Abstract
We prove that all star vector fields, including Lorenz attractors and multi-singular hyperbolic vector fields, admit the intermediate entropy property. To be precise, if X is a star vector field with htop(X) > 0, then for any h ∈ [0, htop(X)), there exists an ergodic invariant measure μ of X such that hμ(X)= h. Moreover, we show that the topological entropy is lower semi-continuous for star vector fields.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. M. Abramov, On the entropy of a flow, Doklady Akademii Nauk SSSR 128 (1959), 873–875.
N. Aoki, The set of Axiom A diffeomorphisms with no cycles, Boletim da Sociedade Brasileira de Matemática 23 (1992), 21–65.
V. Araujo, M. Pacifico, E. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Transactions of the American Mathematical Society 361 (2009), 2431–2485.
F. Béguin, S. Crovisier and F. Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy—Rees technique, Annales Scientifiques de l’École Normale Supérieure 40 (2007), 251–308.
C. Bonatti and A. da Luz, Star flows and multisingular hyperbolicity, Journal of the European Mathematical Society, to appear, https://arxiv.org/abs/1705.05799.
A. da Luz, Star flows with singularities of different indices, https://arxiv.org/abs/1806.09011.
S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Inventiones Mathematicae 164 (2006), 279–315.
S. Gan and D. Yang, Morse—Smale systems and horseshoes for three-dimensional singular flows, Annales Scientifiques de l’École Normale Supérieure 51 (2018), 39–112.
F. R. Gantmacher, Applications of the Theory of Matrices, Interscience, New York, 1959.
L. Guan, P. Sun and W. Wu, Measures of intermediate entropies and homogeneous dynamics, Nonlinearity 9 (2017), 3349–3361.
J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and its Applications, Applied Mathematical Sciences, Vol. 19, Springer, New York, 1976, pp. 368–381.
F. Hahn and Y. Katznelson, On the entropy of uniquely ergodic transformations, Transactions of the American Mathematical Society 126 (1967), 335–360.
S. Hayashi, Diffeomorphisms in \({{\cal F}^1}(M)\) satisfy Axiom A, Ergodic Theory and Dynamical Systems 12 (1992), 233–253.
M. Herman, Construction d’un difféomorphisme minimal d’entropie topologique non nulle, Ergodic Theory and Dynamical Systems 1 (1981), 65–76.
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer, Berlin—New York, 1977.
A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Institut des Hautes Études Scientifiques. Publications Mathematiques 51 (1980), 137–173.
A. Katok and L. Mendoza, Dynamical systems with nonuniformly hyperbolic behavior, in Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Vol. 54, Cambridge University Press, Cambridge, 1995, pp. 657–700.
M. Li, S. Gan and L. Wen, Robustly transitive singular sets via approach of extended linear Poincaré flow, Discrete and Continuous Dynamical Systems 13 (2005), 239–269.
S. Liao, Obstruction sets. II, Beijing Daxue Xuebao 2 (1981), 1–36.
S. Liao, On (η, d)-contractable orbits of vector fields, Systems Science and Mathematical Sciences 2 (1989), 193–227.
R. Mañé, An ergodic closing lemma, Annals of Mathematics 116 (1982), 503–540.
R. Metzger and C. A. Morales, On sectional-hyperbolic systems, Ergodic Theory and Dynamical Systems 28 (2008), 1587–1597.
C. A. Morales, M. J. Pacifico and E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Annals of Mathematics 160 (2004), 375–432.
M. J. Pacifico, F. Yang and J. Yang, Entropy theory for sectional hyperbolic flows, https://arxiv.org/abs/1901.07436.
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astŕisque 187-188 (1990).
C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Mathematica 23 (1971), 115–122.
Y. Shi, S. Gan and L. Wen, On the singular hyperbolicity of star flows, Journal of Modern Dynamics 8 (2014), 191–219.
P. Sun, Measures of intermediate entropies for skew product diffeomorphisms, Discrete and Continuous Dynamical Systems 27 (2010), 1219–1231.
P. Sun, Zero-entropy invariant measures for skew product diffeomorphisms, Ergodic Theory and Dynamical Systems 30 (2010), 923–930.
P. Sun, Density of metric entropies for linear toral automorphisms, Dynamical Systems 27 (2012), 197–204.
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer, New York—Berlin, 1982.
L. Wen, Differentiable Dynamical Systems, Graduate Studies in Mathematics, Vol. 173, American Mathematical Society, Providence, RI, 2016.
W. Wu, D. Yang and Y. Zhang, On the growth rate of periodic orbits for vector fields, Advances in Mathematics 346 (2019), 170–193.
J. Yang, Topological entropy of Lorenz-like flows, https://arxiv.org/abs/1412.1207.
Author information
Authors and Affiliations
Corresponding author
Additional information
M. Li is supported by NSFC 11571188 and the Fundamental Research Funds for the Central Universities.
Y. Shi is supported by NSFC 11701015 and Young Elite Scientists Sponsorship Program by CAST.
S. Wang is partially supported by NSFC 11771026, 11471344 and acknowledges PIMS-CANSSI postdoctoral fellowship.
X. Wang is the corresponding author, and supported by NSFC 11701366 and Shanghai Sailing Program 17YF1409300.
Rights and permissions
About this article
Cite this article
Li, M., Shi, Y., Wang, S. et al. Measures of intermediate entropies for star vector fields. Isr. J. Math. 240, 791–819 (2020). https://doi.org/10.1007/s11856-020-2080-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-020-2080-2