9, 403 -425, (1989).

Books and chapters of books.

(iii) Dynamics beyond uniform hyperbolicity, (with Ch. Bonatti and M. Viana), Encyclopaedia of Mathematical Sciences (Mathematical Physics), 102}, Springer Verlag

(ii) Routes to wild dynamics, 127--143, Contibuciones Matemáticas, Homenaje al Porfesor Enrique Outerelo Dominguez, (editor J. Ruiz) Editorial Complutense (UCM), (2004).

(i) The intermittancy route to chaotic dynamics, (with I. Rios and M. Viana), Global Analysis of Dynamical Systems, Festschrift dedicated to F. Takens for his 60th birthday, Broer, Krauskopf and Vegter editors, IOP, 309-327, 2001.


(f) Nonhyperbolic ergodic measures, Proc. International Conference of Mathematicians,

(e)Blender-horseshoes in center-unstable Hénon-like families (with S. Pérez) arXiv:1804.02577v1, Springer Proceedings in Mathematics and Statistics (to appear)

(d) Porcupine-like horseshoes: topological and ergodic aspects, (with K. Gelfert), Progress and Challenges in Dynamical Systems Proceedings of the International Conference Dynamical Systems: 100 Years after Poincare, September 2012, Gijon, Spain, Springer Proceedings in Mathematics and Statistics.

(c) Cycles and systems of iterated functions: the role of secondary bifurcations, cd-rom do First Joint Meeting rsme-ams, Sevilla, Spain, june 18-20, (2003), (Dynamical Systems, session 15th, page 4).

(b) The end of domination, Abstracts Booklet of the conference New Directions in Dynamical Systems, Kyoto, Japan, 204-211,(2002).

(a) Transitividad robusta de conjuntos y difeomorfismos, Actas VII Reunióon de Topólogos Espanoles, 1-20, (2000).

Notes of courses

(I) Cycles, partial hyperbolicity, and transitivity, V Escuela Internacional de Sistemas Dinámicos, Facultad de Ciencias de la Universidad de Chile (january, 3-7, 2005).

Slides (talks)

* Nonhyperbolic ergodic measures, Intenational Conference of Mathematicians, Rio de Janeiro Brazil, 2018.

* Rich model cycles in skew product dynamics, in in Dynamics, Bifurcations, and Strange Attractors, dedicated to the memory L. P. Shilnikov, 1-5th July 2013, Nizhny Novgorod, Russia.

* Procupines-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions, in in Research Institute of Mathematical Sciences, Kyoto, Japan;

* Ergodic measures with only (and many) zero exponents, in in Skew Product Dynamics and Multifractal Analysis, 1-5th October 2012 in Luisenthal, Germany.

* Non-hyperbolicities, in Encontro de Matemática Brasil-França, IMPA, Rio de Janeiro, Brazil, September (2009).

* Non-hyperbolicities, in III Congreso Latino Americano de Matemáticos, Santiago de Chile, Chile, August (2009).

* Heterodimensional tangencies, in 2005 International Conference on Dynamical Systems, Peking University and USTC, Huang Shan, Anhui, China, July (2005).

* Heterodimensional cycles, in Segundo Congreso Latinoamericano de Matemáticos (Sesión Sistemas Dinámicos), Cancún, Mexico, June (2004).

* Collision of hyperbolic homoclinic classes, in Summer School and Conference on Dynamical Systems, ICTP, Trieste, Italy, July (2004).

* Tame and wild dynamics, in Dynamical Methods in Differential Equations, Medina del Campo, Valladolid, Spain (September 2002).