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Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper, we will give a systematically study to the packing topological entropy for a continuous $G$-action dynamical system $(X,G)$, where $X$ is a compact metric space and $G$ is a countable discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset $Z$ of $X$, the packing topological entropy of $Z$ equals the supremum of upper local entropy over all Borel probability measures for which the subset $Z$ has full measure. And then we obtain an entropy inequality concerning amenable packing entropy. Finally we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu$ coincides with the metric entropy if either $\mu$ is ergodic or the system satisfies a kind of specification property., Comment: 35 pages; many mistakes in previous version are corrected and section 5 is rewrotten

Transparent amorphous oxide semiconductors (TAOSs) based transparent thin-film transistors (TTFTs) with high field effect mobility are essential for developing advanced flat panel displays. Among TAOSs, amorphous (a-) SnO$_2$ has several advantages against current a-InGaZnO4 such as higher field effect mobility and being indium free. Although a-SnO$_2$ TTFT has been demonstrated several times, the operation mechanism has not been clarified thus far due to the strong gas sensing characteristics of SnO$_2$. Here we clarify the operation mechanism of a-SnO$_2$ TTFT by electric field thermopower modulation analyses. We prepared a bottom-gate top-contact type TTFT using 4.2-nm-thick a-SnO$_2$ as the channel without any surface passivation. The effective thickness of the conducting channel was ~1.7 + - 0.4 nm in air and in vacuum, but a large threshold gate voltage shift occurred in different atmospheres; this is attributed to carrier depletion near at the top surface (~2.5 nm) of the a-SnO$_2$ due to its interaction with the gas molecules and the resulting shift in the Fermi energy. The present results would provide a fundamental design concept to develop a-SnO$_2$ TTFT.

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system., Comment: 18 pages

In this short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along tempered F{\o}lner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some F{\o}lner sequence) equals its topological entropy. This answers questions by Zheng and Chen (Israel Journal of Mathematics 212 (2016), 895-911) and Simpson (Theory Comput. Syst. 56 (2015), 527-543).

In this paper, we prove variational principles between metric mean dimension and rate distortion function for countable discrete amenable group actions which extend recently results by Lindenstrauss and Tsukamoto., Comment: 36 pages. Comments are welcome

Let $G$ be a countable infinite amenable group and $P$ be a polyhedron. We give a construction of minimal subshifts of $P^G$ with arbitrarily mean topological dimension less than $\dim P$., Comment: 15 pages

The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight to analyze the entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate the joinings among entropy zero systems and prove the disjointness property among entropy zero systems using the dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.