HOME

TheInfoList



OR:

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 tw ...
of the circle to be topologically conjugate 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 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. ...
. 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 number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
, 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.


See also

*
Circle map In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynam ...


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 Un ...

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