# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Circle homeomorphisms with two break points. (English) Zbl 1147.37025
Summary: Let $f$ be a circle class $P$ homeomorphism with two break points 0 and $c$. If the rotation number of $f$ is of bounded type and $f$ is ${C}^{2}\left({S}^{1}\setminus \left\{0,c\right\}\right)$ then the unique $f$-invariant probability measure is absolutely continuous with respect to the Lebesgue measure if and only if 0 and $c$ are on the same orbit and the product of their $f$-jumps is 1. We indicate how this result extends to class $P$ homeomorphisms of rotation number of bounded type and with a finite number of break points such that $f$ admits at least two break points 0 and $c$ not on the same orbit and that the jump of $f$ at $c$ is not the product of some $f$-jumps at breaks points not belonging to the orbits of 0 and $c$.

##### MSC:
 37E10 Maps of the circle 37C15 Topological and differentiable equivalence, conjugacy, invariants, moduli, classification