Smoothness of conjugacies of diffeomorphisms of the circle with rotations.

*(English. Russian original)*Zbl 0701.58053
Russ. Math. Surv. 44, No. 1, 69-99 (1989); translation from Usp. Mat. Nauk 44, No. 1(265), 57-82 (1989).

The problem dealt with in the paper is old and famous. It was proved by Denjoy that if \(T_ f\) is a homeomorphism of the unit circle, satisfying \(T_ f\in C^ 1\) and Var(ln T\({}'_ f)<+\infty\), and with irrational rotation number \(\rho\), then \(T_ f\) is conjugate to the rotation \(T_{\rho}\) through the angle \(\rho\). This means that there exists a homeomorphism of the circle \(T_ g\) such that \(T_ g\circ T_ f\circ T_ g^{-1}=T_{\rho}.\)

The problem of smoothness of \(T_ g\) has been considered by many authors: V. I. Arnol’d proved that for typical \(\rho\) and analytic \(T_ f\) close enough to \(T_{\rho}\), \(T_ g\) is analytic. He also gave examples where \(T_ g\) is not even absolutely continuous. J. Moser generalized Arnol’d’s result for smooth enough \(T_ f.\)

The first result not demanding the closeness of \(T_ f\) and \(T_{\rho}\) was proved by M. Hermann. Let \(D_{\delta}=\{\rho \in {\mathbb{R}}\setminus {\mathbb{Q}}:\) there exists c(\(\rho\)) such that for all p/q\(\in {\mathbb{Q}}\), \(| \rho -p/q| \geq c(\rho)q^{-2-\delta}\}.\) He proved that if \(T_ f\in C^ k\), \(k\geq 3\) (not necessarily an integer) and \(\rho \in D=\cap_{\delta >0}D_{\delta}\), then \(T_ g\in C^{k-1-\epsilon}\), for any \(\epsilon >0\). This result was improved by J.-C. Yoccoz: if \(T_ f\in C^ k\), \(k\geq 3\) and \(\rho \in D_{\delta}\), \(k>2\delta +1,\) then \(T_ g\in C^{k-1-\delta -\epsilon}\), for any \(\epsilon >0.\)

The authors prove analogous results assuming less about the smoothness of \(T_ f\). Write \(\rho =[k_ 1,k_ 2,...,k_ n,...]\) if \(\rho\) can be expanded into the continued fraction \(\rho =1/[k_ 1+1/[k_ 2+...]\). The authors prove the following assertion: Let \(T_ f\in C^{k+\nu}\), \(\nu >0\), \(\rho =[k_ 1,k_ 2,...]\). If there exists K such that \(k_ n\leq K\) for all \(n\geq 1\), then \(T_ g\in C^{1+\nu}\). If there exists \(T>0\) such that \(k_ n\leq const\cdot n^ T\), for \(n\geq 1\), then \(T_ g\in C^{1+\nu -\epsilon}\), for any \(\epsilon >0\). Also, if \(\rho \in D_{\delta}\), \(\delta <\nu\), then \(T_ g\in C^{1+\nu -\delta}.\)

The proofs are quite elementary (which does not mean easy) and use some ideas of renormalization group technique.

Later improvements of the results in this field can be found in preprints by Y. Katznelson and D. Ornstein [Ergodic Theory Dyn. Syst. 9, No.4, 643-680 (1989); “The absolute continuity of the conjugation of certain diffeomorphisms of the circle”, Stanford Univ. Press, Stanford, CA, 1987] and a paper by J. Stark [Nonlinearity 1, No.4, 541-575 (1988)].

The problem of smoothness of \(T_ g\) has been considered by many authors: V. I. Arnol’d proved that for typical \(\rho\) and analytic \(T_ f\) close enough to \(T_{\rho}\), \(T_ g\) is analytic. He also gave examples where \(T_ g\) is not even absolutely continuous. J. Moser generalized Arnol’d’s result for smooth enough \(T_ f.\)

The first result not demanding the closeness of \(T_ f\) and \(T_{\rho}\) was proved by M. Hermann. Let \(D_{\delta}=\{\rho \in {\mathbb{R}}\setminus {\mathbb{Q}}:\) there exists c(\(\rho\)) such that for all p/q\(\in {\mathbb{Q}}\), \(| \rho -p/q| \geq c(\rho)q^{-2-\delta}\}.\) He proved that if \(T_ f\in C^ k\), \(k\geq 3\) (not necessarily an integer) and \(\rho \in D=\cap_{\delta >0}D_{\delta}\), then \(T_ g\in C^{k-1-\epsilon}\), for any \(\epsilon >0\). This result was improved by J.-C. Yoccoz: if \(T_ f\in C^ k\), \(k\geq 3\) and \(\rho \in D_{\delta}\), \(k>2\delta +1,\) then \(T_ g\in C^{k-1-\delta -\epsilon}\), for any \(\epsilon >0.\)

The authors prove analogous results assuming less about the smoothness of \(T_ f\). Write \(\rho =[k_ 1,k_ 2,...,k_ n,...]\) if \(\rho\) can be expanded into the continued fraction \(\rho =1/[k_ 1+1/[k_ 2+...]\). The authors prove the following assertion: Let \(T_ f\in C^{k+\nu}\), \(\nu >0\), \(\rho =[k_ 1,k_ 2,...]\). If there exists K such that \(k_ n\leq K\) for all \(n\geq 1\), then \(T_ g\in C^{1+\nu}\). If there exists \(T>0\) such that \(k_ n\leq const\cdot n^ T\), for \(n\geq 1\), then \(T_ g\in C^{1+\nu -\epsilon}\), for any \(\epsilon >0\). Also, if \(\rho \in D_{\delta}\), \(\delta <\nu\), then \(T_ g\in C^{1+\nu -\delta}.\)

The proofs are quite elementary (which does not mean easy) and use some ideas of renormalization group technique.

Later improvements of the results in this field can be found in preprints by Y. Katznelson and D. Ornstein [Ergodic Theory Dyn. Syst. 9, No.4, 643-680 (1989); “The absolute continuity of the conjugation of certain diffeomorphisms of the circle”, Stanford Univ. Press, Stanford, CA, 1987] and a paper by J. Stark [Nonlinearity 1, No.4, 541-575 (1988)].

##### MSC:

37-XX | Dynamical systems and ergodic theory |