Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

References

• F. Abdenur, C. Bonatti, S. Crovisier, and L.J. Díaz, Generic diffeomorphisms on compact surfaces, Fundam. Math. 187 (2005), pp. 127–159.
   Web of Science ®Google Scholar
• F. Abdenur, C. Bonatti, and S. Crovisier, Nonuniform hyperbolicity for C1-generic diffeomorphisms, Isr. J. Math. 183 (2011), pp. 1–60.
   Web of Science ®Google Scholar
• C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math. 158(1) (2004), pp. 33–104.
   Google Scholar
• T. Bousch, La condition de Walters, Ann. Sci. Éc. Norm. Supér. 34(4) (2001), pp. 287–311.
   Google Scholar
• R. Bowen, Entropy-expansive maps, Trans. Am. Math. Soc. 164 (1972), pp. 323–331.
   Web of Science ®Google Scholar
• R. Bowen, Periodic orbits for hyperbolic flows, Am. J. Math. 94 (1972), pp. 1–30.
   Web of Science ®Google Scholar
• J. Brémont, Entropy and maximizing measures of generic continuous functions, C. R. Math. Acad. Sci. Ser. I 346 (2008), pp. 199–201.
   Google Scholar
• D. Burguet, C2 surface diffeomorphisms have symbolic extensions, Invent. Math. 186 (2011),pp. 191–236.
   Web of Science ®Google Scholar
• J. Buzzi, Cr surface diffeomorphisms with no maximal entropy measure, Ergod. Theory Dyn. Syst. 34 (2014), pp. 1770–1793.
   Web of Science ®Google Scholar
• G. Contreras, Ground states are generically a periodic orbit, Invent. Math. 205 (2016),pp. 383–412.
   Web of Science ®Google Scholar
• S. Crovisier, M. Sambarino, and D. Yang, Partial hyperbolicity and homoclinic tangencies,J. Eur. Math. Soc. 17 (2015), pp. 1–49.
   Web of Science ®Google Scholar
• S. Crovisier and E. Pujals, Strongly dissipative surface diffeomorphisms, Comment. Math. Helv. 93 (2018), pp. 377–400.
   Web of Science ®Google Scholar
• L. Díaz, T. Fisher, M. Pacifico, and J. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst. 32 (2012), pp. 4195–4207.
   Web of Science ®Google Scholar
• J. Franks, Rotation vectors and fixed points of area preserving surface diffeomorphisms, Trans. Am. Math. Soc. 348 (1996), pp. 2637–2662.
   Web of Science ®Google Scholar
• J-M. Gambaudo and E. Ghys, Commutators and diffeomorphisms of surfaces, Ergod. Theory Dyn. Syst. 24 (2004), pp. 1591–1617.
   Web of Science ®Google Scholar
• S. Gan and D. Yang, Morse-Smale systems and horseshoes for three dimensional singular flows, Ann. Sci. Éc. Norm. Supér. 51 (2018), pp. 39–112.
   Web of Science ®Google Scholar
• K. Gelfert, Horseshoes for diffeomorphisms preserving hyperbolic measures, Math. Z. 283 (2016), pp. 685–701.
   Web of Science ®Google Scholar
• J. Guckenheimer and R. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), pp. 59–72.
   Google Scholar
• J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Applications. (Applied Mathematical Series, 19). Springer, New York, 1976. pp. 368–381.
   Google Scholar
• S. Hayashi, Connecting invariant manifolds and the solution of the C1 stability and Ω-stability conjectures for flows, Ann. Math. 145 (1997), pp. 81–137; 150 (1999), 353–356.
   Web of Science ®Google Scholar
• W. Huang, Z. Lian, X. Ma, L. Xu, and Y. Zhang, Ergodic optimization theory for axiom A flows. Available at arXiv:1904.10608.
   Google Scholar
• O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst. 15 (2006), pp. 197–224.
   Web of Science ®Google Scholar
• O. Jenkinson, Every ergodic measure is uniquely maximizing, Discrete Contin. Dyn. Syst. 16 (2006), pp. 383–392.
   Web of Science ®Google Scholar
• O. Jenkinson and I. Morris, Lyapunov optimizing measures for C1 expanding maps of the circle, Ergod. Theory Dyn. Syst. 28 (2008), pp. 1849–1860.
   Web of Science ®Google Scholar
• A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), pp. 137–173.
   Google Scholar
• M. Li, S. Gan, and L. Wen, Robustly transitive singular sets via approach of an extended linear Poincaré flow., Discrete Contin. Dyn. Syst. 13(2) (2020), pp. 239–269.
   Google Scholar
• S. Liao, Some uniformity properties of ordinary differential systems and a generalization of an existence theorem for periodic orbits, Beijing Daxue Xuebao 2 (1985), pp. 1–19.
   Google Scholar
• G. Liao, M. Viana, and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc. 15 (2013), pp. 2043–2060.
   Web of Science ®Google Scholar
• E. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20 (1963), pp. 130–141.
   Web of Science ®Google Scholar
• R. Mañé, An ergodic closing lemma, Ann. Math. 116 (1982), pp. 503–540.
   Web of Science ®Google Scholar
• C. Morales, M. Pacifico, and E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. Math. 160 (2004), pp. 1–58.
   Web of Science ®Google Scholar
• I. Morris, Ergodic optimization for generic continuous functions, Discrete Contin. Dyn. Syst. 27(1) (2010), pp. 383–388.
   Web of Science ®Google Scholar
• M. Morro, R. Sant'Anna, and P. Varandas, Ergodic optimization for hyperbolic flows and Lorenz attractors, Ann. Henri Poincaré 21(10) (2020), pp. 3253–3283.
   Web of Science ®Google Scholar
• S. Newhouse, Continuity properties of entropy, Ann. Math. 129 (1989), pp. 215–235.
   Web of Science ®Google Scholar
• M. Pacifico, F. Yang, and J. Yang, Entropy theory for sectional hyperbolic flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 38(4) (2021), pp. 1001–1030.
   Web of Science ®Google Scholar
• J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque 261 (2000), pp. 339–351.
   Google Scholar
• M. Pollicott and R. Sharp, Livŝic theorems, maximizing measures and the stable norm, Dyn. Syst. 19 (2004), pp. 75–88.
   Web of Science ®Google Scholar
• C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compos. Math. 23 (1971),pp. 115–122.
   Web of Science ®Google Scholar
• F. Rodriguez Hertz, M.A. Rodriguez Hertz, A. Tahzibi, and R. Ures, Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Commun. Math. Phys. 306 (2011),pp. 35–49.
   Web of Science ®Google Scholar
• D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), pp. 83–87.
   Google Scholar
• O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Am. Math. Soc. 26 (2013), pp. 341–426.
   Web of Science ®Google Scholar
• Y. Shi, S. Gan, and L. Wen, On the singular-hyperbolicity of star flows, J. Mod. Dyn. 8(2) (2014), pp. 191–219.
   Web of Science ®Google Scholar
• Y. Shi, F. Yang, and J. Yang, A countable partition for singular flows, and its application on the entropy theory. Available at arXiv:1908.01380.
   Google Scholar
• K. Sigmund, Generic properties of invariant measures for axiom a diffeomorphisms, Invent. Math. 11 (1970), pp. 99–109.
   Web of Science ®Google Scholar
• L. Wen, On the C1 stability conjecture for flows, J. Differ. Equ. 129 (1996), pp. 334–357.
   Web of Science ®Google Scholar
• L. Wen and Z. Xia, C1 connecting lemmas, Trans. Am. Math. Soc. 352 (2000), pp. 5213–5230.
   Web of Science ®Google Scholar
• R. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), pp. 73–99.
   Google Scholar
• W. Wu, D. Yang, and Y. Zhang, On the growth rate of periodic orbits for vector fields, Adv. Math. 346 (2019), pp. 170–193.
   Web of Science ®Google Scholar
• Y. Yomdin, Volume growth and entropy, Isr. J. Math. 57 (1987), pp. 285–300.
   Web of Science ®Google Scholar