# Twist maps of the annulus

(I first recall the definitions, but specialists can probably go directly to the question.)

A twist map of the annulus $A=(\mathbb R/\mathbb Z)\times \mathbb R$ is an orientation preserving homeomorphism $f=(f_1,f_2):A\to A$ that satisfies the "twist condition": for every $x_1$, the function $f_1(x_1,x_2)$ is strictly monotone in $x_2$. Here $f_1$ and $f_2$ are the two coordinate components of $f$, and monotonicity in $\mathbb R/\mathbb Z$ should be taken to mean the monotonicity of a lifting of the corresponding function to $\mathbb R$.

Twist maps of the annulus have been studied a lot in the measure-preserving case, notably in the development of Mather-Aubry theory. For a map $f$ to be measure preserving means that for every Lebesgue-measurable set $U$, $\mu(U)=\mu(f^{-1}(U))$, where $\mu$ denotes Lebesgue measure.

A twist map $f$ is topologically conjugate to another map $g$ if there exists a homeomorphism $\phi : A\to A$ such that $f=\phi^{-1}\circ g\circ \phi$.

Question. Do there exist twist maps of the annulus that are not topologically conjugate to a map that preserves the measure? In particular, are there any twist maps where Mather-Aubry theory does not hold?

I assume that the expression "that preserves the measure" means that preserves area (a.k.a. the Lebesgue, or Haar measure) on $\mathbb{R}/\mathbb{Z}\times\mathbb{R}$.
In this case the answer is no. The most simple example of a twist map which is not topologically conjugate to an area-preserving one is given by a dissipative twist diffeomorphism (i.e. $0<|\mathrm{det} f_x|<1$, for every $x\in A$) exhibiting a non-trivial attractor. These attractors were first studied by Birkhoff, so now they are called "Birkhoff attractors" in the literature.