We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of ${\mathcal B}$-free subshifts satisfying them, extending the work of Dymek [Automorphisms of Toeplitz ${\mathcal B}$-free systems. Bull. Pol. Acad. Sci. Math. 65(2) (2017), 139–152]. Additionally, we provide an example of a ${\mathcal B}$-free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 of S. Ferenczi et al [Sarnak’s conjecture: what’s new. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213). Eds. S. Ferenczi, J. Kułaga-Przymus and M. Lemańczyk. Springer, Cham, 2018, pp. 163–235].

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

The effect of weed interference on corn yield and the critical period for weed control (CPWC) were determined in Germany and Benin. Treatments with weed control starting at different crop growth stages and continuously kept weed-free until harvest represented the “weed-infested interval.” Treatments that were kept weed-free from sowing until different crop growth stages represented the “weed-free interval.” Michaelis–Menten, Gompertz, logistic and log–logistic models were employed to model the weed interference on yield. Cross-validation revealed that the log–logistic model fitted the weed-infested interval data equally well as the logistic and slightly better than the Gompertz model fitted the weed-free interval. For Benin, economic calculations considered yield revenue and cost increase due to mechanical weeding operations. Weeding once at the ten-leaf stage of corn resulted already profitable in three out of four cases. One additional weeding operation may optimize and assure profit. Economic calculations for Germany determined a CPWC starting earlier than the four-leaf stage, challenging the decade-long propagated CPWC for corn. Differences between Germany and Benin are probably due to the higher yields and high costs in Germany. This study provides a straightforward method to implement economic data in the determination of the CPWC for chemical and nonchemical weed control strategies.

Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.

We carry the argument used in the proof of Denjoy's theorem over to the quasiperiodically forced case. Thus we derive that, if a system of quasiperiodically forced circle diffeomorphisms with bounded variation of the derivative has no invariant graphs with a certain kind of topological regularity, then the system is topologically transitive.

Let X\subset\mathbb{R}^2 be a finite union of bounded polytopes and let T:X\to X be piecewise affine and eventually expanding. Then the Perron–Frobenius operator \mathcal{L} of T is quasicompact as an operator on the space of functions of bounded variation on \mathbb{R}^2 and its isolated eigenvalues (including multiplicities) are just the reciprocals of the poles of the dynamical zeta function of T. In higher dimensions the result remains true under an additional generically satisfied transversality assumption.

For S-unimodal maps $f$, we study equilibrium states maximizing the free energies $F_t(\mu) := h(\mu) - t\int \log|f'|\,d\mu$ and the pressure function $P(t):=\sup_\mu F_t(\mu)$. It is shown that if $f$ is uniformly hyperbolic on periodic orbits, then $P(t)$ is analytic for $t\approx 1$. On the other hand, examples are given where no equilibrium states exist, where equilibrium states are not unique and where the notions of equilibrium state for $t=1$ and of observable measure do not coincide.