Measures of intermediate entropies for star vector fields
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MEASURES OF INTERMEDIATE ENTROPIES FOR STAR VECTOR FIELDS BY
Ming Li∗ School of Mathematical Sciences and LPMC, Nankai University Tianjin 300071, P. R. China e-mail: [email protected] AND
Yi Shi∗∗ School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China e-mail: [email protected] AND
Shirou Wang† Department of Mathematical and Statistical Sciences, University of Alberta Edmonton T6G2G1, Alberta, Canada e-mail: [email protected] AND
Xiaodong Wang†† School of Mathematical Sciences, Shanghai Jiao Tong University Shanghai 200240, P. R. China e-mail: [email protected]
∗ 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. Received May 10, 2019 and in revised form October 9, 2019
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M. LI, Y. SHI, S. WANG AND X. WANG
Isr. J. Math.
ABSTRACT
We prove that all star vector fields, including Lorenz attractors and multisingular 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.
1. Introduction The concept of entropy was introduced by Kolmogorov in 1958, which has been the most important invariant in ergodic theory and dynamical systems during the past 60 years. It reflects the complexity of the dynamical system. In some circumstances, the positivity of entropy forces the system to have typical structure. For instance, Katok proved the following milestone theorem. Theorem ([16]): Let f be a C r (r > 1) surface diffeomorphism and μ be an ergodic measure of f with hμ (f ) > 0. Then for any ε > 0, there exists a hyperbolic horseshoe Λε satisfying htop (f, Λε ) > hμ (f ) − ε. A similar result holds for higher dimensional diffeomorphisms having a hyperbolic measure μ with positive measure entropy; see [17]. Combined with the variational principle, this theorem implies the lower semi-continuity of the topological entropy for C r (r > 1) surface diffeomorphisms. Moreover, a hyperbolic horseshoe is conjugate to a full shift. Thus every C r (r > 1) surface diffeomorphism f has the intermediate entropy property, i.e., for any constant h ∈ [0, htop (f )), there exists an ergodic measure μ of f satisfying hμ (f ) = h. Katok raised the following conjecture. Conjecture: Every C r (r ≥ 1) diffeomorphism f on a manifold satisfies the intermediate entropy property. Not all systems admit the intermediate entropy property. There are uniquely ergodic homeomorphisms [12] with positive topological entropy; see also [4]. It seems that the system is required to have
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