Transitively-saturated property, Banach recurrence and Lyapunov regularity

Abstract

Abstract The topological entropy of upper recurrent, lower recurrent points and Birkhoff regularity were considered in Tian (2016 Adv. Math . 288 464–526) but Banach upper recurrence points are not considered. In this paper we mainly consider whether the points that are Banach upper recurrent but not upper recurrent can carry full topological entropy. Moreover, we use statistical -limit sets to obtain entropy results on the refined orbit distribution of Banach recurrence. In this process, we strengthen Pfister and Sullivan’s result of Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst . 27 929–56) from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property). We also establish an abstract framework on the combination of Banach recurrence and multifractal analysis of general observable functions and in particular, how these results can be applied in the combination of Banach recurrence Lyapunov regularity to obtain a generalized and refined theory of Barreira and Doutor (2009 J. Math. Pures Appl . 92 1–17); Barreira and Gelfert (2006 Commun. Math. Phys . 267 393–418); Barreira and Saussol (2001 Trans. Am. Math. Soc . 353 3919–44); Feng and Huang (2010 Commun. Math. Phys . 297 1–43); Feng (2003 J. Math . 138 353–76); Feng (2009 Israel J. Math . 170 355–94); Feng and Lau (2002 Math. Res. Lett . 9 363–78) etc, on mixed Lyapunov multifractal analysis such as irregular sets and level sets of Lyapunov exponents.