I study dynamical systems, especially complex dynamical systems which are defined by holomorphic functions over complex numbers.
It is well known that simple dynamical systems can create quite complicated behavior, called "chaos". A slight change of parameters also causes a drastic change of dynamical behavior, and this is called "bifurcation".
In order to study complex dynamical systems, one can use powerful tools such as complex analysis, hyperbolic geometry or the theory of Teichmuller spaces.
I study the nature of Julia sets which arises as invariant sets of complex dynamical systems. My recent interest is the theory of renormalization, which is defined by the return maps of dynamical systems.
Complex dynamical systems generate many beautiful and mathematically interesting "fractal sets" such as Julia sets and Mandelbrot set. Please see my gallery. These sets are generated by simple function $p_c(z) = z^2 + c$ ($z, c \in \mathbb{C}$).