Discrete Conley index theory for zero dimensional basic sets

In this article the discrete Conley index theory is used to study diffeomorphisms on closed differentiable n-manifolds with zero dimensional hyperbolic chain recurrent set. A theorem is established for the computation of the discrete Conley index of these basic sets in terms of the dynamical information contained in their associated structure matrices. Also, a classification of the reduced homology Conley index of these basic sets is presented using its Jordan real form. This, in turn, is essential to obtain a characterization of a pair of connection matrices for a Morse decomposition of zero-dimensional basic sets of a diffeomorphism.

1. Introduction. The algebraic and topological techniques in dynamical systems have played a significant role as can be verified, for example, in Morse Theory and more recently in Conley's Index Theory. This theory was introduced by C.Conley in the late seventies and has been further developed in a variety of contexts. Initially defined for continuous dynamical systems on compact sets, see [5], [17] and [18] and later for discrete dynamical systems.
The Conley index was originally defined for isolated invariant sets S of continuous flows and defined to be the homotopy type of the pointed space obtained from a filtration pair associated to S. It was then proven that the Conley index is independent of the choice of the filtration pair and is a homotopy invariant. This is precisely the difficulty in the discrete case since there are examples where the homotopy type of the filtration pair of S may depend on the choice of the pair.
Motivated by this difficulty several definitions were presented for the discrete case. See [7], [11], [16] and [21]. The version we adopt herein is due to Richeson [15] and called the reduced homological Conley index. The reason behind this choice is twofold: firstly it is computationally easier to work with and secondly, it contains the information that is needed in the definition of connection matrix pairs. This index will be defined more precisely in Section 2. It is defined for an isolated invariant set S of a continuous map f as The Conley index has the crucial property of continuation which guarantees the invariance of the index under small C 0 perturbations of the system. See [7] and [15]. The continuation property together with the fact that fitted diffeomorphisms are C 0 -dense on the set of diffeomorphisms on closed manifolds, see [20], motivated our investigation of the discrete Conley index for zero dimensional basic sets.
Another crucial tool in Conley index theory is the connection matrix. Franzosa in [8], [9] and [10] developed the connection matrix theory for Morse decompositions of flows. This theory has proven to be useful in the study of the connection orbits between Morse sets as well as the detection of bifurcation behaviour, see [13]. Independently and around the same time, Bartiomiejczyk and Dzedzej in [1] and Richeson in [15], considered a generalization of this theory for discrete dynamical systems. In [1], the connection matrix ∆ uses solely the information contained in CH * of the discrete Conley index Con * = (CH * , χ * ). On the other hand, the generalization presented in [15] is actually a pair of connection matrices (∆, a) where ∆ uses the information given by CH * and a uses the information given by the isomorphism χ * . A disadvantage of the connection matrix presented in [1] is that it only identifies degree one connections whereas the benefit in considering the pair of connection matrices in [14] and [15] is that the existence of both degree zero and degree one connections can be detected and thus becomes a richer dynamical invariant.
Herein, we work with the pair of connection matrices (∆, a). As with the Conley index, the main property of the connection matrix pair is its invariance under continuation. This confers to it a robust feature, more specifically invariance under C 0 perturbations of the system. The advantage being that, in order to study the pair of connection matrices of diffeomorphisms on a closed manifold M , Diff(M ) it suffices to consider fitted diffeomorphisms which are C 0 -dense in Diff(M ), see [20]. In fact, we will consider an even larger class that contains the fitted diffeomorphisms, i.e. diffeomeorphisms which admit a finite zero-dimensional basic set decomposition. For more details see [6].
In this article, Section 2 provides the background information needed throughout this work. In Section 3 one of the main results, Theorem 3.2, establishes a relation between a homological invariant, namely the Conley index, and a dynamical invariant obtained from the structure matrix of a zero dimensional basic set. In Section 4 a characterization of the isomorphism of the Conley index of a zero dimensional basic set by using the real Jordan form of its structure matrix is obtained in Theorem 4.1. Furthermore, in Section 5, by using Theorem 3.2, an alternative formulation for the results in [6] on the homology zeta function and on Morse inequalities of diffeomorphisms with hyperbolic chain recurrents sets on closed manifolds is obtained. Theorem 3.2 will also be essential in obtaining important characterization results for connection matrices of diffeomorphisms with zero dimensional basic sets. In Section 6, we prove in Theorem 6.2 that the connection matrix pair (∆, a) has, in some sense, complementary information, that is, in some cases a captures dynamical information and ∆ does not and vice-versa. Furthermore, in Section 7, Theorem 7.1 establishes that the matrix a gives no information for time-one maps, i.e, any detectable dynamical behaviour captured by the connection matrix pair (∆, a) is registered solely in ∆. In Section 8, another main result is obtained in Theorem 8.2 which is a global characterization for diffeomorphisms whose hyperbolic chain recurrent set is a union of zero-dimensional basic sets.

Background.
2.1. Diffeomorphism with hyperbolic chain recurrent set and zero dimensional basic sets. In this section we will present some basic definitions and theorems on the dynamical behaviour of diffeomorphisms f : M → M where M is a closed manifold with metric d(·, ·).
Define the chain recurrent set, R(f ), as the set of all chain recurrent points of f . By Smale's spectral decomposition, Theorem 2.3, one has that if R(f ) is hyperbolic it admits a basic set decomposition.
Definition 2.2. Let f : M → M be a diffeomorphism. A compact f -invariant set Λ is said to have a hyperbolic structure provided that the tangent bundle of M restricted to Λ can be written as a Whitney sum T Λ M = E u ⊕ E s of subbundles Df -invariant, and there are constants C > 0, λ ∈ (0, 1), such that [19]). Let f : M → M be a diffeomorphism such that its chain recurrent set R(f ) has a hyperbolic structure.
Then R(f ) is a finite disjoint union of compact invariant sets Ω 1 , Ω 2 , . . . , Ω n and each Ω i contains an orbit of the system which is dense in Ω i .
The Ω i 's of Theorem 2.3 are called basic sets of f , and the fiber dimension of the bundle E u Ωi is called the index of Ω i . Given a zero dimensional basic set Ω of a diffeomorphism f : M → M with hyperbolic chain recurrent set, one can associate to the dynamics of f in Ω, a matrix with 0, 1 and −1 entries, called the structure matrix.
In order to define the structure matrix one needs the relation between a zero dimensional basic set and its subshift of finite type. We refer the reader to [3]. The subshift of finite type σ determined by S and → is called a vertex shift associated to the matrix A if one numbers the elements of S from 1 to n (n is the cardinality of S) and the matrix A n×n is given as follows: Bowen proves in [3] that if Ω is a zero dimensional basic set of a diffeomorphism f with hyperbolic R(f ), then f restricted to Ω is topologically conjugate to a vertex shift σ(G) : Σ G → Σ G by h : Σ G → Ω that has the property that the locally constant function Definition 2.5. Define the structure matrix A for a basic set Ω by A jk = ∆ k G jk .
Next we present a result of Bowen and Franks [4] which associates a structure matrix to a homological invariant. Definition 2.6. Let f : M → M be a diffeomorphism with hyperbolic chain recurrent set with basic sets {Ω i } n i=0 , then a filtration associated to f is a collection of submanifolds M 0 ⊂ M 1 ⊂ · · · ⊂ M n = M such that The existence of a filtration follows easily from the existence of a smooth Lyapunov function φ : M → R for the diffeomorphism f , [6].
2. One says that h and h are conjugate if there exists an isomorphism  Define the maximal invariant subset of N , Inv(N ) as the set of points x ∈ N such that there exists a complete orbit {x n } ∞ −∞ ⊂ N with x 0 = x and f (x n ) = x n+1 for all n. One says that a set S is an isolated invariant set if there exists an isolating neighborhood N (i.e., a compact N with Inv(N ) ⊂ int(N )) such that S = Inv(N ).
Given an isolated invariant set S there exists a pair of compact spaces (N, L) with L ⊂ N contained in the interior of the domain of f with the following properties: [7]. This compact pair is called filtration pair. is the collapsed set that is taken as base point(see [7]). This map is the pointed space map associated to P . Henceforth, whenever we mention the Conley index we mean the reduced homological Conley index defined above.

Connection matrix pair.
In preparation for the connection matrix pair definition we need to present some basic concepts. See [14] and [15] for more details.
We say that a relation < is a partial order on a finite set P provided for all π, π ∈ P 1. π < π never holds, and 2. π < π and π < π implies π < π . On the other hand, < is a total ordering, if, in addition, for any π = π we have that π < π or π < π.
Suppose P is a finite set with partial order <. 1. A subset I ⊂ P is called an interval if π < π < π and π, π ∈ I then π ∈ I. Denote by I = I(P, <) the set of all intervals in P. 2. An interval I is called attracting if π < π and π ∈ I implies π ∈ I.
Denote by A = A(P, <) the set of all attracting intervals in P. 3. An ordered collection of disjoint intervals (I 1 , . . . , I n ) is called an adjacent n-tuple if I 1 ∪ · · · ∪ I n ∈ I(P) and for any π ∈ I i and π ∈ I j with i < j we have that π ≮ π. Denote by I n = I n (P, <) the set of all adjacent n-tuples.
For any interval I ⊂ P define the set where C(M π , M π ; S) is the set of connecting orbits from M π to M π , and we define it to be A well known property of Morse decompositions is that if (I, J) ∈ I 2 (P, <) then (M I , M J ) is an attractor-repeller decomposition of M IJ .
Given a Morse decomposition M(P, <) = {M π ⊂ S | π ∈ P} there exists a collection of compact sets in X such that it is possible to extract a filtration pair for each M I , [15]. Such a collection, which we define subsequently, is called a Morse set filtration.
is called a Morse set filtration for M(P, <) provided that for any attracting intervals I, J ⊂ P the following conditions hold: We now proceed to define a connection matrix pair (∆, a) which makes use of the notion of braid isomorphisms. The basic idea is that for a pair of graded matrices (∆, a) to be a connection matrix pair, the graded module braids with endomorphism obtained for both (∆, a) and for the Conley indices of the Morse sets be isomorphic.
3. For all (I, J, K) ∈ I 3 the braid diagram in Figure 1 commutes.
Next we define the connection matrix pair for a Morse decomposition M(P, <) of an isolated invariant set S.
is isomorphic to the graded module braid with endomorphism where C∆(I) = π∈I CH * (M π ), ∆(I) is a cochain complex and a(I) is the associated cochain map.
In others words, (∆, a) is a connection matrix pair for M(P, <) of S if for all (I, J, K) ∈ I 3 we have that the following graded module braids with endomorphisms 1 and 2 are isomorphic, see Figure 2 and Figure 3.  We denote the set of connection matrix pairs for M(P, <) by CM(M(P, <)).
The proposition that follows is due to Richeson, see [14], which we state in terms of Morse sets.
Proposition 1. Let f : M → M be a diffeomorphism and (∆, a) a connection matrix pair for a Morse decomposition M(P, <) of an isolated invariant set S of f . Let π, π ∈ P with π < π be adjacent elements. If one of the following occurs: 3. The Conley index and the structure matrix. In this section we prove the relation between the Conley index isomorphism of a zero dimensional basic set with the nonnilpotent part of its structure matrix.
are basic sets of f and {M i } n i=0 is an associated filtration. Then Then M i is compact and

Remark 1.
For an endomorphism f we denote the generalized kernel by gKer(f ) and the generalized image by gIm(f ). Suppose f has a hyperbolic chain recurrent set and Ω i is a zero dimensional basic set of index u. If A is an n × n structure matrix for Ω i , and F is a field, then ) are conjugate for all k and consequently f + * u and (f P ) + * u are also conjugate. We will show that (f P ) Therefore, χ u (Ω i ) = (f P ) + * u and this proves item 1. It is well known that the map f * k is nilpotent for all k = u (see Theorem 2.8).
Hence, (f P ) * k is also nilpotent for all k = u. Thus, Remark 2. An immediate consequence of our Theorem 3.2 and the Lemma 6.2 of Remark 3. Note that any Smale diffeomorphism on a closed manifold satisfies Since, in general, computing the nonnilpotent part of a matrix can involve an increasing number of computations as the matrix gets larger, a simple example is provided in order to illustrate the use of Theorem 3.2. We will contrast this theorem with Theorem 4.1 which makes the calculations more straightforward.
Observe that the advantage of this approach is that we do not make use of the filtration pair nor the pointed space map in order to compute the Conley index. We make use solely of the dynamical information contained in the structure matrix of Smale's Horseshoe.

The Conley index and the real Jordan form of a structure matrix.
In this section, we show that the isomorphism χ * , of the Conley index of a zero dimensional basic set of a diffeomorphism with hyperbolic chain recurrent set, is conjugate to the nonnilpotent part of an associated structure matrix. From a practical point of view, working with the structure matrix in its real Jordan form makes the computation of the nilpotent and nonnilpotent part of the matrix effortless. The information of the nonnilpotent part of this matrix is easily retrieved by a quick analysis of the non zero eigenvalues and the nilpotent part by the zero eigenvalues.
We prove the following theorem: Jordan blocks corresponding to zero eigenvalue. In other words, χ u (Ω) is conjugate to if r = n and λ i = 0, ∀i = 1, . . . , n.
where for each i = 1, . . . , r we have that s i is the geometric multiplicity of the eigenvalue λ i , n i is the algebraic multiplicity of the eigenvalue λ i and J λi has the form where each block J λi,j is a real Jordan block associated to the eigenvalue λ i which has one of the following forms: be the real Jordan form of A, where each s i is the geometric multiplicity of the eigenvalue λ i and each J λi,j is the real Jordan block associated to the eigenvalue λ i . Denote by n i,j the size of the real Jordan block J λi,j with j = 1, . . . , s i and i = 1, . . . , r. By the construction of the real Jordan form, we have that n i,1 n i,2 · · · n i,si . Since, n = r i=1 si j=1 n i,j , then KerJ k =[KerJ k λ1,1 ⊕ · · · ⊕ KerJ k λ1,s1 ] ⊕ · · · ⊕ [KerJ k λr,1 ⊕ · · · ⊕ KerJ k λr,sr ], ∀k > 0, gKerJ =[gKerJ λ1,1 ⊕ · · · ⊕ gKerJ λ1,s1 ] ⊕ · · · ⊕ [gKerJ λr,1 ⊕ · · · ⊕ gKerJ λr,sr ], gImJ =[gImJ λ1,1 ⊕ · · · ⊕ gImJ λ1,s1 ] ⊕ · · · ⊕ [gImJ λr,1 ⊕ · · · ⊕ gImJ λr,sr ] Therefore, Hence, in order to calculate the nonnilpotent part, J + , of the real Jordan form J of A, it suffices to compute the nonnilpotent part of the real Jordan blocks that make up J. In other words, it suffices to compute the nonnilpotent part of the following real Jordan blocks associated to an eigenvalue λ: Firstly, we show that J(1) + = J(1). Since λ = 0, we have that Note that the matrix J(2) is nilpotent, consequently gKerJ(2) = F p , which implies that J(2) + = 0.

KETTY A. DE REZENDE AND MARIANA G. VILLAPOUCA
Finally, we will prove that J(3) + = J(3). Indeed, since In what follows we present an example which confirms the practicability of the computation of the Conley index using Theorem 4.1, specially in the cases of larger strucuture matrices.  Observe that if we were to compute the nonnilpotent part of the structure matrix in this example it would have been necessary many more calculations than the ones needed by using Theorem 4.1.

Conley index, homology zeta function and morse inequalities.
In this section we present two basic tools using the Conley index, namely the homology zeta function and the Morse inequalities which are generally used as an obstruction to the existence of diffeomorphisms realizing an a priori collection of dynamical data (see [2]). Our motivation in this section is different since our aim is to show how to calculate the homology zeta function and Morse Inequalities using the Conley index presented in Subection 2.2. The example of a diffeomorphism on a torus at the end of this section illustrates this calculation.
Proof. Let {M i } be a filtration associated to f . We proved in Lemma 3.1, that On the other hand, there exists a base By using our Proposition 2 and Theorem 6.5 of [6], we obtain the following proposition.
Proposition 3. If f : M → M has hyperbolic chain recurrent set and their basic sets are homologically split at q over R, then there exists an integer polynomial P (t) such that The construction of the following diffeomorphism of Example 5.1 can be found in [6].
The diffeomorphism f will have an attracting fixed point p, a repelling fixed point ∞ in the deleted disc D ∞ and is constructed in such a way that the two strips H = h 1 ∪ h 2 form a set of hyperbolic handles of f . Thus, The structure matrices of p, Λ and ∞ are respectively A 0 = (1), A 1 = 0 1 −1 1 and A 2 = (1). Hence, by Theorem 3.2, the Conley indices of the basic sets of f are Con q (p) = (R, Id), q = 0 (0, 0), q = 0 , Con q (∞) = (R, Id), q = 2 (0, 0), q = 2 Moreover, by Proposition 2, the homology zeta functions associated to the basic sets are

Now, since
(R, Id), q = 2 (0, 0), otherwise and the basic sets of f are homologically split at 1 over R we have that, by Proposition 3, there exists an integer polynomial P (t) such that Furthermore, we have that P (t) = 1, i.e., Of course the existence of this polynomial was expected since we started off with a diffeomorphism on a torus satisfying the hypothesis of Proposition 3. We also observe, in this example, that χ 1 (Λ) is not conjugate to χ 1 (p) ⊕ χ 1 (∞) and hence by Proposition 1 C(∞, p) = ∅.
6. Properties of connection matrix pairs. In this section we prove properties satisfied by connection matrix pairs that allows the detection of connecting orbits between Morse sets. Definition 6.1. Let A and B be two fixed endomorphism. We say that two endomorphism g and f are related by ∼ AB , g ∼ AB f , if and only if, The above is an equivalence relation. We denote the equivalence class ∼ AB by [g] where g is an element of the class. (a) If < is the f -defined order 1 and π, π ∈ P are two elements that are not related by <, then ∆(π, π ) = ∆(π , π) = 0 and [a(π, π )] = [a(π , π)] = 0. (b) If π < π in P are adjacent elements such that the submatrix ∆(ππ ) of ∆ is null and the entry [a(π , π)] of a is non-zero, then C(M π , M π ) = ∅. (c) Let π < π be two adjacent elements in P such that If ∆(π , π) = 0, then a(π , π) = 0.
Under the hypothesis of item (b) of Theorem 6.2 and Proposition 1 it is required that f be a diffeomorphism, however it is easy to see that continuity suffices.
It is interesting to see that in the case of two-manifolds, depending on the choice of the Morse decomposition, we obtain information only for ∆, as is shown in the following proposition.  It is easy to see that ∆(3, 1) = 0, a(2, 1) = a(3, 1) = a(3, 2) = 0.
In [12], a uniqueness theorem for connection matrices associated to a Morse-Smale flow without periodic orbits is presented. In what follows we prove a similar result for the connection matrix pair of a Morse-Smale diffeomorphisms without periodic orbits.  (∆, a), over a field F, for M(P, <) is unique in the equivalence class of each entry a(π, π ) of the matrix a given by the relation ∼ χ(M π )χ(Mπ) presented in Definition 6.1.
Proof. Denote by W u (π) and W s (π) the unstable and stable manifolds of M π , respectively.
Since π, π are not adjacent elements, then there is an element π ∈ P such that π < π < π. Let i, j and k be such that CH i (M π ) = F, CH j (M π ) = F and CH k (M π ) = F.
So π < π , W s (π ) and W u (π ) intersect transversally, and we have that i < j. Likewise, we have that j < k.
Hence, i j − 1 (k − 1) − 1 = k − 2 and consequently i = k. Therefore, is null for all q. are also isomorphic. Therefore, ∆ is a connection matrix, in continuous sense, for the Morse decomposition M(P, <) of S.
We would like to know what happens when the hypothesis of Theorem 7.1 changes from ∆(π , π) = 0 to ∆(π , π) = 0. We do not know what happens to the entry a(π , π) of the matrix a, since in this case we do not have the identification a(ππ ) ≡ (a(ππ )) * .
In the case of Morse-Smale flows, the connection matrix a will not give any information on connecting orbits since in this case there do not exist zero degree connections.  <). Then the entries a(i, j) of a with i = j are zeros, i.e., the graded matrix a has the following form Moreover, each entry a q (i, i) is conjugate to the identity if q equals the index of the basic set M i and conjugate to zero otherwise.

Connection matrix pair for zero-dimensional basic sets decomposition.
In this section we prove Theorem 8.2 which characterizes the connection matrix pair for diffeomorphisms with zero-dimensional hyperbolic chain recurrent under an index duality hypothesis which requires that Con q (Λ) = Con q (Λ). We prove in Corollary 1 that this condition is satisfied for basic sets of Morse-Smale diffeomorphisms and for Λ(k) = Note that (i, j) < (i , j ) whenever i < i , where < is a total order 0 < 1 < · · · < n.
In the following Theorem the characterize a pair of connection matrices for a Morse decomposition of zero-dimensional basic sets.
Theorem 8.2. Let M be closed manifold and f : M → M be a diffeomorphism with hyperbolic chain recurrent set. Suppose that for each basic set Λ of f we have that Λ is zero-dimensional and Con q (Λ) = Con q (Λ), ∀q. Let M(P, <) be the Morse decomposition of basic sets ordered by their index. Then a connection matrix pair (∆, a) for M(P, <) has the following form as in Figure 7, where for each i = 1, . . . , s q we have that A + i is the nonnilpotent part of the structure matrix A i associated to the basic set Λ q,i .   1 · · · q q + 1 · · · n 0 0 · · · 0 0 0 The entries of the highlighted square are maps of CH q (Ω q,i ) to CH q (Ω q,j ) for some i, j ∈ {1, . . . , s q } Figure 7. Connection matrix pair for zero-dimensional basic sets decomposition Proof. By Theorem 3.2, we have where u(π) is the index of the basic set M π . We will compute the entries of ∆, Let π, π ∈ P with π < π , then we have one of the following cases: Case 1: If M π has index u and M π has index u + r with r = 1, then ∆ q (π , π) = 0.
We will compute the entries of a a q (π , π) : Let π, π ∈ P with π < π , we have one of the following cases: Case 1: If M π has index u and M π has index s with u = s, then a(π , π) = 0.
In fact, if q = u, then CH q (M π ) = 0 which implies that a q (π , π) = 0. Conversely, if q = u, then CH u (M π ) = 0. Since u = s, then a u (π , π) = 0. Case 2: If M π and M π have the same index u, then a q (π , π) = 0, ∀q = u and a u (ππ ) ∼ A(M π ) + 0 a u (π , π) A(M π ) + where A(M π ) + is the nonnilpotent part of the structure matrix A(M π ) associated to the basic set M π and A(M π ) + is the nonnilpotent part of the structure matrix A(M π ) associated to the basic set M π .
Proof. Since N L ∼ = S k1 ∨ · · · ∨ S ks , we have H * (N L ) = H * (N L ) e (f P ) * ∼ (f P ) * . So, gKer((f P ) * ) = gKer((f P ) * ) which implies that CH * (N L ) = H * (N L ) gKer((f P ) * ) = H * (N L ) gKer((f P ) * ) = CH * (N L ) and therefore Con q (S) = Con q (S), ∀q.  The connection matrix pair presented in Corollary 2, where the collection of matrices of ∆ form a block auxiliary diagonal below the main block diagonal containing the blocks of the collection of matrices of a is very convenient to work with.
We foresee that this will be useful for algebraic calculations, for example to explore continuation properties.   Observe that if we only have the information on the pair of connection matrices of this system it is possible, by using Theorem 6.2, to conclude the existence of a connecting orbit between M 3 and M 1 once it is known that ∆(1, 3) ≡ 0 and [a(3, 1)] = 0. Analogously, we conclude the existence of a connecting orbit between M 3 and M 2 . Note that these connections are degree zero connections detected by the matrix a in the matrix connection pair while the matrix ∆ being null provides no information on the existence of connections. This example illustrates the sharp difference between the connection matrix theory in the discrete dynamical setting and in the continuous dynamical setting. In the latter case, the connection matrix would not have detected connections. A consequence of Corollary 3, is that this connection is still detected by the pair of connection matrices after the system undergoes a C 0 perturbation.