(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?

Thank you in advance!

Edit: Just to clarify, I am interested in maps that are surjective and send points close to each boundary to points close to the same boundary.

(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?

Thank you in advance!

(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?

Thank you in advance!

(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?

Thank you in advance!

Edit: Just to clarify, I am interested in maps that are surjective and send points close to each boundary to points close to the same boundary.