Properties of compact center-stable submanifolds (Q1745299)

The foundation of the theory of partially hyperbolic diffeomorphisms was developed in [\textit{M.W. Hirsch}, \textit{C.C. Pugh}, and \textit{M. Shub}, Invariant manifolds. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0355.58009)]. There, two cases were considered: first, it was assumed that the dynamics normal to an invariant compact submanifold of the phase space is hyperbolic; in later chapters, diffeomorphisms with a partially hyperbolic splitting on the entire phase space were discussed, which may overlap with the former case. In recent work, additional configurations were explored: thus, in [\textit{F. Rodriguez Hertz}, \textit{M.A. Rodriguez Hertz}, and \textit{R. Ures}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33, No. 4, 1023--1032 (2016; Zbl 1380.37067)], a partially hyperbolic system on the \(3\)-torus was constructed which admits a compact submanifold -- a \(2\)-torus -- that is tangent to the center and stable eigenspaces; further examples of systems with compact center-stable submanifolds were supplied in [\textit{A. Hammerlindl}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35, No. 3, 713--728 (2018; Zbl 1417.37115)], both in dimension \(3\) and in higher dimensions. Here, the author rigorously establishes a wealth of general properties of such systems; in particular, he shows that any partially hyperbolic diffeomorphism may admit at most finitely many compact center-stable submanifolds, as well as that every such manifold must be periodic, generalising previous results of [\textit{F. Rodriguez Hertz}, \textit{J. Rodriguez Hertz}, and \textit{R. Ures}, Math. Res. Lett. 23, No. 6, 1819--1832 (2016; Zbl 1375.37095)] in the process. He then proceeds to derive sufficient conditions for the existence of compact center-stable submanifolds in partially hyperbolic systems. Moreover, he considers the question of whether two of these submanifolds can have a non-empty intersection; specifically, he proves that compact center-stable submanifolds in strongly partially hyperbolic [\textit{A. Hammerlindl} and \textit{R. Potrie}, Ergodic Theory Dyn. Syst. 38, No. 2, 401--443 (2018; Zbl 1393.37038)] \(3\)-dimensional systems must be pairwise disjoint. Finally, he constructs an example that illuminates why an intersection is possible in higher dimensions. The author's results translate, \textit{mutatis mutandis}, to systems with compact center-unstable manifolds; moreover, they represent a crucial step towards a classification of partially hyperbolic diffeomorphisms in dimension \(3\) which admit center-stable tori, which will be the topic of an upcoming manuscript.