Order isomorphisms of countable dense real sets which are universal - PDF document

Order isomorphisms of countable dense real sets which are universal entire functions (preliminary report) Paul Gauthier (extinguished professor) New Developments in Complex Analysis and Function Theory Heraklion, 2-6 July 2018 1

ORDER An order is a way of giving meaning to an expression of the form x < y . Examples: On people, Height, weight, age and income are orders. Nationality, religion, color and gender are not. orders. 2

ORDER ISOMORPHISMS Every well-ordered set is order-isomorphic to a unique ordinal. Note: Q not well-ordered. Definition. An ordered set is dense , if between every two elements, there is a third. Note: Q is dense. Cantor 1895 If A and B are countable dense ordered sets without first or last elements, then there is an order isomorphism f : A → B . Corollary If A and B are countable dense subsets of R , then there is an order homeomorphism f : R → R with f ( A ) = B . 3

Corollary If A and B are countable dense subsets of R , then there is an order homeomorphism f : R → R with f ( A ) = B . Corollary (suggested by Solynin talk) If A and B are countable dense subsets of the circle T , then there is an order homeomorphism f : T → T with f ( A ) = B . 4

Corollary to Cantor 1895 If A and B are countable dense subsets of R , then there is an order homeomorphism f : R → R with f ( A ) = B . Stäckel 1895 If A and B are countable dense subsets of R , then there is an entire function f f : R → R f ( A ) ⊂ B . with Erdös 1957 A , B countable dense, ∃ entire function f with f ( A ) = B ? Yes A , B ⊂ C , Maurer 1967. A , B ⊂ R , Barth-Schneider 1970. Franklin 1925 If A and B are countable dense subsets of R , then there is an order bianalytic mapping f : R → R , with f ( A ) = B . 5

Corollary to Cantor 1895 If A and B are countable dense subsets of R , then there is an order homeomorphism f : R → R with f ( A ) = B . Corollary (suggested by Solynin talk) If A and B are countable dense subsets of T , then there is an order homeomorphism f : T → T with f ( A ) = B . Franklin 1925 A , B ⊂ R countable dense. Then ∃ order bianalytic f : R → R , with f ( A ) = B . Question (suggested by Solynin talk) A , B ⊂ T countable dense. Is there is a diffeomorphism f : T → T with f ( A ) = B . 6

Franklin 1925 (again) For A and B countable dense subsets of R , there exists a bianalytic map f : R → f ( R ) ⊂ R , such that: f restricts to a bijection of A onto B (hence, f ( R ) = R ). Morayne 1987 If A and B are countable dense subsets of C n (respec- tively R n ), n > 1 , there is a measure preserving biholo- morphic mapping of C n (respectively bianalytic mapping of R n ) which maps A to B . Rosay-Rudin 1988 Same result for C n only. Remarks Franklin’s proof invokes the statement that the uniform limit of analytic functions is analytic, which is false (in view of Weierstrass approximation theorem, for exam- ple). For C 1 , Morayne, Rosay-Rudin results are false. For n = 1 , Morayne conclusion ⇒ Franklin, but Morayne proof fails for n = 1 . 7

Theorem. For A and B countable dense subsets of R , there exists an entire function f of finite order such that: f ( R ) = R ; f ′ ( x ) > 0 , for x ∈ R and f | A : A → B is an order isomorphism . Proof. A = { α 1 , α 2 , . . . } ; B = { β 1 , β 2 , . . . } .   ∞ n   � �     f ( z ) = lim n →∞ f n ( z ) = lim  z + λ j h j ( z )  = z + λ j h j ( z ) ,       n →∞     j = 1 j = 1 h n ( z ) = e − z 2 n − 1 � ( z − α k ) , h 1 = 1; and for n = 2 , 3 , . . . , k = 1 λ j ’s small and real ⇒ f ( R ) ⊂ R . λ j ’s small ⇒ f entire of finite order and f ′ ( x ) > 0 , ∀ x ∈ R , z = α k , k = 2 , . . . , n − 1 . h n ( z ) = 0 , iff Choose λ n so f n ( α n ) = β n . � 8

I OVERSIMPLIFIED Choose enumerations A = { a 1 , a 2 , . . . } and B = { b 1 , b 2 , . . . , } . The sequences { α n } and { β n } are rearrangements of { a n } and { b n } chosen recursively. First, choose α 1 , λ 1 , β 1 , β 2 � β 1 , so f 1 ( α 1 ) = β 1 . Suppose we have respectively distinct α 1 , . . . , α 2 n − 1 ; λ 1 , . . . , λ 2 n − 1 ; β 1 , . . . , β 2 n α 2 k − 1 = ( first a i ) ∈ A \ { α j : j < 2 k − 1 } , k = 1 , . . . , n β 2 k = ( first b i ) ∈ B \ { β j : j < 2 k } , k = 1 , . . . , n j = 1 , . . . , 2 n − 1 f ( α j ) = β j , Choose α 2 n , λ 2 n , β 2 n + 1 , α 2 n + 1 , λ 2 n + 1 , β 2( n + 1) with f 2 n ( α 2 n ) = β 2 n f 2 n + 1 ( α 2 n + 1 ) = β 2 n + 1 9

α 1 λ 1 β 1 − − β 2 − − − · · · · · · · · · α 2 n − 1 λ 2 n − 1 β 2 n − 1 [ α 2 n λ 2 n ] β 2 n α 2 n + 1 [ λ 2 n + 1 β 2 n + 1 ] − − β 2( n + 1) 10

How to find [ α 2 n , λ 2 n ] such that 2 n − 1 � β 2 n = f 2 n ( α 2 n ) = α 2 n + λ j h j ( α 2 n ) + λ 2 n h 2 n ( α 2 n ) = j = 1 f 2 n − 1 ( α 2 n ) + λ 2 n h 2 n ( α 2 n ) . Put g ( x , y ) = f 2 n − 1 ( x ) + yh 2 n ( x ) . Show g ( · , y n ) : R → R surjective. Fix y n small. So, ∃ α with g ( α, y n ) = β 2 n . Implicit function theorem implies, there is ( α 2 n , λ 2 n ) near ( α, y n ) , with g ( α 2 n , λ 2 n ) = β 2 n and α 2 n ∈ A . 11

Universal Functions Birkhoff 1925 There exists an entire function f which is universal . That is, for each entire function g , there is a sequence a n , such that f ( · + a n ) → g . Most entire functions are universal. No example is known. Voronin Universality Theorem 1975 Zero-free holomorphic functions in strip 1 / 2 < ℜ z < 1 can be approximated by translates of the Riemann zeta-function: ζ ( z + it n ) , t n → ∞ . If the zero-free hyposthesis is superfluous, the Riemann Hypothesis fails. Bagchi 1981 The following are equivalent: i) ∃ t n → ∞ , d { t n } > 0 , ζ ( · + it n ) → ζ in strip; ii) the Riemann Hypothesis is true. 12

Approximation on Closed Sets A chaplet is a locally finite sequence of disjoint closed discs D 1 , D 2 , . . . . Theorem Given a chaplet { D n } , a sequence of positive numbers { ǫ n } and a sequence of functions f n ∈ A ( D n ) = C ( D n ) ∩ Hol ( D n ) , there exists an entire function g , such that, for n = 1 , 2 , . . . , | g ( z ) − f n ( z ) | < ǫ n , for all z ∈ D n . Application. The existence of a universal entire function (Birkhoff’s Theorem). 13

Approximation by Functions of Finite Order With the help of a (not the ) theorem of Arakelian on ap- proximation by entire functions of finite order, we can prove: Theorem. For an arbitrary sequence ǫ k > 0 , there exists a sequence D k = D ( a k , k ) such that for every sequence f k ∈ D k , with | f k ǫ k | < 1 , there exists an entire function f of finite order, such that | f ( z ) − f k ( z ) | < ǫ k , for all z ∈ D k . Corollary. For sequences D n = ( | z | < n ) , ϕ n ∈ A ( D n ) and ǫ n > 0 , there exists a subsequence a k n and an entire function f of finite order, such that, setting f n ( z ) = ϕ n ( z − a k n ) , | f ( z + a k n ) − ϕ n ( z ) | < ǫ k n , z ∈ D n . for all Application (Arakelian). There exist universal entire functions of finite order. 14

Given: countable dense real sets A and B , Theorem (again) There exists an entire function f of finite order : f is an order isomorphism of A onto B ; f ′ ( x ) > 0 , x ∈ R ; Can impose other conditions on an order isomorphism f . Given: increasing sequences a n and b n , without limit points, Theorem (universal-interpolating) There exists a universal entire function f : f is an order isomorphism of A onto B ; f ′ ( x ) > 0 , x ∈ R ; and f ( a n ) = b n , n = 1 , 2 , . . . 15

Proof of universal-interpolating theorem . Lemma. Suppose { D n } a chaplet disjoint from R ; { a n } and { b n } , n = 0 , ± 1 , ± 2 , . . . , strictly increasing sequences of real numbers tending to ∞ , as n → ∞ and ǫ n > 0 . Then, for every sequence g n ∈ A ( D n ) , there exists an entire function Φ , such that | Φ − g n | < ǫ n on D n ; Φ maps R bijectively onto R ; Φ ′ > 0 on R and Φ ( a n ) = b n , n = 0 , ± 1 , ± 2 , . . . . Lemma. Same hypotheses, there exists entire function H , such that H D n ∼ 0 , H R ∼ 1 , H ′ R ∼ 0 Proof of theorem. Replace λ j e − z 2 j − 1 ∞ � � ( z − α k ) f ( z ) = z + j = 1 k = 1 by j − 1 ∞ λ j e − Φ 2 ( z ) � � ( Φ ( z ) − Φ ( α k )) . f ( z ) = Φ ( z ) + H ( z ) j = 1 k = 1 16

Franklin 1925 (again) For A and B countable dense subsets of R , there exists a bianalytic map f : R → R , such that: f restricts to an order isomorphism of A onto B . Morayne 1987 If A and B are countable dense subsets of C n (respec- tively R n ), n > 1 , there is a measure preserving biholo- morphic mapping of C n (respectively bianalytic mapping of R n ) which maps A to B . Rosay-Rudin 1988 Same result for C n only. Remarks 1. (n=1, real case) Franklin’s proof incorrect. 2. (n=1, real case) The only measure preserving order isomorphisms of R are translations x �→ x + c . The only possible image of a real set A is the real set B = A + c . (n=1, complex case) Barth/Schneider 1972 If A and B are countable dense subsets of C , there exists an entire function f , such that f ( z ) ∈ B if and only if z ∈ A . 17