Lefschetz Zeta Function Research Articles

K-theoretic torsion and the zeta function

We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.

On the Lefschetz Zeta Function for a Class of Toral Maps

In the present article we study the periodic structure of a class of maps on the n-dimensional torus such that the eigenvalues of the induced map on the first homology are dilations of roots of unity. This family of maps shares some properties with the family of the quasi-unipotent maps. We compute their Lefschetz numbers, and show that the Lefschetz numbers of period m are non-zero, for all m’s, in the case that n is an odd prime. Moreover we give an explicit expression for their corresponding Lefschetz zeta functions. We conjecture that in any dimension the Lefschetz numbers of period m are non-zero, for all m.

On topological entropy, Lefschetz numbers and Lefschetz zeta functions

In the present article we give sufficient conditions for C∞ self-maps on some connected compact manifolds in order to have positive entropy. The conditions are given in terms of the Lefschetz numbers of the iterates of the map and/or its Lefschetz zeta function. We consider the cases where the manifold is a compact orientable and non-orientable surface, the n-dimensional torus, the product of n spheres of dimension ℓ and the product of spheres of different dimensions.

On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms on products of ℓ-spheres

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the space X=Sℓ×⋯×︸n−timesSℓ, with ℓ>1. Among the quasi-unipotent maps are Morse–Smale diffeomorphisms. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on X; we completely describe this set, for families containing infinitely many Morse–Smale diffeomorphisms. The results of the present article are based on the techniques used in [5], in the computation of the Lefschetz zeta function for quasi-unipotent self maps on the n-dimensional torus.

On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus. II: The general case

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n-dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n-dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7]. The methods used are based on the arithmetical properties of the number n.

Uniqueness of dynamical zeta functions and symmetric products

A characterization of dynamically defined zeta functions is presented. It comprises a list of axioms, natural extension of the one which characterizes topological degree, and a uniqueness theorem. Lefschetz zeta function is the main (and proved unique) example of such zeta functions. Another interpretation of this function arises from the notion of symmetric product from which some corollaries and applications are obtained.

Fixed point indices of planar continuous maps

We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed pointsthat are isolated as an invariant set for a continuous map $f$ in the plane. In particular, we provethat the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 0$. This characterization allows us tocompute effectively the Lefschetz zeta functions for a wide class of continuous maps in the \(2\)-sphere, to obtainnew results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree--\(d\) maps in the 2-sphere.

On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus

We compute the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus, using arithmetical properties of the number n. In particular we compute the Lefschetz zeta function for quasi-unipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the th cyclotomic polynomial, when is an odd prime. These computations involve fine combinatorial properties of roots of unity. We also show that the Lefchetz zeta functions for quasi-unipotent maps on are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for quasi-unipotent maps on , having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse–Smale diffeomorphisms of .

On the set of periods for the Morse–Smale diffeomorphisms on the disc with N holes

Let be the two-dimensional disc with n holes. We assume that is compact. For every homological class of Morse–Smale diffeomorphisms on without periodic points in its boundary, we provide an algorithm for characterizing its minimal set of Lefschetz periods. We give the complete classification of these sets of periods for . The main tool used for this characterization is the Lefschetz zeta function.

ZETA FUNCTIONS FOR ONE-DIMENSIONAL GENERALIZED SOLENOIDS

We compute zeta functions of 1-solenoids. When our 1-solenoid is nonorientable, we compute Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid and its orientable double cover explicitly in terms of adjacency matrices and branch points. And we show that Artin-Mazur zeta function of orientable double cover is a rational function and a quotient of Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid.

Minimal Lefschetz sets of periods for Morse–Smale diffeomorphisms on the n-dimensional torus

We present a complete description of the minimal Lefschetz set of periods for every homological class of Morse–Smale diffeomorphisms defined on the n-dimensional torus for , by using the Lefschetz zeta function. The techniques applied for obtaining these results also work for n>4.

A note on the dynamical zeta function of general toral endomorphisms

It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated by integer arithmetic only. We also address the functional equation and the relation between the Artin-Mazur and Lefschetz zeta functions.

Topological entropy, homological growth and zeta functions on graphs

In connection with the entropy conjecture it is known that the topological entropy of a continuous graph map is bounded from below by the logarithm of the spectral radius of the induced map on the first homology group. We show that in the case of a piecewise monotone graph map, its topological entropy is equal precisely to the maximum of the mentioned logarithm of the spectral radius and the exponential growth rate of the number of periodic points of negative type. This nontrivially extends a result of Milnor and Thurston on piecewise monotone interval maps. For this purpose we generalize the concept of Milnor–Thurston zeta function incorporating in the Lefschetz zeta function.

Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions

We study transfer operators $\mathcal{M}^{(k)}$ associated to a finite family {ψω} of $\mathcal{C}^{r}$ (r ≥ 1) transversal maps Uω → $\mathbb{R}^n$, where Uω ⊂ $\mathbb{R}^n$, with $\mathcal{C}^{r}$ compactly supported weights gω, acting on k-forms in $\mathbb{R}^n$. Using the definitions of sharp trace Tr≯ and flat trace Tr≭, the following formula holds between power series: $\mathrm{Det}^{\#}(1-z\mathcal{M})=\Pi_{k=0}^n \mathrm{Det}^{\flat}(1-z\mathcal{M}^{(k)})^{(-1)^k}$. Following ideas of Kitaev [17], we define kneading operators $\mathcal{D}_k$(z), which are kernel operators. Our main result is the equality (as formal power series) $$ \mathrm{Det}^{\#}(1-z\mathcal{M})=\prod_{k=0}^{n-1} \mathrm{Det}^{\flat}\big(1+\mathcal{D}_k(z)\big)^{(-1)^{k+1}}.$$ We also show that a finite power of $\mathcal{D}_k$(z) is trace-class on L2. This (partially) generalizes results obtained by Baladi, Kitaev, Ruelle, and Semmes in dimension one, complex and real [8], [10]).

Closed Orbits of Gradient Flows and Logarithms of Non-Abelian Witt Vectors

We consider the flows generated by generic gradients of Morse maps of a closed connected manifold $M$ to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called the eta function, and lies in a suitable quotient of the Novikov completion of the group ring of the fundamental group of $M$. Its abelianization coincides with the logarithm of the twisted Lefschetz zeta function of the flow. For $C^0$-generic gradients we obtain a formula expressing the eta function in terms of the torsion of a special homotopy equivalence between the Novikov complex of the gradient flow and the completed simplicial chain complex of the universal cover of $M$.

Gromov invariants and symplectic maps

Given a symplectomorphism f of a symplectic manifold X, one can form the `symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$ with other symplectic manifolds. This is done by expressing the Gromov invariants in terms of the Lefschetz zeta function of f and, in special cases, in terms of the Alexander polynomials of knots. The result is a large set of interesting non-Kahler symplectic manifolds with computational ways of distinguishing them. In particular, this gives a simple symplectic construction of the `exotic' elliptic surfaces recently discovered by Fintushel and Stern and of related `exotic' symplectic 6-manifolds.

Trace formulae, Zeta functions, congruences and Reidemeister torsion in Nielsen theory

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and torsion free. We connect the Reidemeister zeta function of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the induced map on the unitary dual of the group. As a consequence we obtain a relation between a special value of the Reidemeister zeta function and a certain Reidemeister torsion. We also prove congruences for Reidemeister numbers of iterates of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group which are the same as those found by Dold for Lefschetz numbers.

Topology of an attraction domain, dynamical zeta functions and Reidemeister torsion

In Section 1 of this paper we consider a flow with a circular chain-recurrent set and describe, with the help of the Reidemeister torsion, the connection between the topology of the attraction domain of an attractor and the dynamics of the flow on the attractor. We show in Theorem 3 that the Reidemeister torsion of a level surface of a Lyapunov function and of the attraction domain of an attractor is calculated as a special value of the twisted Lefschetz zeta function build via closed orbits in the attractor. In Section 2 we continue the study of analytical properties of the Nielsen zeta function. The Nielsen zeta function Nf (z) has a positive radius of convergence which has a sharp estimate in terms of the topological entropy of the map f [15]. In Theorem 5 of Section 2 we propose another proof of the positivity of the radius and give an exact algebraic lower estimate for the radius using the Reidemeister trace formula for the generalized Lefschetz numbers.

Periodic orbits of transversal maps

One of the most useful theorems for proving the existence of fixed points, or more generally, periodic points of a continuous self-map f of a compact manifold, is the Lefschetz fixed point theorem. When studying the periodic points of f it is convenient to use the Lefschetz zeta function Zf(t) of f, which is a generating function for the Lefschetz numbers of all iterates of f. The function Zf(t) is rational in t and can be computed from the homological invariants of f. See Section 2 for a precise definition. Thus there exists a relation, based on the Lefschetz fixed point theorem, between the periodic points of a self-map of a manifold f:M → M and the properties of the induced homomorphism f*i on the homology groups of M. This relation has been used in several papers, namely [F1], [F2], [F3] and [M]. In these papers, sufficient conditions are given for the existence of infinitely many periodic points in the case when all the zeros and poles of the associated Lefschetz zeta function are roots of unit. Here we restrict ourselves to maps defined on manifolds with a certain homology type. For transversal maps f defined on this class of manifolds, it is possible to extend the techniques introduced in [F1], [F3] and [M] in order to obtain information on the set of periods of f. We recover the above mentioned results of J. Franks and T. Matsuoka, and derive new results on the set of periods of f when the associated Lefschetz zeta function has zeros or poles outside the unit circle.

Algebraic properties of the Lefschetz zeta function, periodic points and topological entropy

The Lefschetz zeta function associated to a continuous self-map f of a compact manifold is a rational function P/Q. According to the parity of the degrees of the polynomials P and Q, we analyze when the set of periodic points of f is infinite and when the topological entropy is positive.