arXiv subjects

Kanji Inui

Publications and source records attributed to Kanji Inui.

6 recordsLinked to original sources

arXiv2026

Rate distortion dimension of Gibbs measures for functions depending on the first two coordinates on the full shift of Ahlfors regular spaces

The study on shift spaces in ergodic theory has extended beyond the classical setting, but there is room to discuss an extension of the Kolmogorov-Sinai entropy in the ergodic theoretical point of view. On the other hand, the rate distortion dimension recently attracted attention in mean dimension theory because it behaves like the Kolmogorov-Sinai entropy on dynamical systems in the ``large" spaces in which the usual entropy is in general infinite. According to these background, we investigate the connection between the Gibbs measure on the product spaces and the variational principle based on the rate distortion dimension. Concretely we calculate the rate distortion dimension of the Gibbs measure for a H\"older continuous potential depending on the first two coordinates and it satisfies the simplest case of themodynamic formalism based on the rate distortion dimension: the extension of the maximal measure of topological entropy. Notably, to the best of our knowledge, this is the first study to introduce Gibbs measures into mean dimension theory, and this connection allows the mean dimension with potential to be computed more naturally through the notion of the zero-temperature limit, in the spirit of thermodynamic formalism.

math.DS↗

arXiv2026

Pointwise regularity and irregularity of energy densities on $N$-dimensional Sierpinski gaskets

We study the pointwise regularity of energy densities associated with harmonic functions on the $N$-dimensional Sierpinski gasket $(N\ge 2)$ with respect to the Kusuoka measure. For any nonconstant harmonic function, we prove that every Borel representative of the density is discontinuous at every point of a set of full Kusuoka measure. In sharp contrast, on each one-dimensional edge of the gasket -- itself a set of zero Kusuoka measure -- the density admits a canonical pointwise version, which is $\gamma_N$-H\"older continuous on that edge with the explicit and optimal exponent $\gamma_N=\log_2 \{(\sqrt{4N+5}+1)/(\sqrt{4N+5}-1)\}$.

math.AP↗

arXiv2023

Understanding the limit sets generated by general iterated function systems on unbounded spaces

In this paper, we reformulate the definition of the iterated function systems (denoted by general IFSs in this paper) and show the existence and uniqueness (in some sense) of the limit sets generated by the general IFSs, to unify the definitions of the limit sets introduced before. Note that the general IFSs are defined on (possibly unbounded) complete metric spaces and we instead assume a ``natural" condition of general IFSs to show the main result. To obtain the main result, we apply techniques in the Banach fixed point theorem to the general IFSs with the ``natural" condition. Besides, we consider an example of general IFSs.

math.DS↗

arXiv2022

Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set is zero and the packing measure of the limit set with respect to the Hausdorff dimension is positive. In this paper, we show that the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal, and the proper-dimensional packing measure of the limit set is finite.

math.DS↗

arXiv2018

Hausdorff measures and packing measures of the limit sets of CIFSs of generalized complex continued fractions

We consider a family of CIFSs of the generalized complex continued fractions with a complex parameter space. We show that for each CIFS of the family, the Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff dimen/sion is zero and the packing measure of the limit set of the CIFS with respect to the Hausdorff dimension is positive (main result). This is a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the main result by showing some estimates for the unique conformal measure of each CIFS of the family and by using some geometric observations.

math.DS↗

arXiv2018

The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions

We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs. We show that the Hausdorff dimension function of the family of the CIFSs of generalized complex continued fractions is continuous in the parameter space and is real-analytic and subharmonic in the interior of the parameter space. As a corollary of these results, we also show that the Hausdorff dimension function has a maximum point and the maximum point belongs to the boundary of the parameter space.

math.DS↗