Different approaches to Talk based on the work made in collaboration with: the global periodicity problem Anna Cima (UAB), Armengol Gasull V´ ıctor Ma˜ nosa (UPC), Francesc Ma˜ nosas (UAB), UAB ICDEA 2012, Barcelona Armengol Gasull (UAB) Global periodicity problem 1 / 39 Armengol Gasull (UAB) Global periodicity problem 2 / 39 1.The problem. An example 1.The problem A map F : U → U is said globally m -periodic if F m = Id and m is the Consider the Lyness recurrence smallest natural with this property. The functional equation x j +2 = a + x j +1 F m = Id , a ∈ C , x j is called functional Babbage equation. or equivalently, the DDS generated by � � y , a + y F a ( x , y ) = . x QUESTION: For which values of a the map F a is globally periodic?. Recall that it is said that F is globally periodic if there exists m = m ( a ) such that a = F a ◦ . . . ( m ) . . . ◦ F a = Id F m The answer is well known: a = 0 and m = 6 and a = 1 and m = 5. Charles Babbage (1791-1871) Babbage’s difference engine-2 Armengol Gasull (UAB) Global periodicity problem 3 / 39 Armengol Gasull (UAB) Global periodicity problem 4 / 39
1. The problem 2. Techniques for studying global periodicity Given a class of maps F , we want to find and characterize the globally periodic cases. Find special properties of the dynamical system induced by F . Recall that the difference equation The local linearization given by the Montgomery-Bochner Theorem. x j + n = f ( x j , x j +1 , . . . , x j + n − 1 ) can be studied considering the DDS given by the map Theory of normal forms. F ( x 1 , . . . , x n ) = ( x 2 , . . . , x n − 1 , f ( x 1 , . . . , x n )) . Properties of the so called vanishing sums. These are polynomial identities with integer coefficients involving only roots of the unity. The goal of this talk is to present some of the techniques that we are using to approach the above problem and some of our results. Armengol Gasull (UAB) Global periodicity problem 5 / 39 Armengol Gasull (UAB) Global periodicity problem 6 / 39 3. Properties of the globally periodic maps-I 3. Properties of the globally periodic maps-II First we present some classical results when U = R n : Recall that a map F defined on an open set U , C k - linearizes if there exists MONTGOMERY-BOCHNER THEOREM a C k -homeomorphism, ψ : U → ψ ( U ) ⊂ R n , for which If x is a fixed of a C k ( k ≥ 1) m -periodic map F , then F C k -linearizes ψ ◦ F ◦ ψ − 1 in a neighborhood of x . The linearization ψ is explicitly given by is the restriction of a linear map to ψ ( U ) . The map ψ is called a m − 1 � ψ = 1 (( DF ) x ) − i F i = Id + · · · linearization of F on U . m i =0 Any continuous globally periodic map on R 2 , C 0 -linearizes. (Ker´ ekj´ art´ o, 1919 and 1920) Montgomery and L. Zippin, 1955. IDEA OF THE PROOF: It is easy to see that For n ≥ 3 there are globally periodic continuous maps on R n that do not linearize. (Bing, 1952 and 1964) ψ ◦ F = L ◦ ψ where L is the linear map L ( y ) = ( DF ) x y , because F m = L m = Id. For n ≥ 7 there are globally periodic differentiable maps on R n without fixed points. (Kister, 1969) Armengol Gasull (UAB) Global periodicity problem 7 / 39 Armengol Gasull (UAB) Global periodicity problem 8 / 39
3. Properties of the globally periodic maps-III 3. Properties of the globally periodic maps-IV We use the following well-known properties for globally periodic maps: If a map F is m -periodic then F k , for any integer k is also periodic. An easy, but useful, consequence of the Mongomery-Bochner Theorem is: If a map F : U → U is periodic then it has to be bijective in U . Proposition If a rational map is periodic of period m in open subset of R n then it Let F be a differentiable map having a fixed point x 0 . Assume that F is has to be periodic –also of period m – in the whole real or complex m -periodic and let k be the minimum positive k such that space, except at the points where F or its iterates are not well defined. (( DF ) x 0 )) k = Id. Then k = m . If F : U → U is a periodic map of period m and x ∈ U is a fix point of F then (( DF ) x ) m = Id . For instance a simple corollary is: The eigenvalues of ( DF ) x have to be m roots of the unity. Corollary The matrix ( DF ) x diagonalices. If a smooth map F ( x , y ) = ( − x + ..., − y + ... ) is m -periodic then it is an involution ( m = 2). The fixed points of a periodic map can not be neither attractor nor repeller. They have zero algebraic entropy. Armengol Gasull (UAB) Global periodicity problem 9 / 39 Armengol Gasull (UAB) Global periodicity problem 10 / 39 3. Properties of the globally periodic maps-V 4. “Equivalence”between global periodicity and existence of n first integrals We also have proved the following “more dynamical properties” when the map F is m -periodic: When and why a planar map with two functionally independent first If it is C 1 then the DDS generated by F has an absolutely continuous integrals is periodic? invariant measure ν , that is there exists an integrable map g such that � ν ( F − 1 ( A )) = ν ( A ) . ν ( A ) = g ( x ) dx , and A The map DDS generated by F , defined on a subset of R n , has n functionally independent first integrals. Moreover if the map is bijective this property ”essentially” characterizes the globally periodic maps. Recall that a non-constant function H is called a first integral or invariant of the DDS generated by F if The set of intersection points is invariant by F . H ( F ( x )) = H ( x ) . We need that the number of intersections is bounded. Notice that the level sets of H are invariant by F Armengol Gasull (UAB) Global periodicity problem 11 / 39 Armengol Gasull (UAB) Global periodicity problem 12 / 39
4. “Equivalence”between global periodicity and 4. “Equivalence”between global periodicity and existence of n first integrals-II existence of n first integrals-III For instance the Lyness map F ( x , y ) = ( y , a + y ) x Example. The map has for all a ∈ C the first integral F ( x , y ) = ( x + 2 π, y ) H 1 ( x , y ) = ( x + 1)( y + 1)( x + y + a ) has the two functionally independent first integrals xy H 1 ( x , y ) = y − sin( x ) , H 2 ( x , y ) = y . and when a = 0 it has also the first integral H 2 ( x , y ) = x 4 y 2 + x 2 y 4 + y 4 + y 2 + x 2 + x 4 and is clearly non periodic. x 2 y 2 and when a = 1, y 4 x + ( x 3+ x 2+2 x +1 ) y 3+ ( x 3+5 x 2+3 x +2 ) y 2+ ( x 4+2 x 3+3 x 2+3 x +1 ) y + x 3+2 x 2+ x H 2 ( x , y ) = . x 2 y 2 Armengol Gasull (UAB) Global periodicity problem 13 / 39 Armengol Gasull (UAB) Global periodicity problem 14 / 39 4. “Equivalence”between global periodicity and 4. “Equivalence”between global periodicity and existence of n first integrals-IV existence of n first integrals-V How to construct the n independent first integrals for any globally periodic map? Example. Consider the map F ( x , y ) = ( y , c Proposition xy ) . Let F : U ⊂ R n → U be a globally m –periodic map on U . Let c corresponding to the difference equation x n +2 = . m x n x n +1 � �� � Φ : U m = It holds U × U × · · · × U − → K ( x , y ) → ( y , c xy ) → ( c xy , x ) → ( x , y ) , be a symmetric function. Then, whenever it is not a constant function, so it is 3-periodic. So n = 2 and m = 3 . H φ ( x ) = Φ( x , F ( x ) , . . . , F m − 1 ( x )) The first integrals are constructed using two suitable symmetric functions of the orbit. is a first integral of the DDS generated by F . Armengol Gasull (UAB) Global periodicity problem 15 / 39 Armengol Gasull (UAB) Global periodicity problem 16 / 39
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