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Analytic Automorphisms and Transitivity of Analytic Mappings

In this paper we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms which show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator which is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclisity of composition operators.


Introduction
Let X be a complex topological vector space. Mapping F : X → X is called an analytic automorphism if F is analytic, bijective and F −1 is analytic. If both F and F −1 are polynomials, then F is a polynomial automorphism. In this paper we consider analytic automorphisms and their applications to hypercyclic and transitive operators.
A continuous mapping T : X → X is called hypercyclic on a topological vector space X if there is an element x 0 ∈ X for which the orbit under T, Orb (T, x 0 ) = {x 0 , Tx 0 , T 2 x 0 , . . .} is dense in X. Every such x 0 is called a hypercyclic element of T. A continuous mapping T : X → X is called topologically transitive (or just transitive) if for each pair U, V of non-empty open subsets of X there is some n ∈ N with T n (U) ∩ V = ∅. A transitive map is chaotic if it has a dense set of periodic points. If X is a separable Fréchet space then, according to the Birkhoff transitivity theorem [1, p. 10], T is topologically transitive if and only if it is hypercyclic.
Let T and S be mapping (not necessary linear) such that T = F • S • F −1 for some continuous bijection F. Then S is called conjugate to T. Clearly, that the relation of conjugacy is an equivalence relation and preserves hypercyclicity, transitivity and chaoticity. So, if S is a linear transitive (hypercyclic or chaotic) operator and F is an analytic automorphism, we can expect that T is a nonlinear analytic transitive (hypercyclic or chaotic) operator.
Analytic automorphisms can be applied, also, for linear dynamics. If C Φ is a hypercyclic composition linear operator on the space of all analytic functions H(X) on a topological vector space X and F is an analytic automorphism of X, then C F•Φ•F −1 is a hypercyclic composition linear operator which does not commute, in general, with C Φ [8]. Note that the hypercyclisity of C Φ on H(C) for Φ : x → x + a was proved by Birkhoff in [9] and generalized in [10], [11], [12], for the case H(C n ) and for some infinite dimensional cases in [13], [14], [15], [8].
Section 1 is devoted to study basic properties of polynomial and analytic automorphisms on infinite dimensional complex spaces. We show that an injective polynomial map P : 1 → 1 is not necessary an analytic automorphism even if its Jacobi operator ∂P i ∂x j is a continuous bijection. This gives us a contrast with the finite dimensional case, where the Jacobian Conjecture remains to be unsolved. In addition, we introduce some classes of analytic automorphisms on the space of all complex sequences C N and show that subspace E = {(x 1 , . . . , x n , . . .) ∈ C N : sup is invariant with respect to this class. In Section 2 we prove that if a Fréchet space (not necessary separable) admits a linear topologically transitive (chaotic) operator T, then X admits a topologically transitive (chaotic) polynomial map of any degree as well as topologically transitive (chaotic) analytic map which is not polynomial. Some examples are constructed.
In Section 3, using some results on symmetric polynomials on 1 , we construct some particular example of "nonlinear weighted backward shift" on C N and E , which is transitive.
In Section 4 we prove some applications to linear dynamics. Using known in [16] result that the Fréchet algebra of all entire analytic functions H(E ) on E is isometric to the algebra of bounded type symmetric analytic functions H bs (L ∞ [0, 1]), we show that the composition operator Detailed information about dynamics of linear operators is given in [1,17]. Symmetric analytic functions on p an L p were investigated in [18], [19], [16], [20], [21], [22]. For details on analytic mappings on topological vector spaces we refer the reader to [23].

Analytic automorphisms of subspaces of C N
A typical example of polynomial automorphism of C n to itself is a triangular polynomial map P defined by the following way: P(x 1 , . . . , x n ) = (y 1 , . . . , y n ), where · · · y n = c n x n + P n−1 (x 1 , . . . , x n−1 ), where all c 1 , . . . , c n are nonzero constants and P j are polynomials on C j , j = 1, . . . , n − 1 and P 0 = const. Note that for the case C 2 every polynomial automorphism is a composition of triangular automorphisms, for the case C 3 it is not so and for C n , n > 3 it is unknown (see [24]). Triangular polynomial automorphisms can be considered in infinite-dimensional spaces. Let X be a complex Banach space which can be represented as a direct topological sum of some nontrivial closed subspaces X = X 1 ⊕ · · · ⊕ X n . Then we define P(x) = P(x 1 , . . . , x n ) = (y 1 , . . . , y n ), x i , y i ∈ X i by y 1 = A 1 (x 1 ) and y k = A k (x k ) + P k−1 (x 1 , . . . , x k−1 ), 1 ≤ k ≤ n, where A k is a linear isomorphisms of X k to itself and P k−1 : X 1 ⊕ · · · ⊕ X k−1 is a polynomial. It make sense to consider the case of an infinity direct sum. But in this case we have some surprising effects. Example 1. Let X = p , 1 ≤ p < ∞ and (e k ) be the standard basis. Let P : X → X be a polynomial map such that y = P(x), x = ∑ ∞ k=1 x k e k and It is easy to see that P is continuous and the inverse map on the range of P can be represented by The first observation is that P −1 is not a polynomial because deg(x n (y)) = 2 n−1 for n > 1. Also, it is interesting that P is not onto. For example, there is no x ∈ X such that P(x) = e 1 . Indeed, P −1 (2e 1 ) = (2, −4, 8, . . . , (−1) n+1 2 n , . . .) / ∈ X. Moreover, the range of P is not dense in X. Let us suppose for simplicity that X = 1 . Then the range of P does not contain the ball of radius r < 1/4, centered at 2e 1 . Indeed, if y − 2e 1 < r, then |y 1 | > 7/4 and ∑ k>1 |y k | < 1/4. So, for n > 1, The similar argument works for the general case p .
The famous Jacobian Conjecture (see e. g. [25]) asserts that a polynomial map P : C n → C n , P(x) = (P 1 (x), . . . , P n (x)) is a polynomial automorphism if and only if the jacobian where C is a constant in C. It is easy to check that if P is a polynomial automorphism, then the jacobian is equal to a nonzero constant. The inverse statement is an open problem for n ≥ 2. For the infinite-dimensional case we can replace condition (1) by the following one: for every fixed x the operator ∂P i ∂x j is invertible. However, Example 1 shows that this condition does not imply that P is a polynomial automorphism. Indeed if P is as in Example 1, then 1 1 0 · · · 0 2x 2 1 · · · · · · · · · · · · · · ·      is invertible but P is not a polynomial automorphism and it is not even an analytic automorphism.
It is interesting to ask about a "natural" domain of a triangular analytic map, that is, the map should be well defined on a topological linear space and invertible. The following proposition is evident.
is well-defined and invertible on the space of all sequences C N , where c n = 0 and f n are analytic functions on C n , n ∈ N and f 0 = const.
Let us consider a partial case of the mapping in Proposition 1 which can be given by a recurrence formula y n = a n1 x n + a n2 x n−1 y 1 + · · · + a nk x n+1−k y k−1 + · · · + a nn x 1 y n−1 , n ∈ N, where all numbers a nj = 0, j ∈ N.
Proposition 2. There are constants b i 1 ...i n ∈ C, n ∈ N such that the map (2) can be written by Moreover, if |a ij | ≤ c for some positive c and all i, j ∈ N, then Proof. For n = 1 we have b 1 = a 11 . Let us suppose that the proposition is true for every k < n. Then Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 18 November 2020 doi:10.20944/preprints202011.0464.v1 Since = n for all k < n, the vector y n is of the form (3). Also, from (2) we can see that Corollary 1. Let F be an analytic map on C N as in Proposition 2 and |a ij | ≤ c for some positive c and all i, j ∈ N.
If there is a constant d > 0 such that sup If cd = 1, then sup n∈N |y n | 1/n ≤ sup n∈N n 1/n < 2. In the general case we can compute the sum of the geometric series Let us denote E = {x ∈ C N : sup n |x n | 1/n < ∞}.
Theorem 1. Let F be an analytic map on C N as in Proposition 2 and there are positive constants c and c such that Proof. From Corollary 1 we have that the map F 0 : E → E is into. The inverse map on the range of F 0 can be written by − y n + a n2 x n−1 y 1 + · · · + a nk x n+1−k y k−1 + · · · + a nn x 1 y n−1 , n ∈ N.
Since |a n1 | −1 ≤ c , we can apply to this map Corollary 1. So, the range of F 0 is E and F 0 is onto. Thus F 0 is a bijection.
Note that E admits function space representations. For every x = (x 1 , . . . , x n , . . .) we can assign a complex function f x (t), t ∈ C by It is well known that the map x → f x is a linear bijection from E to the space H(0 C ) of holomorphic germs at the origin in C. Also, H(0 C ) is isomorphic to the space of functions of exponential type, H exp (C) via the Borrel transform x n t n−1 (n − 1)! .

The existence of hypercyclic analytic mappings
Theorem 2. Let X be an infinite dimension Fréchet space (not necessary separable). If X admits a linear topologically transitive (chaotic) operator T, then X admits a topologically transitive (chaotic) polynomial map of any degree as well as topologically transitive (chaotic) analytic map which is not polynomial and which is conjugate to T.
Proof. It is enough to construct an analytic automorphism Φ : We know that there are a lot of ways to define an analytic automorphism. Let X 1 be a 1-dimensional subspace in X and X 2 be a complemented subspace, that is, X = X 1 ⊕ X 2 . Let F be an analytic map on X such that F(x 1 + x 2 ) = F(x 1 ) ∈ X 2 for all x 1 ∈ X 1 and x 2 ∈ X 2 . We set Then Φ −1 = I − F and so Φ is an analytic automorphism. Let us write T(x) = T(x) 1 + T(x) 2 , where T(x) 1 ∈ X 1 and T(x) 2 ∈ X 2 . Then, for every x = (x 1 , since F(x) ∈ X 2 and by the definition of F, F(T(x) − F(x)) = F(T(x)). Thus, to make sure that Φ • T • Φ −1 ia an m-degree polynomial we have to guaranty that T • F − F • T is so. Let us suppose that T(X 1 ) ⊂ X 2 and F(tz 1 ) = f (t)z 2 for some z 1 ∈ X 1 , z 1 = 0 and z 2 ∈ X such that T(z 2 ) = 0, where f is a function of t ∈ C. Then Therefore, if f is an m-degree polynomial or an analytic function which is not a polynomial, then Φ • T • Φ −1 is so. Note first that if we put instead of f a continuous function on C which is not analytic, we can get a continuous nonanalytic topologically transitive (or chaotic) map. Also, Φ • T • Φ −1 n = Φ • T n • Φ −1 , and for the map in Theorem 2, It is known (see e.g. [6]) that the left backward shift T : (x 1 , . . . , x n , . . .) = (x 2 , . . . , x n+1 , . . .) on C N is hypercyclic. Then from the direct calculations we can see that that completes the proof.

Analytic automorphisms and symmetric polynomials
In this section we apply some results on symmetric polynomials of infinite many variables to analytic automorphism. For us it is convenient consider polynomials defined on the complex Banach space 1 . Let us recall some definitions.
A polynomial P : 1 → C is said to be symmetric if it is invariant with respect to permutations of basis vectors e n , n ∈ N. Let us denote by P s ( 1 ) the space of all symmetric polynomials on 1 . It is well known [20] that polynomials z n e n ∈ 1 form an algebraic basis in P s ( 1 ), that is, every polynomial in P s ( 1 ) can be uniquely represented as a finite algebraic combination of polynomials is another algebraic basis of homogeneous polynomials in P s ( 1 ), then Q n = a n1 F n + a n2 F n−1 Q 1 + · · · + a nk F n+1−k Q k−1 + · · · + a nn F 1 Q n−1 , n ∈ N, where all numbers a nj = 0, j ∈ N. Comparing this equation with (2), we can see that there is a bijection between algebraic bases of P s ( 1 ) and analytic automorphisms (2) on C N . We will denote by F Q the map (2) associated with (4). By the Newton formulas, the corresponding analytic automorphism F G can be written as In [8] it is proved that if τ a is a homomorphism of P s ( 1 ), defined on the algebraic basis {F k } by for some a ∈ 1 , a = 0, then Proposition 3. Let y = F G (x) and y = F G (x ) for some x, x ∈ C N . Then Proof. If there are z, a ∈ 1 such that F n (z) = x n and F n (a) = x n for all natural n, then G n (z) = y n and G n (a) = y n , and the statement is true by formula (5). Let us show that it is true for all x, x ∈ C N . Let u = F G (x + x ) and we suppose for contrary that u = v, where v n = n ∑ j=0 y j y n−j .
Let m be the smallest natural number such that u m = v m . We denote by w and b vectors in 1 such that F n (w) = x n and F n (b) = x n for 1 ≤ n ≤ m. Such vectors always exist (see e. g. [18]). Then G n (w) = y n for 1 ≤ n ≤ m because y n depends only on x 1 , . . . , x n . By the same resont, G n (a) = y n for 1 ≤ n ≤ m. So, according to (5), u m = v m . A contradiction.
In [5] the following result was prowed.
Theorem 3. Let X be a locally convex sequence space such that c 00 ⊂ X ⊂ C N with continuous inclusions, and such that c 00 is dense in X. If ω is a strictly positive weight such that the weighted backward shift T ω is continuous on X, then I + T ω is a transitive operator on X.

Corollary 3.
Let ω be a strictly positive weight. The mapping Φ : C N → C N defined by Φ(y) = u, where u n = n ∑ j=0 ω n−j y j y n+1−j , n ∈ N is transitive and so hypercyclic.
Thus, according to Proposition 3, By Theorem 3, I + T ω is transitive. Hence Φ is transitive and so hypercyclic, since C N is a Fréchet space.
Let us consider a modified version of F G , defined by It is easy to see that if we put y n = v n /n, then y = F G (x). Using similar arguments as in Proposition 3 we have the following result.

Proposition 4.
Let v = F(x) and v = F(x ) for some x, x ∈ C N . Then The mapping F is interesting for us because the subspace E ⊂ C N is invariant of F by Theorem 1. So, we ave the following theorem.

Theorem 4.
Let ω be a strictly positive weight. The mapping Ψ : C N → C N defined by Ψ(y) = u, where u n = n ∑ j=0 n j(n − j + 1) ω n−j y j y n+1−j , n ∈ N is hypercyclic. If ω is such that the weighted backward shift T ω is continuous on E , then the restriction Ψ 0 of Ψ to E is transitive.
Proof. As in Corollary 3, we can see that Ψ = F • (I + T ω ) • F −1 and if F 0 is the restriction of F to E , then

Applications to linear dynamics
If we have an analytic automorphism F on a topological vector space X, then the composition operator C F is a linear isomorphism of the space H(X) of analytic functions on X, defined by C F ( f ) = f • F, f ∈ H(X). It is well known that the translation operator f (x) → f (x + a) defined on the space of entire functions H(C n ) is hypercyclic if a = 0. Infinite-dimensional generalizations of this results can be found in [13], [8], [15].
Let us denote by H(E ) the space of all analytic functions on E . In [16] is proved that H(E ) is a Fréchet algebra and isomorphic to algebra According to [16], the isomorphism between H bs (L ∞ [0; 1]) and H(E ) is given by where φ n is the coordinate functional, φ n (x) = x n , x = (x 1 , . . . , x n , . . .) ∈ E . In [21] is proved that H bs (L ∞ [0; 1]) is isomorphic to the algebra of symmetric analytic functions of bounded type on and the isomorphism is given by R n → R n , n ∈ N.

Example 3.
Let F 0 be the restriction of the analytic automorphism (6) to E . That is, y = F(x), where y n = n−1 ∑ j=1 (−1) n+1 x j y n−j n − j + (−1) n+1 x n , x ∈ C N , n ∈ N and T x (x) = x + x . Then, by Proposition 4, i + j ij y j y j + y n + y n .
Thus C F 0 •T x •F −1 0 (y) = C −1 F 0 T x C F 0 is hypercyclic on the Fréchet algebra H(E ).

Discussion and Conclusions
The paper is an invitation to study analytic automorphisms of infinite dimensional topological vector spaces and possible applications in functional analysis. We show that using analytic and polynomial automorphisms it is possible to construct various examples of nonlinear topological transitive mappings. On the other hand, if F is an analytic automorphisms of a topological vector space X, then the composition operator C F is an isomorphism of the algebra of analytic functions H(X). Thus, it is possible to construct new hypercyclic composition operators on spaces of analytic functions. Some applications for linear dynamics on spaces of symmetric analytic functions are proposed.
Note that if we will use injective analytic maps with dense ranges F : Y → X, instead of analytic automorphisms, then we can get quasiconjugate mappings S and T, that is, T • F = F • S. The quasiconjugacy is more general than conjugacy and also preserves topological transitivity and chaoticity [1, pp. 11-12]. However, there are some technical difficulties to construct natural examples of transitive analytic mappings with using the quasiconjugacy. We are going to continue our further investigations in this direction.