Pictures of near-parabolic renormalizations

Fatou coordinates for near-parabolic case (drawn by qfract with these dirty plugins). Those pictures are for dynamics of λQ(z), where λ=exp(-2πiα) and α=1/(a₁-1/a₂) or indicated in each description.

Remark: Fatou coordinates in the pictures are just first-order approximations near the critical orbits. For instance, strips must be asymtotic to vertical strips at +i∞ in the pre-Fatou coordinate, but clearly not in some of these pictures.)

Actual image size is 1280x1280.

• Fatou coordinate for λQ(z) where a₁=a₂=50. (The picture almost depends only on a₁.)

  

• Fatou coordinate for λQ(z) where |α|=1/50, arg(α)=π/4.

  

• Fatou coordinate for λQ(z) where |α|=1/50, arg(α)=-π/4.

  

• Fatou coordinate for λQ(z) where a₁=a₂=50, in the pre-Fatou (unwrapping) coordinate.

  

• Fatou coordinate for λQ(z) where |α|=1/50, arg(α)=0.025π, in the pre-Fatou (unwrapping) coordinate.

  

• Fatou coordinate for λQ(z) where |α|=1/50, arg(α)=-0.025π, in the pre-Fatou (unwrapping) coordinate.

  

• Fatou coordinate for λQ(z) where |α|=1/50, arg(α)=π/4, in the pre-Fatou (unwrapping) coordinate.

  

• Fatou coordinate for λQ(z) where |α|=1/50, arg(α)=-π/4, in the pre-Fatou (unwrapping) coordinate.

  

• Fatou coordinate for the first renormalization of λQ(z) where a₁=50, a₂=50. The third one is in the pre-Fatou (unwrapping) coordinate.

  

• Fatou coordinate for the first renormalization of λQ(z) where a₁=50, a₂=500. The third one is in the pre-Fatou (unwrapping) coordinate.

  

• Fatou coordinate for the first renormalization of λQ(z) where a₁=50, a₂=5000. The third one is in the pre-Fatou (unwrapping) coordinate.