Motivated by fractal geometry of self-affine carpets and sponges, Feng and Huang [J. Math. Pures Appl.106(9) (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng and Huang. The equivalence of the two definitions seems highly non-trivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford–McMullen carpet in purely topological terms.

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^{2}$-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of $\mathbb{Z}^{2}$-subshifts with respect to a subaction of $\mathbb{Z}$. The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^{2}$-subshifts in terms of Kolmogorov–Sinai entropy.

We show that if $(X, T)$ is an extension of an aperiodic subshift (a subsystem of $(\mathop{\{ 1, 2, \ldots , l\} }\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$ for some $l\in \mathbb{N} $) and has mean dimension $\mathrm{mdim} (X, T)\lt (D/ 2), D\in \mathbb{N} $, then it can be equivariantly embedded in $(\mathop{(\mathop{[0, 1] }\nolimits ^{D} )}\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$. The result is sharp. If $(X, T)$ is an extension of an aperiodic zero-dimensional system then it can be equivariantly embedded in $(\mathop{(\mathop{[0, 1] }\nolimits ^{D+ 1} )}\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$.

The main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

We propose a new approach to the value distribution theory of entire holomorphic curves. We define packing density of Brody curves, and show that it has various non-trivial properties. The packing density of Brody curves can be considered as an infinite dimensional version of characteristic number, and it has an application to Gromov’s mean dimension theory.

A Brody curve is a holomorphic map from the complex plane ℂ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its “mean dimension”. We introduce the notion of “mean energy” and show that this can be used to estimate the mean dimension.

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an “infinite energy” situation. We show that we can glue an infinite number of instantons, and that the resulting ASD connections have infinite energy in general. Moreover they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson’s “alternating method”.

Diffusion experiments of radionuclides in compacted sodium bentonite with a dry density of 1.0 g/cm3 were performed in nitrogen gas atmosphere at 90 °C for 208 d and 375 d. The corrosion experiments of crushed radioactive glass, JSS-A, carried out simultaneously to provide the source of the radionuclides for the diffusion experiments. The normalized elemental mass losses of cesium isotopes and 238Pu were lower than those of boron (ca. 10 g/m2) probably because of the difference of sorption and/or precipitation. The apparent diffusion coefficients of 238Pu, 234U and 125Sb were determined to be 2x 10-14 m2/s, 5x 10-12 m2/s and 2x 10-12 m2/s, respectively. The distribution coefficient of Pu estimated from the diffusion data was of the same order as that from batch sorption experiments. The glass corrosion and the plutonium diffusion were described by the geochemical codes PHREEQE, STRAG4 and GESPER. The calculation results well fitted the observed data.

The release of Am-241 during corrosion of the radioactive waste glass, JSS-A, has been studied in the presence of corrosion products and/or uncom-pacted bentonite. The corrosion behaviour of Am-241 has been analyzed using gamma spectrometry. Adsorption of Am-241 on bentonite is observed in all cases. The contents of Am-241 in centrifuged leachates are in most cases less than 1/100 of total values. The normalized elemental mass loss of Am increases initially with corrosion time, and the values in the presence of bentonite and corrosion products are larger than those in the presence of bentonite alone. This tendency is in agreement with results previously found for other elements. The release of Am is low, only about 10–20 % of the corresponding total mass loss.