Lorenzo J. Díaz

Preprints

1. "Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms" Preprint, arXiv:2505.03079 [math.DS], with Katrin Gelfert, Michal Rams, Jinhua Zhang (2025).
2. "A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures" Preprint, arXiv:2502.12505 [math.DS], with Jiagang Yang, Jinhua Zhang (2025).
3. "Heterodimensional cycles of hyperbolic ergodic measures" Preprint, arXiv:2405.12686 [math.DS], with Christian Bonatti, Katrin Gelfert (2024).
4. "The amount of nonhyperbolicity for partially hyperbolic diffeomorphisms" Preprint, arXiv:2405.12051 [math.DS], with Katrin Gelfert, Jinhua Zhang (2024).
5. "Loosely Bernoulli zero exponent measures for elliptic matrix cocycles" Preprint, arXiv:2311.09351 [math.DS], with K. Gelfert, M. Rams (2023).
6. "Non-hyperbolic Iterated Function Systems: attractors and stationary measures" Preprint, arXiv:1605.02752 [math.DS], with Edgar Matias (2016).

Research articles

1. "Nonhyperbolic dynamics by mingling, blending, and flip-flopping" Topology Appl., with K. Gelfert (2023). (pdf)
2. "Nontransverse heterodimensional cycles: stabilisation and robust tangencies" Trans. Amer. Math. Soc., with Sebastián A. Pérez (2023). (pdf)
3. "Mingled hyperbolicities: ergodic properties and bifurcation phenomena (an approach using concavity)" Discrete Contin. Dyn. Syst., with Katrin Gelfert, Michał Rams (2022). (pdf)
4. "Homoclinic tangencies leading to robust heterodimensional cycles" Math. Z., with Pablo G. Barrientos, Sebastián A. Pérez (2022). (pdf)
5. "Variational principle for nonhyperbolic ergodic measures: skew products and elliptic cocycles" Comm. Math. Phys., with K. Gelfert, M. Rams (2022). (pdf)
6. "Robust existence of nonhyperbolic ergodic measures with positive entropy and full support" Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), with Christian Bonatti, Dominik Kwietniak (2021). (pdf)
7. "Weak* and entropy approximation of nonhyperbolic measures: a geometrical approach" Math. Proc. Cambridge Philos. Soc., with Katrin Gelfert, Bruno Santiago (2020). (pdf)
8. "Non-hyperbolic iterated function systems: semifractals and the chaos game" Fund. Math., with Edgar Matias (2020). (pdf)
9. "The structure of the space of ergodic measures of transitive partially hyperbolic sets" Monatsh. Math., with K. Gelfert, T. Marcarini, M. Rams (2019). (pdf)
10. "Hyperbolic graphs: critical regularity and box dimension" Trans. Amer. Math. Soc., with K. Gelfert, M. Gröger, T. Jäger (2019). (pdf)
11. "Entropy spectrum of Lyapunov exponents for nonhyperbolic step skew-products and elliptic cocycles" Comm. Math. Phys., with K. Gelfert, M. Rams (2019). (pdf)
12. "Attracting graphs of skew products with non-contracting fiber maps" Math. Z., with Edgar Matias (2019). (pdf)
13. "Hénon-like families and blender-horseshoes at nontransverse heterodimensional cycles" Internat. J. Bifur. Chaos Appl. Sci. Engrg., with Sebastian A. Pérez (2019). (pdf)
14. "Stability of the Markov operator and synchronization of Markovian random products" Nonlinearity, with Edgar Matias (2018). (pdf)
15. "A criterion for zero averages and full support of ergodic measures" Mosc. Math. J., with Christian Bonatti, Jairo Bochi (2018). (pdf)
16. "Nonhyperbolic step skew-products: ergodic approximation" Ann. Inst. H. Poincaré C Anal. Non Linéaire, with K. Gelfert, M. Rams (2017). (pdf)
17. "Topological and ergodic properties of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products" Tr. Mat. Inst. Steklova, with K. Gelfert, M. Rams (2017). (pdf)
18. "What is... a blender?" Notices Amer. Math. Soc., with Ch. Bonatti, S. Crovisier, A. Wilkinson (2016). (pdf)
19. "Robust criterion for the existence of nonhyperbolic ergodic measures" Comm. Math. Phys., with Jairo Bochi, Christian Bonatti (2016). (pdf)
20. "Skew product cycles with rich dynamics: from totally non-hyperbolic dynamics to fully prevalent hyperbolicity" Dyn. Syst., with Salete Esteves, Jorge Rocha (2016). (pdf)
21. "Generation of spines in porcupine-like horseshoes" Nonlinearity, with T. Marcarini (2015). (pdf)
22. "Abundant rich phase transitions in step-skew products" Nonlinearity, with K. Gelfert, M. Rams (2014). (pdf)
23. "Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations" Nonlinearity, with Shin Kiriki, Katsutoshi Shinohara (2014). (pdf)
24. "Robust vanishing of all Lyapunov exponents for iterated function systems" Math. Z., with Jairo Bochi, Christian Bonatti (2014). (pdf)
25. "Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication" Ergodic Theory Dynam. Systems, with Ch. Bonatti, S. Crovisier, N. Gourmelon (2013). (pdf)
26. "Almost complete Lyapunov spectrum in step skew-products" Dyn. Syst., with K. Gelfert, M. Rams (2013). (pdf)
27. "Entropy-expansiveness for partially hyperbolic diffeomorphisms" Discrete Contin. Dyn. Syst., with Todd Fisher, Maria José Pacifico, José L. Vieitez (2012). (pdf)
28. "Abundance of C1-robust homoclinic tangencies" Trans. Amer. Math. Soc., with Christian Bonatti (2012). (pdf)
29. "Stabilization of heterodimensional cycles" Nonlinearity, with C. Bonatti, S. Kiriki (2012). (pdf)
30. "Fragile cycles" J. Differential Equations, with Ch. Bonatti (2012). (pdf)
31. "Porcupine-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions" Fund. Math., with Katrin Gelfert (2012). (pdf)
32. "Rich phase transitions in step skew products" Nonlinearity, with Katrin Gelfert, Michał Rams (2011). (pdf)
33. "Symbolic extensions and partially hyperbolic diffeomorphisms" Discrete Contin. Dyn. Syst., with Todd Fisher (2011). (pdf)
34. "Non-hyperbolic ergodic measures with large support" Nonlinearity, with Christian Bonatti, Anton Gorodetski (2010). (pdf)
35. "Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes" Ergodic Theory Dynam. Systems, with Anton Gorodetski (2009). (pdf)
36. "Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes" Ergodic Theory Dynam. Systems, with V. Horita, I. Rios, M. Sambarino (2009). (pdf)
37. "Robust heterodimensional cycles and C1-generic dynamics" J. Inst. Math. Jussieu, with Christian Bonatti (2008). (pdf)
38. "Super-exponential growth of the number of periodic orbits inside homoclinic classes" Discrete Contin. Dyn. Syst., with Christian Bonatti, Todd Fisher (2008). (pdf)
39. "How to put a stop to domination once and for all (and what comes afterwards)" Partially hyperbolic dynamics, laminations, and Teichmüller flow, (2007). (pdf)
40. "Uma introdução aos sistemas dinâmicos via frações contínuas" Publicações Matemáticas do IMPA, with Danielle de Rezende Jorge (2007). (pdf)
41. "Periodic points and homoclinic classes" Ergodic Theory Dynam. Systems, with F. Abdenur, Ch. Bonatti, S. Crovisier, L. Wen (2007). (pdf)
42. "How do hyperbolic homoclinic classes collide at heterodimensional cycles?" Discrete Contin. Dyn. Syst., with Jorge Rocha (2007). (pdf)
43. "Pseudo-orbit shadowing in the C1 topology" Discrete Contin. Dyn. Syst., with Flavio Abdenur (2007). (pdf)
44. "Heterodimensional tangencies" Nonlinearity, with A. Nogueira, E. R. Pujals (2006). (pdf)
45. "Generic diffeomorphisms on compact surfaces" Fund. Math., with Flavio Abdenur, Christian Bonatti, Sylvain Crovisier (2005). (pdf)
46. "Collision, explosion and collapse of homoclinic classes" Nonlinearity, with Bianca Santoro (2004). (pdf)
47. "Non-wandering sets with non-empty interiors" Nonlinearity, with Flavio Abdenur, Christian Bonatti (2004). (pdf)
48. "Robustly transitive sets and heterodimensional cycles" Astérisque, with Christian Bonatti, Enrique R. Pujals, Jorge Rocha (2003). (pdf)
49. "A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources" Ann. of Math. (2), with C. Bonatti, E. R. Pujals (2003). (pdf)
50. "Critical saddle-node horseshoes: bifurcations and entropy" Nonlinearity, with Isabel L. Rios (2003). (pdf)
51. "On maximal transitive sets of generic diffeomorphisms" Publ. Math. Inst. Hautes Études Sci., with Christian Bonatti (2002). (pdf)
52. "Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms" J. Inst. Math. Jussieu, with Christian Bonatti, Raúl Ures (2002). (pdf)
53. "Heterodimensional cycles, partial hyperbolicity and limit dynamics" Fund. Math., with J. Rocha (2002). (pdf)
54. "Critical saddle-node cycles: Hausdorff dimension and persistence of tangencies" Ergodic Theory Dynam. Systems, with Raul Ures (2002). (pdf)
55. "Partially hyperbolic and transitive dynamics generated by heteroclinic cycles" Ergodic Theory Dynam. Systems, with Jorge Rocha (2001). (pdf)
56. "Pas de shadowing lemma pour des dynamiques partiellement hyperboliques" C. R. Acad. Sci. Paris Sér. I Math., with Christian Bonatti, Gaël Turcat (2000). (pdf)
57. "Partial hyperbolicity and robust transitivity" Acta Math., with Enrique R. Pujals, Raúl Ures (1999). (pdf)
58. "Connexions hétéroclines et généricité d'une infinité de puits et de sources" Ann. Sci. École Norm. Sup. (4), with Christian Bonatti (1999). (pdf)
59. "Cubic tangencies and hyperbolic diffeomorphisms" Bol. Soc. Brasil. Mat. (N.S.), with Ch. Bonatti, F. Vuillemin (1998). (pdf)
60. "Large measure of hyperbolic dynamics when unfolding heteroclinic cycles" Nonlinearity, with Jorge Rocha (1997). (pdf)
61. "Non-critical saddle-node cycles and robust non-hyperbolic dynamics" Dynam. Stability Systems, with Jorge Rocha (1997). (pdf)
62. "Strange attractors in saddle-node cycles: prevalence and globality" Invent. Math., with J. Rocha, M. Viana (1996). (pdf)
63. "Persistent nonhyperbolic transitive diffeomorphisms" Ann. of Math. (2), with Christian Bonatti (1996). (pdf)
64. "Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations" Nonlinearity, (1995). (pdf)
65. "Robust nonhyperbolic dynamics and heterodimensional cycles" Ergodic Theory Dynam. Systems, (1995). (pdf)
66. "Discontinuity of the Hausdorff dimension of hyperbolic sets" C. R. Acad. Sci. Paris Sér. I Math., with Christian Bonatti, Marcelo Viana (1995). (pdf)
67. "Persistent homoclinic tangencies and the unfolding of cycles" Ann. Inst. H. Poincaré C Anal. Non Linéaire, with Raúl Ures (1994). (pdf)
68. "Nonconnected heterodimensional cycles: bifurcation and stability" Nonlinearity, with Jorge Rocha (1992). (pdf)
69. "Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms" Ergodic Theory Dynam. Systems, with Marcelo Viana (1989). (pdf)

Books and chapters of books

• Dynamics beyond uniform hyperbolicity , with Ch. Bonatti and M. Viana, Encyclopaedia of Mathematical Sciences (Mathematical Physics), 102, Springer Verlag, (2005).
• Routes to wild dynamics , 127--143, Contibuciones Matemáticas, Homenaje al Profesor Enrique Outerelo Dominguez, (editor J. Ruiz) Editorial Complutense (UCM), (2004).
• 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).

Proceedings

• Nonhyperbolic Ergodic Measures , Proceedings of the International Congress of Mathematicians (ICM 2018), World Scientific, pp. 1887-1907, (2019).
• Blender-horseshoes in Center-unstable Hénon-like Families , with Sebastián A. Pérez, New Trends in One-Dimensional Dynamics, Springer Proceedings in Mathematics & Statistics, vol. 285, Springer, Cham, pp. 137-163, (2019).
• Porcupine-like Horseshoes: Topological and Ergodic Aspects , with Katrin Gelfert, Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics, vol. 54, Springer, Berlin, Heidelberg, (2013).
• Cycles and Systems of Iterated Functions: The Role of Secondary Bifurcations , with Jorge Rocha and Bianca Santoro, Proceedings of the First Joint Meeting RSME-AMS, Sevilla, Spain, (2003).
• The End of Domination , Abstracts Booklet of the conference New Directions in Dynamical Systems, Kyoto, Japan, pp. 204-211, (2002).
• Transitividad Robusta de Conjuntos y Difeomorfismos , Actas VII Reunión de Topólogos Españoles, pp. 1-20, (2000).