Dynamical Zeta Function Research Articles

On the Generalised Transfer Operators of the Farey Map with Complex Temperature

We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This is an important problem in the thermodynamic formalism approach to dynamical systems, which in this particular case is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices. This formulation gives a new way to study numerically the spectrum of the Laplace–Beltrami operator and the properties of the Selberg Zeta function for the modular surface.

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Abstract The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ (z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽 p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽 p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold Sigma with Betti number b_1, the order of vanishing of the Ruelle zeta function at zero equals 4-b_1, while in the hyperbolic case it is equal to 4-2b_1. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle SSigma with harmonic 1-forms on Sigma .

Flat vector bundles and analytic torsion on orbifolds

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.

Complex Valued Analytic Torsion and Dynamical Zeta Function on Locally Symmetric Spaces

AbstractWe show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent series at the zero point is related to the regularised determinant of the flat Laplacian of Cappell–Miller. When the flat vector bundle is close to an acyclic and unitary one, we show that the dynamical zeta function is regular at the zero point and that its value is equal to the complex valued analytic torsion of Cappell–Miller. This generalises the author’s previous results for unitarily flat vector bundles as well as Müller and Spilioti’s results on hyperbolic manifolds.

Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted Burnside–Frobenius Theorem

Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted Burnside–Frobenius Theorem

Alphabets, rewriting trails and periodic representations in algebraic bases

For \(\beta > 1\) a real algebraic integer (the base), the finite alphabets \({\mathcal {A}} \subset {\mathbb {Z}}\) which realize the identity \({\mathbb {Q}}(\beta ) = \mathrm{Per}_{{\mathcal {A}}}(\beta )\), where \(\mathrm{Per}_{{\mathcal {A}}}(\beta )\) is the set of complex numbers which are \((\beta , {\mathcal {A}})\)-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base \(\beta \) and Lehmer’s problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in \({\mathbb {Q}}(\beta )\), generalizing Schmidt’s theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases \(\gamma _s := \gamma _{n, m_1 , \ldots , m_s}\) such that \(\gamma _{s}^{-1}\) is the unique root in (0, 1) of an almost Newman polynomial of the type \(-1+x+x^n +x^{m_1}+\cdots + x^{m_s}\), \(n \ge 3\), \(s \ge 1\), \(m_1 - n \ge n-1\), \(m_{q+1}-m_q \ge n-1\) for all \(q \ge 1\). For \(\beta > 1\) a reciprocal algebraic integer close to one, the poles of modulus \(< 1\) of the dynamical zeta function of the \(\beta \)-shift \(\zeta _{\beta }(z)\) are shown, under some assumptions, to be zeroes of the minimal polynomial of \(\beta \).

Analytic Torsion, Dynamical Zeta Function, and the Fried Conjecture for Admissible Twists

We show an equality between the analytic torsion and the absolute value at zero of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author’s previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

Morse-Smale flow, Milnor metric, and dynamical zeta function

We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.

Dynamical zeta functions of Reidemeister type

In this paper we study dynamical representation theory zeta functions counting numbers of fixed irreducible representations for iterations of group endomorphism. The rationality and functional equation for these zeta functions are proven for several classes of groups. We prove Pólya-Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta functions for a large class of automorphisms of infinitely generated Abelian groups. We also establish the connection between the Reidemeister zeta function and dynamical representation theory zeta functions under restriction of endomorphism to a subgroup.

Functional equations of Selberg and Ruelle zeta functions for non-unitary twists

We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable s in some right-half plane of $${\mathbb {C}}$$. In Spilioti (Ann Glob Anal Geom 53(2):151–203, 2018), it was proved that they admit a meromorphic continuation to the whole complex plane. In this paper, we establish functional equations for them, relating their values at s with those at $$-s$$. We prove also a determinant representation of the zeta functions, using the regularized determinants of certain twisted differential operators.

Hyperbolic Dynamics Meet Fourier Analysis, an Invitation to the Book by V. Baladi: “Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps”

Hyperbolic Dynamics Meet Fourier Analysis, an Invitation to the Book by V. Baladi: “Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps”

On a Multifractal Pressure for Countable Markov Shifts

In a recent article [J. d' Analyse Math 131, 207, 2017], Olsen intoduced a generalized notion of multifractal pressure, and also a multifractal dynamical zeta function, which essentilly consists in considering not all configurations, but those which are ”multifractally relevant”. In this way more precise information about the multifractal spectrum analyzed is encoded by the multifractal pressure and the multifratcal zeta function. He applied the theory for dynamical systems modelled by finite alphabet shifts, in particular for self conformal iterated systems. Here we continue with this line considering dynamical systems given by countable Markov shifts.

Dynamics on abelian varieties in positive characteristic

We study periodic points for endomorphisms $\sigma$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $\zeta_\sigma$ of $\sigma$ is either rational or transcendental, the first case happening precisely when $\sigma^n-1$ is a separable isogeny for all $n$. We call this condition very inseparability and show it is equivalent to the action of $\sigma$ on the local $p$-torsion group scheme being nilpotent. The "false" zeta function $D_\sigma$, in which the number of fixed points of $\sigma^n$ is replaced by the degree of $\sigma^n-1$, is always a rational function. Let $1/\Lambda$ denote its largest real pole and assume no other pole or zero has the same absolute value. Then, using a general dichotomy result for power series proven by Royals and Ward in the appendix, we find that $\zeta_\sigma(z)$ has a natural boundary at $|z|=1/\Lambda$ when $\sigma$ is not very inseparable. We introduce and study tame dynamics, ignoring orbits whose order is divisible by $p$. We construct a tame zeta function $\zeta^*_{\sigma}$ that is always algebraic, and such that $\zeta_\sigma$ factors into an infinite product of tame zeta functions. We briefly discuss functional equations. Finally, we study the length distribution of orbits and tame orbits. Orbits of very inseparable endomorphisms distribute like those of Axiom A systems with entropy $\log \Lambda$, but the orbit length distribution of not very inseparable endomorphisms is more erratic and similar to $S$-integer dynamical systems. We provide an expression for the prime orbit counting function in which the error term displays a power saving depending on the largest real part of a zero of $D_\sigma(\Lambda^{-s})$.

Analytic torsion, dynamical zeta functions, and the Fried conjecture

We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of Fried. This article should be read in conjunction with an earlier paper by Moscovici and Stanton.

Selberg and Ruelle zeta functions for non-unitary twists

In this paper we study the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd-dimensional manifold. These are functions of a complex variable s in some right half-plane of \(\mathbb {C}\). Using the Selberg trace formula for arbitrary finite dimensional representations of the fundamental group of the manifold, we establish the meromorphic continuation of the dynamical zeta functions to the whole complex plane. We explicitly describe the singularities of the Selberg zeta function in terms of the spectrum of certain twisted Laplace and Dirac operators.

Semigroup Actions of Expanding Maps

We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the semigroup action and prove the existence of stationary probability measures.

The Quest for the Ultimate Anisotropic Banach Space

We present a new scale $U^{t,s}_p$ (with $s<-t<0$ and $1 \le p <\infty$) of anisotropic Banach spaces, defined via Paley-Littlewood, on which the transfer operator associated to a hyperbolic dynamical system has good spectral properties. When $p=1$ and $t$ is an integer, the spaces are analogous to the "geometric" spaces considered by Gou\"ezel and Liverani. When $p>1$ and $-1+1/p<s<-t<0<t<1/p$, the spaces are somewhat analogous to the geometric spaces considered by Demers and Liverani. In addition, just like for the "microlocal" spaces defined by Baladi-Tsujii, the spaces $U^{t,s}_p$ are amenable to the kneading approach of Milnor-Thurson to study dynamical determinants and zeta functions. In v2, following referees' reports, typos have been corrected (in particular (39) and (43)). Section 4 now includes a formal statement (Theorem 4.1) about the essential spectral radius if $d_s=1$ (its proof includes the content of Section 4.2 from v1). The Lasota-Yorke Lemma 4.2 (Lemma 4.1 in v1) includes the claim that $\cal M_b$ is compact. Version v3 contains an additional text "Corrections and complements" showing that s> t-(r-1) is needed in Section 4.

The dynamical zeta function for commuting automorphisms of zero-dimensional groups

For a $\mathbb{Z}^{d}$-action $\unicode[STIX]{x1D6FC}$ by commuting homeomorphisms of a compact metric space, Lind introduced a dynamical zeta function that generalizes the dynamical zeta function of a single transformation. In this article, we investigate this function when $\unicode[STIX]{x1D6FC}$ is generated by continuous automorphisms of a compact abelian zero-dimensional group. We address Lind’s conjecture concerning the existence of a natural boundary for the zeta function and prove this for two significant classes of actions, including both zero entropy and positive entropy examples. The finer structure of the periodic point counting function is also examined and, in the zero entropy case, we show how this may be severely restricted for subgroups of prime index in $\mathbb{Z}^{d}$. We also consider a related open problem concerning the appearance of a natural boundary for the dynamical zeta function of a single automorphism, giving further weight to the Pólya–Carlson dichotomy proposed by Bell and the authors.

Prime number theorems and holonomies for hyperbolic rational maps

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of $S^1$. We will show that these zeta functions are non-vanishing on a half plane $\Re(s) > \delta - \epsilon$, where $\delta$ is the Hausdorff dimension of the Julia set of f.