Theorem 1. Let the countable abelian group G act nonsingularly and aperiodically on Lebesgue space (X, p.). Then for each finite subset A c G and e > 0 3 finite B c G and F tz X with [bF: bEB} disjoint and PKfl meAB a)F] > 1-e. Theorem 2. Every nonsingular action of a countable abelian group on a Lebesgue space is hyperfinite.

We review recent results which relate spectral theory of discrete one-dimensional Schrödinger operators over strictly ergodic systems to uniform existence of the Lyapunov exponent. In combination with suitable ergodic theorems this allows one to establish Cantor spectrum of Lebesgue measure zero for a large class of quasicrystal Schrödinger operators. The results can also be used to study non-u...

We investigate a one-parameter family of interval maps arising in the study of the geometric Lorenz ow for non-classical parameter values. Our conclusion is that for all parameters in a set of positive Lebesgue measure, the map has a positive Lyapunov exponent. Furthermore, this set of parameters has a density point which plays an important dynamic role. The presence of both singular and critic...

We establish a continuous embedding $W^{s(\cdot),2}(\Omega)\hookrightarrow L^{\alpha(\cdot)}(\Omega)$, where the variable exponent $\alpha(x)$ can be close to critical $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\bar{\Omega}$. Subsequently, this is used prove multiplicity of solutions nonlocal degenerate Kirchhoff problems singular exponent. Moreover, we also ob...

‎We present a Picone's identity for the‎ ‎$mathcal{A}_{p(x)}$-Laplacian‎, ‎which is an extension of the classic‎ ‎identity for the ordinary Laplace‎. ‎Also‎, ‎some applications of our‎ ‎results in Sobolev spaces with variable exponent are suggested.