Let f0(z) = z +z

2 +. . . be a holomorphic function of one variable, which has the origin as

a parabolic fixed point. Then for any α ∈ R \ {0} sufficiently close to 0, f(z) = fα(z) =

e

2πiαf0(z) has another fixed point σ near 0, and one can find a “fundamental domain” S

bounded by an arc ` and its image f(`), where ` is an arc joining 0 and σ. If we glue the

boundary curves ell and f(`) by f, we obtain a bi-infinite cylinder isomorphic to C/Z, then by the exponential map z 7→ e

2πiz the space is identified with C

∗ = C\{0}. The first

return map to S defines a near- parabolic renormalization or cylindrical renormalization

Rf which is defined on the neighborhood of the origin with multiplier e

−2πi

1

α . We will

define a space of functions for which the action of this renormalization is hyperbolic

(joint with Hiroyuki Inou). They key to the proof is to consider the limiting situation

where “parabolic renormalization” is defined for maps with a parabolic fixed point and to

construct an invariant space for it. The invariant space will be characterized by the partial

covering properties of the maps. As an application, Buff and Ch´eritat have obtained a

result which says there exists a quadratic polynomial whose Julia set has positive Lebesgue

measure.