We shall investigate topological properties of a uniquely determined compact set K such that K = ~a~Afx(K), where eachfx is a weak contraction of a complete metric space and A = { 1, 2, " ', m} or A = N. Such a set K is said to be self-similar. Many classical peculiar sets can be represented in this form. We shall also discuss the interesting problem presented by R . F. Williams.

In this paper, we generalize the Takagi function which is known as a nowhere differentiable continuous function, and we clarify the close relation between the Takagi function and Lebesgue's singular function with de Rham's functional equation.

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At Naka‐dake cone, Aso caldera, Japan, volcanic activity is raised cyclically, an example of which was a phreatomagmatic eruption in September 2015. Using a three‐dimensional model of electrical resistivity, we identify a magma pathway from a series of northward dipping conductive anomalies in the upper crust beneath the caldera. Our resistivity model was created from magnetotelluric measurements conducted in November–December 2015; thus, it provides the latest information about magma reservoir geometry beneath the caldera. The center of the conductive anomalies shifts from the north of Naka‐dake at depths >10 km toward Naka‐dake, along with a decrease in anomaly depths. The melt fraction is estimated at 13–15% at ~2 km depth. Moreover, these anomalies are spatially correlated with the locations of earthquake clusters, which are distributed within resistive blocks on the conductive anomalies in the northwest of Naka‐dake but distributed at the resistive sides of resistivity boundaries in the northeast.

A three-dimensional (3-D) lithospheric-scale electrical resistivity model, developed using network-magnetotelluric (network-MT) data, contains structures associated with arc magmatism beneath Kyushu Island in the Southwest Japan arc. Kyushu Island, where the Philippine Sea Plate (PSP) subducts beneath the Eurasian plate, can be divided into northern and southern volcanic regions separated by a nonvolcanic region. Many active Quaternary volcanoes occur along the volcanic front (VF) associated with the PSP in the two volcanic regions. Our 3-D electrical resistivity model shows three different shapes of upwelling fluid-like conductive anomalies, indicative of either slab-derived aqueous fluid and/or partial melt beneath the volcanic and nonvolcanic regions. A conductive anomaly in the northern volcanic region, located at some distance from the subducting PSP, extends from the surface to depths of <100 km, whereas another conductive anomaly in the southern volcanic region, located along the subducting PSP at >70 km depth, extends from the surface to depths of >100 km. In the nonvolcanic region, the upper region of a relatively conductive anomaly extends upward to a depth of~50 km along the subducting plate. The degrees of magmatism and the relative contribution of slab-derived fluids to the magmatism vary spatially in the one nonvolcanic and two volcanic regions.

Our 3‐D electrical resistivity model clearly detects particular subsurface features for magmatism associated with subduction of the Philippine Sea Plate (PSP) in three regions: a southern and a northern volcanic region, and a nonvolcanic region on the island of Kyushu. We apply 3‐D inversion analyses for geomagnetic transfer function data of a short‐period band, in combination with results of a previous 3‐D model that was determined by using Network‐Magnetotelluric response function data of a longer‐period band as an initial model in the present inversion to improve resolution at shallow depths; specifically, a two‐stage inversion is used instead of a joint inversion. In contrast to the previous model, the presented model clearly reveals a conductive block on the back‐arc side of Kirishima volcano at shallow depths of ~50 km; the block is associated with hydrothermal fluids and hydrothermal alteration zones related to the formation of epithermal gold deposits. A second feature revealed by the model is another conductive block regarded as upwelling fluids, extending from the upper surface of the PSP in the mantle under Kirishima volcano in the southern volcanic region. Third, a resistive crustal layer, which confines the conductive block in the mantle, is distributed beneath the nonvolcanic region. Fourth, our model reveals a significant resistive block, which extends below the continental Moho at the fore‐arc side of the volcanic front and extends into the nonvolcanic region in central Kyushu.

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