Variation of physical measures in nontrivial mixed partially hyperbolic systems

We construct a C∞ nontrivial mixed partially hyperbolic system and explicitly identify its skeleton. This example shares characteristics with the classical examples studied by Smale [Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967), pp. 747–817. https://doi.org/10.1090/S0002-9904-1967-11798-1], A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, (1995), https://doi.org/10.1017/CBO9780511809187. Shub [Topologically transitive diffeomorphisms of T4, in Proceedings of the Symposium on Differential Equations and Dynamical Systems, D. Chillingworth, ed., Springer Berlin Heidelberg, 1971, pp. 39–40. ISBN 978-3-540-36662-1. https://doi.org/10.1007/BFb0070139], Kan [Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Am. Math. Soc. (N.S.) 31(1) (1994), pp. 68–74. https://doi.org/10.1090/S0273-0979-1994-00507-5], and Dolgopyat–Viana–Yang [Geometric and measure-theoretical structures of maps with mostly contracting center, Commun. Math. Phys. 341(3) (2016), pp. 991–1014. https://doi.org/10.1007/s00220-015-2554-y]. Moreover, the support of each physical measure contains three hyperbolic fixed points with mutually distinct unstable indices. By appropriately perturbing the skeleton, we provide an example where the number of physical measures varies upper semicontinuously. The general framework of mixed partially hyperbolic systems has been studied in Mi et al. [A note on partially hyperbolic systems with mostly expanding centers, Proc. Am. Math. Soc. 145(12) (2017), pp. 5299–5313. https://doi.org/10.1090/proc/13701] and Mi and Cao [Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers, Math. Z. 299(3-4) (2021), pp. 2519–2560. https://doi.org/10.1007/s00209-021-02766-y].