Commuting endomorphisms of the circle. (English) Zbl 0787.58027

The main theorem of this paper states that if \(f_ 1\) and \(f_ 2\) ar two commuting, orientation-preserving maps of the circle with a common fixed point, \(f_ 1,f_ 2\in C^ r\), they are both expanding, \(f_ 1\) is \(p\)-to-1, \(f_ 2\) is \(q\)-to-1 where \(p\) and \(q\) generate a nonlacunary semigroup then there exists a diffeomorphism \(g\in C^ r\) such that \(gf_ 1g^{-1} = T_ p\), \(gf_ 2g^{-1} = T_ q\) where \(T_ px = px\mod 1\). A semigroup of \(\mathbb{N}\) is said to be nonlacunary if it is not contained in a singly generated semigroup.


37A99 Ergodic theory
Full Text: DOI


[1] DOI: 10.2307/2373276 · Zbl 0201.56305
[2] Sacksteder, Springer Lecture Notes in Mathematics 318 pp 235– (1972)
[3] Johnson, Israel J. none pp none– (none)
[4] Parry, Astérisque none pp none– (none)
[5] Krzyzewski, Astérisque 50 pp 205– (1977)
[6] Ruelle, Encyclopedia of Mathematics and its Applications 5 (1978)
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