In mathematics, the Denjoy theorem gives a sufficient condition for a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...

of the circle to be

topologically conjugate In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated func ...

to a diffeomorphism of a special kind, namely an

irrational rotation In the mathematical theory of dynamical systems, an irrational rotation is a map : T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1 where ''θ'' is an irrational number. Under the identification of a circle wit ...

. proved the theorem in the course of his topological classification of

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

s of the circle. He also gave an example of a ''C''¹ diffeomorphism with an irrational

rotation number In mathematics, the rotation number is an invariant of homeomorphisms of the circle. History It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theore ...

that is not conjugate to a rotation.

Statement of the theorem

Let ''ƒ'': ''S''¹ → ''S''¹ be an

orientation-preserving The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed ...

diffeomorphism of the circle whose rotation number ''θ'' = ''ρ''(''ƒ'') is

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...

. Assume that it has positive derivative ''ƒ''^′(''x'') > 0 that is a continuous function with

bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...

on the interval

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

and every nontrivial interval ''I'' of the circle intersects its forward image ''ƒ''°^''q''(''I''), for some ''q'' > 0 (this means that the non-wandering set of ''ƒ'' is the whole circle).

Complements

If ''ƒ'' is a ''C''² map, then the hypothesis on the derivative holds; however, for any irrational rotation number Denjoy constructed an example showing that this condition cannot be relaxed to ''C''¹, continuous differentiability of ''ƒ''.

Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...

showed that the conjugating map need not be

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

, even for an analytic diffeomorphism of the circle. Later Michel Herman proved that nonetheless, the conjugating map of an analytic diffeomorphism is itself analytic for "most" rotation numbers, forming a set of full Lebesgue measure, namely, for those that are badly approximable by rational numbers. His results are even more general and specify differentiability class of the conjugating map for ''C''^''r'' diffeomorphisms with any ''r'' ≥ 3.

References

* * {{Citation , first1=M.R. , last1=Herman , title=''Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations'', journal= Publ. Math. IHES, language=French , volume=49, year=1979, pages=5–234, doi=10.1007/BF02684798 , zbl=0448.58019, s2cid=118356096 , url=http://www.numdam.org/item/PMIHES_1979__49__5_0/ * Kornfeld, Sinai, Fomin, ''Ergodic theory''.

External links

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...

''Denjoy Theorem''
Dynamical systems Diffeomorphisms Theorems in topology Theorems in dynamical systems

Statement of the theorem

Complements

See also

References

External links