×

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.

MSC:

37A99 Ergodic theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.