We prove the central limit theorem (CLT), the first-order Edgeworth expansion and a mixing local central limit theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb {R}$. The class of observables in the CLT and the MLCLT on $\mathbb {R}$ include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber [Sampling the Lindelöf hypothesis with the Cauchy random walk. Proc. Lond. Math. Soc. (3) 98 (2009), 241–270] and Steuding [Sampling the Lindelöf hypothesis with an ergodic transformation. RIMS Kôkyûroku Bessatsu B34 (2012), 361–381] who have proven the strong law of large numbers for sampling the Lindelöf hypothesis.

In this paper, we describe finite order entire solutions of nonlinear partial differential equation

\begin{equation*} \prod _{i=1}^n\left (a_{i 1} u_{z_1}+a_{i 2} u_{z_2}+\cdots +a_{i n} u_{z_n}\right )=1 \end{equation*}

We consider twist diffeomorphisms of the torus, $f:\mathrm {T^2\rightarrow T^2,}$ and their vertical rotation intervals, $\rho _V(\widehat {f})=[\rho _V^{-},\rho _V^{+}],$ where $\widehat {f}$ is a lift of f to the vertical annulus or cylinder. We show that $C^r$-generically, for any $r\geq 1$, both extremes of the rotation interval are rational and locally constant under $C^0$-perturbations of the map. Moreover, when f is area-preserving, $C^r$-generically, $\rho _V^{-}<\rho _V^{+}$. Also, for any twist map f, $\widehat {f}$ a lift of f to the cylinder, if $\rho _V^{-}<\rho _V^{+}=p/q$, then there are two possibilities: either $\widehat {f}^q(\bullet )-(0,p)$ maps a simple essential loop into the connected component of its complement which is below the loop, or it satisfies the curve intersection property. In the first case, $\rho _V^{+} \leq p/q$ in a $C^0$-neighborhood of $f,$ and in the second case, we show that $\rho _V^{+}(\widehat {f}+(0,t))>p/q$ for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the $C^r$-generic case, assuming that $\rho _V^{-}<\rho _V^{+}=p/q,$ we present some consequences of the existence of the free loop for $\widehat {f}^q(\bullet )-(0,p)$, related to the description and shape of the attractor–repeller pair that exists in the annulus. The case of a $C^r$-generic transitive twist diffeomorphism (if such a thing exists) is also investigated.

We study the geometry induced on the local orbit spaces of Killing vector fields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G=\textrm {Spin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms with one-dimensional fibres from $G_2$-manifolds to Einstein manifolds.

We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz–Thurston type. We prove that for any point z in the interval, there is a local escape rate and hitting time statistics that is one of three types. While it is key that we cover all points z, the particular interest here is when z is periodic and in the postcritical orbit that yields the third part of the trichotomy. We also prove generalized asymptotic escape rates of the form first shown by Bruin, Demers and Todd.

In 1954, B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018, G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.

We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non-linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $\Lambda^3(\mathcal{T})$, where ${\mathcal{T}}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms.

Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular, we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing–Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterization of algebraically special Riemannian metrics of type D in dimensions higher than four.

The results on Γ-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the Γ-limit, which is then used to study deterministic and stochastic homogenization problems for this type of functional.

Using bi-contact geometry, we define a new type of Dehn surgery on an Anosov flow with orientable weak invariant foliations. The Anosovity of the new flow is strictly connected to contact geometry and the Reeb dynamics of the defining bi-contact structure. This approach gives new insights into the properties of the flows produced by Goodman surgery and clarifies under which conditions Goodman’s construction yields an Anosov flow. Our main application gives a necessary and sufficient condition to generate a contact Anosov flow by Foulon–Hasselblatt Legendrian surgery on a geodesic flow. In particular, we show that this is possible if and only if the surgery is performed along a simple closed geodesic. As a corollary, we have that any positive skewed $\mathbb {R}$-covered Anosov flow obtained by a single surgery on a closed orbit of a geodesic flow is orbit equivalent to a positive contact Anosov flow.

Seasonal changes and cyclical human activities (such as periodic fishing bans, Wolbachia-based mosquito population control, and school term breaks) have significant impacts on population dynamics. We propose a general switching dynamical model to describe these periodic changes. The existence, uniqueness and stability of positive periodic solutions are thoroughly investigated. The results are stated in terms of an introduced threshold value. To demonstrate their practicability, the obtained results are applied to two biological situations.

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

We consider the drifting p-Laplace operator

\begin{equation*}\Delta_{p,v}u=e^{-v} \text{div}\,(e^v |\nabla u|^{p-2}\nabla u)\end{equation*}

and discuss generalized weighted Hardy-type inequalities associated with the measure $d\mu=e^{v(x)}dx$. As an application, we obtain several Liouville-type results for positive solutions of the non-linear elliptic problem with singular lower order term

\begin{equation*}-\Delta_{p,v} u\ge c(x) u^{p-1}+B \frac{|\nabla u|^p}{u} \quad \text{in}\ \Omega,\end{equation*}

where Ω is a bounded or an unbounded exterior domain in ${\mathbb{R}}^N$, $N \gt p \gt 1$, $B+p-1 \gt 0$, as well as of the non-autonomous quasilinear elliptic problem

\begin{equation*}-\Delta_{p,v} u+b(x)|\nabla u|^{p-1} \ge c(x) u^{p-1}\quad \text{in}\ \Omega,\end{equation*}

with general weights $b\ge0$ and c > 0. Liouville-type results are also discussed for a class of higher order differential equations.

We solve generalizations of Hubbard’s twisted rabbit problem for analogs of the rabbit polynomial of degree $d\geq 2$. The twisted rabbit problem asks: when a certain quadratic polynomial, called the Douady rabbit polynomial, is twisted by a cyclic subgroup of a mapping class group, to which polynomial is the resulting map equivalent (as a function of the power of the generator)? The solution to the original quadratic twisted rabbit problem, given by Bartholdi and Nekrashevych, depended on the 4-adic expansion of the power of the mapping class by which we twist. In this paper, we provide a solution to a degree-d generalization that depends on the $d^2$-adic expansion of the power of the mapping class element by which we twist.

We extend a result of Lopes and Thieullen [Sub-actions for Anosov flows. Ergod. Th. & Dynam. Sys. 25(2) (2005), 605–628] on sub-actions for smooth Anosov flows to the setting of geodesic flow on locally CAT($-1$) spaces. This allows us to use arguments originally due to Croke and Dairbekov to prove a volume rigidity theorem for some interesting locally CAT($-1$) spaces, including quotients of Fuchsian buildings and surface amalgams.

We establish some interactions between uniformly recurrent subgroups (URSs) of a group G and cosets topologies $\tau _{\mathcal {N}}$ on G associated to a family $\mathcal {N}$ of normal subgroups of G. We show that when $\mathcal {N}$ consists of finite index subgroups of G, there is a natural closure operation $\mathcal {H} \mapsto \mathrm {cl}_{\mathcal {N}}(\mathcal {H})$ that associates to a URS $\mathcal {H}$ another URS $\mathrm {cl}_{\mathcal {N}}(\mathcal {H})$, called the $\tau _{\mathcal {N}}$-closure of $\mathcal {H}$. We give a characterization of the URSs $\mathcal {H}$ that are $\tau _{\mathcal {N}}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when G belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\mathcal {A}_G$ and prove that for certain coset topologies on G, almost all subgroups $H \in \mathcal {A}_G$ have the same closure. For groups in which amenability is detected by a set of laws (a property that is variant of the Tits alternative), we deduce a criterion for $\mathcal {A}_G$ to be a singleton based on residual properties of G.

We investigate the pullback measure attractors for non-autonomous stochastic p-Laplacian equations driven by nonlinear noise on thin domains. The concept of complete orbits for such systems is presented to establish the structures of pullback measure attractors. We first present some essential uniform estimates, as well as the existence and uniqueness of pullback measure attractors. A novel technical proof method is shown to overcome the difficulty of the estimates of the solutions in $W^{1,p}$ on thin domains. Then, we prove the upper semicontinuity of these measure attractors as the $(n + 1)$-dimensional thin domains collapse onto the lower n-dimensional space.

This paper develops methods for simplifying systems of partial differential equations (PDEs) that have families of conservation laws which depend on arbitrary functions of the independent or dependent variables. Cases are identified in which such methods can be combined with reduction using families of symmetries to give a multiple reduction; this is analogous to the double reduction of order for ordinary differential equations (ODE) with variational symmetries. Applications are given, including a widely used class of pseudoparabolic equations and several mean curvature equations.

We study the behaviour of (resonant) dynamic B-tipping in a forced two-dimensional nonautonomous system, close to a nonsmooth saddle-focus (NSF) bifurcation. The NSF arises when a saddle-point and a focus meet at a border collision bifurcation. The emphasis is on the Stommel 2-box model, which is a piecewise-smooth continuous dynamical system, modelling thermohaline circulation. This model exhibits an NSF as parameters vary. By using techniques from the theory of nonsmooth dynamical systems, we are able to provide precise estimates for the general tipping behaviour close to the bifurcation as parameters vary. In particular, we consider the combination of both slow drift and also periodic changes in the parameters, corresponding, for example, to the effects of slow climate change and seasonal variations. The results are significantly different from the usual B-tipping point estimates close to a saddle-node bifurcation. In particular, we see a more rapid rate of tipping in the slow drift case, and an advancing of the tipping point under periodic changes. The latter is made much more pronounced when the periodic variation resonates with the natural frequency of the focus, leading both to much more complicated behaviour close to tipping and also significantly advanced tipping in this case.