For a holomorphic map f with a parabolic fixed point with derivative 1 and non-zero second derivative, the attracting and repelling Fatou coordinates are defined so that they conjugate the dynamics to the translation z 7→ z + 1 on half-neighborhoods of the fixed points, on attracting and repelling sides respectively. The horn map is the correspondence between the two Fatou coordinates induced by the orbits going from the repelling side to the attracting side. Via the quotient by z 7→ z+1 and the exponential map z 7→ e2πiz , and after a suitable normalization, the horn map defines a holomorphic map g near the origin with derivative 1. This construction defines the parabolic renormalization, which plays a crucial role in the study of perturbed map z 7→ e 2πiαz+. . . , where α is an irrational number whose continued fraction coefficients are large. In this talk, we will present our result (joint with Hiroyuki Inou) which asserts that there exists a concrete space of holomorphic maps invariant under the parabolic renormalization and persistent under a perturbation. We also obtain the result that the near-parabolic or cylinder renormalization is hyperbolic on the space of maps with small rotation angles at the fixed point.