Fixed point indices of iterates of a low-dimensional diffeomorphism at a fixed point which is an isolated invariant set

Let f be an Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document}-diffeomorphism, where n=2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2,3$$\end{document}, for which {0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0\}$$\end{document} is an isolated invariant set. We determine all possible forms of the sequences of fixed point indices of iterates of f at 0, {ind(fn,0)}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\mathrm{ind}(f^n, 0)\}_n$$\end{document}, confirming in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^3$$\end{document} the conjecture of Ruiz del Portal and Salazar (J Differ Equ 249, 989–1013, 2010).


Introduction.
The fixed point index of a map f at a point p is a well-known topological device that is used in fixed and periodic point theory. In studying the dynamics of f , the whole sequence of all indices of iterates {ind(f n , p)} n often provides deep insight into the structure of periodic points as well as local behavior of f near fixed and periodic points. However, establishing the form of a possible sequence of indices for a given class of maps is usually an uneasy problem. The pioneering work in this direction for a low-dimensional case was the result of Brown [3] (cf. also [11]), who studied a planar orientation preserving homeomorphism and showed that indices can take only two values. The full description of the forms of {ind(f n , p)} n for planar homeomorphisms was provided in [19].
In 1997 Le Calvez and Yoccoz studied the behavior of an orientation preserving planar homeomorphism f which, near a fixed point p, satisfies the following condition: there is a neighborhood U of p such that each orbit (different from {p}) leaves U either in positive or negative time (cf. condition (2.2), also expressed as "{p} is an isolated invariant set"). Under this assumption G. Graff Arch. Math.
they found the forms of local fixed point indices [17]. One of the most powerful application of this result (which uses the fact that indices for some iterates are non-positive) is a solution of the old Ulam problem from The Scottish Book of non-existence of minimal homeomorphisms of a punctured sphere S 2 (cf. Problem 115 in [18]). A higher dimensional analog of Le Calvez and Yoccoz theorem was obtained in [16] for an orientation preserving R 3 -homeomorphism f . In this case the sequence of indices must be periodic although there are no other restrictions (such as non-positivity). In the recent paper [13] the forms of {ind(f n , p)} n were established also for an orientation reversing R 3 -homeomorphism f .
Another line of research, started by Shub and Sullivan [22], was related to finding the forms of indices for smooth maps, and resulted in a complete description of possible forms of indices in this case (cf. [1,5,7,8,10,12]).
In this paper we consider R 2 -and R 3 -diffeomorphisms with a fixed point which is an isolated invariant set. In other words, we work with the class of maps which satisfies both above mentioned conditions: isolation of a fixed point as an invariant set and smoothness. We establish the form of indices of iterates in such situation, answering in the positive the conjecture posed for R 3 -diffeomorphisms by Ruiz del Portal and Salazar in [21,Theorem 4.4]. In Concluding Remarks we discuss also the problem of finding all possible forms of indices for diffeomorphisms and non-injective maps in higher dimensions.

Preliminary facts and definitions.
Let f : R m → R m be a continuous map, with 0 an isolated fixed point for each f n , n = 1, 2, . . .. Then the following congruences, called Dold relations, hold (cf. [6]): where μ is the arithmetic Möbius function, i.e. μ : N → Z is defined by the following properties: μ(1) = 1, μ(k) = (−1) r if k is a product of r different primes, μ(k) = 0 otherwise (cf. [4]). Let us remark that the Dold relations are true also in more a general setting for self-maps of ENR and an isolated set of fixed points.
Using the Dold relations one may represent sequences of indices of iterations in the simple form of k-periodic expansion, i.e. by an integral combination of some basic periodic sequences described in Definition 2.1 below. Definition 2.1. For a given k we define a basic sequence {reg k (n)} ∞ n=1 by the formula: Observe that reg k is a periodic sequence: (0, . . . , 0, k, 0, . . . , 0, k, . . .), where the non-zero entries appear for indices of the sequence divisible by k.
Furthermore, by the Dold relations (2.1), the coefficients a n are always integers.
3. Let f : X → X be a homeomorphism, p be a fixed point of f . We will say that {p} is an isolated invariant set if there is a neighborhood W of p such that The condition (2.2) is crucial when studying some important classes of discrete dynamical systems, as it makes it possible to apply Conley index theory to study the dynamics near p (or more generally near an invariant set of f ).

Fixed point indices of iterates of R 2 -diffeomorphisms at a fixed point which is an isolated invariant set.
Before analyzing the problem in dimension 3, for the sake of completeness in this section we deal with an easier 2-dimensional case. We start with giving the complete list of indices of iterates of C 1 maps in dimension 2.
Moreover, every sequence of integers which is of one of the forms (α)−−(δ) can be realized as a sequence of local indices of iterates of a C 1 self-map of R 2 .
The additional demanding that a map is a diffeomorphism for which {0} is an isolated invariant set leads to further restrictions on the indices. (1) its indices of iterates at 0 satisfy the following restrictions with respect to the cases listed in Theorem 3.1: in the case (α) a 1 ≤ 1, in the case (β) a d ≤ 0, in the case (γ) a 2 ≤ 0, and in the case (δ) a 2 ≤ 1.
(2) Each of the sequences satisfying the restrictions described in (1) can be realized as an Arch. Math.
Proof. We start with showing part (1). From the proof of Theorem 3.1 given in [1] we may identify the forms of indices in dependence on the orientation of a diffeomorphism. Namely, if f is orientation preserving, then it has the form (α) or (β), while when it is orientation reversing, it has the forms (β ) c β (n) = reg 1 (n) + a 2 reg 2 (n), (γ), or (δ). Notice that if 0 is a sink or a source that preserves the orientation, then ind(g n , 0) = reg 1 (n). If 0 is a source that reverses the orientation, then (ind(g n , 0)) n = (−1, 1, −1, 1, . . .), i.e. ind(g n , 0) = −reg 1 (n) + reg 2 (n). Now we use Le Calvez' and Yoccoz' theorem (for another argument for deducing the bounds for the coefficients, see Main Theorem in [20] which was proved by the use of Conley index methods, see also Remark 6.4).
Let g be orientation preserving. If 0 is neither a sink nor a source, then its indices may be expressed in the language of periodic expansion as ind(g n , 0) = reg 1 (n) − sreg q (n), where s, q > 0 (cf. [17]). We admit q = 1 for which we get Taking into account that an orientation preserving sink or source in the plane has the sequence of indices of the form reg 1 (n), we get the restrictions for a 1 in the case (α). If q > 1, then we obtain immediately the restrictions a d < 0 in the case (β), while the case a d = 0 reduces to the case (α).
We prove now part (2) showing the realizations of each case listed in Theorem 3.2 by a planar diffeomorphism f with Fix(f ) = Per(f ) = {0} an isolated invariant set.
Let us consider a planar smooth flow h q : R 2 × R → R 2 such that its phase portraits consist of 2|q − 1| = 2(|q| + 1) hyperbolic regions for q < 1, or with 0 as a source for q = 1. The exact formulas for such flows are given in [10].
The classical Poincaré-Bendixson formula states that the index of the discretization of each of the above flows is equal to 1 − h/2 where h is the number of hyperbolic regions (and there are no elliptic regions). Thus, taking H q = h q (·, ·, 1) : R 2 → R 2 , we get that H q has {0} as an isolated invariant set and ind(H q , 0) = q. Furthermore, ind(H n q , 0) = qreg 1 (n). Thus for q := a 1 we get the realization of the sequences of the form (α). Now, for a given natural number d ≥ 2 and integer a d < 0, we define the map O as the 2π d rotation around 0 and put q : which gives the realization of the sequences of the form (β).
To realize sequences of the form (γ), let us consider the discretization G l of a flow having one hyperbolic region which coincides with the first quadrant = −2l and the same holds for even iterates. Taking l := −a 2 we obtain the realization of the sequences of the form (γ).
Finally, consider K l as the discretization of a flow having in the upper half-plane 2l hyperbolic regions, symmetric about the x-axis and such that on the x-axis it is a source at 0. We define K l = s • K l . Then ind(K n l , 0) = ind(K n l |R×{0} , 0) = −1 for odd n and, again by the Poincaré-Bendixson formula, ind(K 2 l , 0) = 1− 2·2l 2 = 1−2l = −1−2(l −1) and the same holds for even n. As a consequence, ind(K n l , 0) = −reg 1 (n) − (l − 1)reg 2 (n) which provides the realization of the sequences of the form (δ) for l := −a 2 + 1 ≤ 0 if a 2 ≤ 0. A source that changes the orientation gives the indices −reg 1 (n) + reg 2 (n), realizing (δ) with a 2 = 1.

Fixed point indices of iterates of R 3 -diffeomorphisms at a fixed point which is an isolated invariant set.
The following theorem provides a complete description of sequences of local indices of iterates for C 1 self-maps of R 3 [10].  (1) Let f be a C 1 self-map of R 3 . Then the sequence of local indices of iterates {ind(f n , 0)} ∞ n=1 has one of the following forms: In all cases d ≥ 3 and a i ∈ Z.
(2) Every sequence of integers which is of one of the forms (A)-(G) can be realized as a sequence of local indices of iterates of a C 1 self-map of R 3 .
In Thus, some forms of indices remain unrealized in the considered class of diffeomorphisms. This provokes the question whether Theorem 4.2 is optimal, which was expressed in the following way: We confirm Conjecture 4.3, i.e. we will prove the following statement: of basic sequences (reg k ) n which may appear in the periodic expansion of (ind(f n , 0)) n and that one may identify all possible k by a use of the derivative Df (0) of f at 0. Let us denote by Δ the set of degrees of all primitive roots of unity which are contained in σ(Df (0)), the spectrum of Df (0). By σ + we denote the number of real eigenvalues of Df (0) greater than 1 and by σ − -the number of real eigenvalues of Df (0) less than −1, in both cases counting with multiplicity. We present now the result of Chow, Mallet-Paret, and Yorke in two theorems below, using the language of k-periodic expansion (cf. [9]).
If σ − is odd and k ∈ 2O odd \O, then a k = −a k/2 .  expansion of (ind(f n , 0)) n , both basic sequences reg 2 and reg d or reg d and reg 2d (d ≥ 3). Using the terminology introduced in Section 5.1, we get that necessarily {2, d} ⊂ O. Now consider two subcases. Case I (i) σ − is even. Then either σ − = 2 or σ − = 0. In the first case, there could be only one root of unity in σ(Df (0)) (of degree 1 or 2) thus d ∈ O, contradiction. In the second case Δ = {2, d}, which implies that f is a local diffeomorphism and that it changes the orientation. By the formula (5.1) we can obtain the case (E) here. Applying Theorem 5.3 (3) we get that a k ≤ 0 for all odd k > 1, which gives all the restrictions listed in Theorem 4.2 for the case (E). Case I (ii) σ − is odd. The case σ − = 3 must be rejected by the same dimension reasoning as used in Case I (i) (otherwise d ∈ O). Let us consider the case σ − = 1. Then the following equality is satisfied:
It turns out that the same result is valid in dimension 3. Namely, Theorem 4.4 remains true also for non-injective 3-dimensional maps: Proof. We can literally repeat the reasoning from the proof of Theorem 4.4, obtaining in some cases that f must be an orientation reversing local diffeomorphism and then applying Theorem 5.3 (3) or in other cases using the restrictions from Theorem 5.1 and Theorem 5.2, which are valid also for noninjective smooth maps. Remark 6.6. The natural question is to ask about the forms of indices for the considered class of maps in higher dimensions. Theorems 5.1 and 5.2 give some restrictions on indices of iterates at 0 for C 1 maps in R n . The explicit list of possible sequences of indices was given in [8,Theorem 3.1]. Additionally, in the same paper there were constructed realizations by diffeomorphisms (Theorem 3.2) with {0} being an isolated invariant set for sequences having all coefficients a k non-negative (a 1 ≤ 1). Roughly speaking, this was done by realizing each a k reg k (n) on different 2-dimensional subspaces of R n as a discretization of some hyperbolic flows h q described in Section 3 and extending the obtained map to R n without changing the indices.