Ergodic Gaussian maps transported from rotations

For $\omega\in\mathbb R^d$ , the almost-everywhere formula

T_\omega(x)_j=\Phi^{-1}\bigl((\Phi(x_j)+\omega_j)\bmod1\bigr)

defines an invertible Gaussian-preserving transformation modulo null sets. It is ergodic exactly when $k\cdot\omega\notin\mathbb Z$ for every nonzero $k\in\mathbb Z^d$ .

Boundary convention

The inverse CDF is defined on $(0,1)$ , so points whose rotated coordinates equal zero require care. Delete all torus points whose integer-time orbit meets a coordinate boundary. This is a countable union of null sets with invariant conull complement. Transport the rotation on that complement; the discarded Gaussian-null set may be fixed to obtain an everywhere-defined measurable bijection on the completed space.

Fixed functions and phases

The transported Fourier functions

h_k(x)=\exp\left(2\pi i\sum_j k_j\Phi(x_j)\right)

form an orthonormal basis. Inverse pullback satisfies $\alpha_\omega(h_k)=e^{-2\pi ik\cdot\omega}h_k$ . Nonlinear $T_\omega$ therefore still produces a complex-linear operator $\alpha_\omega$ on functions.

For $d=1$ , an irrational increment gives an ergodic example. For increment $1/2$ , $h_2$ is nonconstant and fixed. In dimension two, equal irrational increments fix $h_{(1,-1)}$ , while $(\sqrt2,\sqrt3)$ gives an ergodic system.

Noninvertible Gaussian example

Replacing rotation by doubling gives $x\mapsto\Phi^{-1}(2\Phi(x)\bmod1)$ almost everywhere. It is mixing and noninvertible, inherited from the doubling map. Forward composition supplies a state-preserving endomorphism rather than inverse pullback.