We study piecewise injective, but not necessarily globally injective, contracting maps on a compact subset of ${\mathbb R}^d$. We prove that, generically, the attractor and the set of discontinuities of such a map are disjoint, and hence the attractor consists of periodic orbits. In addition, we prove that piecewise injective contractions are generically topologically stable.

We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the $C^0$ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J. 146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math. 348 (2019), 353–377] and Zhang (2020).

A probability measure-preserving action of a discrete amenable group G is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb {Z}$, an action is dominant if and only if it has positive entropy and that for any G, positive entropy implies dominance. In this paper, we show that the converse also holds for any G, that is, that zero entropy implies non-dominance.

An ergodic dynamical system $\mathbf {X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown by Glasner et al [On some generic classes of ergodic measure preserving transformations. Trans. Moscow Math. Soc. 82(1) (2021), 15–36] that Bernoulli systems with finite entropy are dominant. In this work, we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf {X}$ has zero entropy, then it is not dominant.

We consider a class of Schrödinger operators on ${\open R}^N$ with radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in $L^2({\open R}^N)$, we show that the following properties are generic with respect to the potential:

1. (P1) the eigenvalues below the essential spectrum are nonresonant (i.e., rationally independent) and so are the square roots of the moduli of these eigenvalues;

2. (P2) the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set.

We prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Mañé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.