In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Denjoy theorem gives a sufficient condition for a diffeomorphism

Diffeomorphism

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

 of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation

Irrational rotation

In mathematical theory of dynamical systems, an irrational rotation is a mapwhere θ is an irrational number. Under the identification of a circle with R/Z, or with the interval [0, 1] with the boundary points glued together, this map becomes a rotation of a circle by a proportion θ of a full...

. Arnaud Denjoy

Arnaud Denjoy

Arnaud Denjoy was a French mathematician.Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to integrate all derivatives...

 proved the theorem in the course of his topological classification of homeomorphisms of the circle. He also gave an example of a C¹ diffeomorphism with an irrational rotation number

Rotation number

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

 that is not conjugate to a rotation.

Statement of the theorem

Let ƒ: S¹ → S¹ be an orientation-preserving diffeomorphism of the circle whose rotation number θ = ρ(ƒ) is irrational

Irrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

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

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 with bounded variation

Bounded variation

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

 on the interval [0,1). Then ƒ is topologically conjugate to the irrational rotation by θ. Moreover, every orbit is dense

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

 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 C² 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 C¹, continuous differentiability of ƒ.

Vladimir Arnold

Vladimir Arnold

Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...

 showed that the conjugating map need not be smooth

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

, even for an analytic

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

 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

Lebesgue measure

In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

, namely, for those that are badly approximable

Diophantine approximation

In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....

 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.

External links

• John Milnor
  John Milnor
  John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
  
  , Denjoy Theorem