We study the fractal properties of shallow volcanic tremor and gas piston events associated with magma degassing at Kilauea Volcano, Hawaii, using data from two dense short-baseline arrays of seismographs deployed near the active crater of Puu Oo on the east rift of the volcano. We found an upper bound on the fractal dimension of a strange attractor common to phase portraits of both tremor and gas piston activities in the range of 3.1--4.5 with a mean of 3.75 over a scale length of the wave field of the order of 100 m. The dimension of the attractor fluctuates over the course of tremor and gas piston episodes within a range comparable to the spatial variations of fractal dimensions found in the wave field. The existence of a categorically stable attractor characterizing both types of activities strongly suggests that the excitation mechanism of tremor is similar to that of gas piston events, which in turn are correlated with visual observations at the volcanic vent. The low value derived for the dimension of the attractor in phase space points to significant self-organization in the process of generation of tremor and offers general constraints on the dimensionality of attractors derived from models of acoustic emission associated with magma flow, vesiculation, and degassing.