We calculate the fractal dimension D (t) of the Julia set associated with the partition function zeros of the ferromagnetic Ising model on the Cayley tree. In the low-temperature phase the Julia set is the unit circle (D (t) = 1) in the complex fugacity plane. In the paramagnetic phase the Julia set is a Fatou dust (Cantor set) on the unit circle whose fractal dimension is a decreasing function of the temperature. Like an order parameter D (t) displays a singular behaviour with a critical exponent β_D = 1/2.