On universality for area-preserving maps of the plane

We present detailed evidence that one-parameter families of area-preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter. We relate this behaviour to a fixed point equation of the form

$\text{Λ}{}^{\mathrm{?1}}\text{°Φ°Φ°Λ = Φ}$

and

$\text{det}\text{DΦ = 1}$

, Φ:$\text{R}$²→$\text{R}$². In particular we argue that the scaling transformation Λ:$\text{R}$²→$\text{R}$² is conjugate to the transformation Λ₀:(x, y)→(λx, μy), with λ² ≠ μ, and in fact λ² >μ. We present some numerical evidence that

$\text{δ = 8.721}$

…,

$\text{?}\text{1}\text{λ}\text{= 4.018}$

…,

$\text{1}\text{μ}\text{= 16.36}$

…, where δ is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2^k described above.