A note on the dynamical zeta function of general toral. Artin mazur zeta function ruelle zeta function ihara zeta function of a graph airy zeta function, related to the zeros of the airy function. Artin mazur zeta function barnes zeta function bessel zeta function beurling zeta function cotangent zeta function dedekind zeta function epstein zeta function fibonacci zeta function goss zeta function hasseweil zeta function height zeta function igusa zeta function ihara zeta function lefschetz zeta function lerch zeta function local zeta. Note that we are, as usual, not counting xed points with multiplicity, which in. It is wellknown that the artin mazur dynamical zeta function of a hyperbolic or quasihyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. Zeta functions and transfer operators for piecewise. Dynamical zeta functions, nielsen theory and reidemeister torsion. Transcendence of the artinmazur zeta function for polynomial. Hence the signs of all nonzero eam need to be determined. The artinmazur zeta function for graph maps article pdf available january 2004. It is well known that artinmazur zeta functions of subshift of nite types can be calculated by bowenlanford formula. Next, we give basic notions about symbolic dynamics.
When m is quasihyperbolic, this is done in 23, lemma 2. This zeta function was introduced by artin and mazur in 2, where it is studied for x a manifold and f a diffeomorphism. It would appropriate here to mention also the milnorthurston zeta function and kneading matrices and the hofbauerkeller zeta function for interval maps 5. Artin mazur zeta function of a piecewise monotone in terv al map. As artin reciprocity shows, when g is an abelian group these l functions have a second description as dirichlet l functions when k is the rational number field, and as hecke l functions in general. I highly recommend any of artins books as they are some of the best exposition you can find on the subjects he writes about. The artin mazur zeta function of a dynamically affine rational map in positive characteristic andrew bridy abstract. Artin and mazur replace the frobenius of v with a diffeomorphism f of a smooth compact manifold m such that its iterates fk all have isolated. Selberg zeta function of a riemann surface witten zeta function of a lie group spectral zeta function of an operator dynamical systems. The result is a closed formula that can be calculated.
In the special case when fis hyperbolic technically, fis an axiom a diffeomorphism restricted to a basic set, one finds that this zeta function is rational smale, guckenheimer, manning, bowen, fried. In this paper, we study the artinmazur zeta function of a generalization of the wellknown. Mazurs name is on one of the grand generalizations of the riemann zeta functions, the artinmazur zeta function which has exploded into a major tool under the lead. Artin was born in hamburg, germany, and brought up in indiana. Mr 0176482 david ruelle, dynamical zeta functions and transfer operators 2002 pdf. The artin mazur zeta function was historically the. The artinmazur zeta function of a dynamically affine rational map. This book requires little more than calculus for its background. Dynamical zeta functions and the distribution of orbits. It has evident analogs for all other zeta functions for some of which it has been proven. We introduce the negative zeta function for piecewise graph maps and we prove that it is rational if and only if so is the artin mazur zeta function in the case of a map with finitely many stable. On dynamical zeta functions for polynomial maps over padic. Emil artin relies on coneptual approaches to illuminating the gamma function s features instead of tedioius calculations.
This is a special case of an artinmazur zeta function, which is defined for certain dynamical systems and in general counts the. In this paper, we prove the asymptotic behavior of these functions for a periodicfinitetype shift. The exposition is selfcontained, and required a preliminary knowledge of only the ele. The milnorthurston theorem states that the artinmazur zeta function is the inverse of the kneading determinant of analogues edit the artinmazur zeta function is formally similar to the local zeta function, when a diffeomorphism on a compact manifold replaces the frobenius mapping for an algebraic variety over a finite field. On the other hand there exist maps for which artin mazur zeta function is transcendental 11. We determine the rationality or transcendence of the artin mazur zeta function of a dynamically a.
Artinmazur power series t is a convenient way of enumerating the periodic opints of f. Artinmazur zeta function, lefschetz zeta func tion, ruelle zeta function for discrete dynamical systems, ruelle. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1. The prime orbit and mertens orbit counting functions describe the growth of closed orbits in a discrete dynamical system in a certain way. The zeta function of the beta transformation volume 14 issue 2 leopold flatto, jeffrey c. This form corresponds in fact closely to the euler product formula for the riemann zeta function. In mathematics, the artinmazur zeta function, named after michael artin and barry mazur, is a.
It is wellknown that the artinmazur dynamical zeta function of a hyperbolic or quasihyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. The artin mazur zeta function suppose t is a map t. Orbit growth of periodicfinitetype shifts via artinmazur. The burnside ring and the universal zeta function of. We introduce the negative zeta function for piecewise graph maps and we prove that it is rational if and only if so is the artinmazur zeta function in the case of a map with finitely many stable. Selberg zetafunction of a riemann surface witten zeta function of a lie group spectral zeta function of an operator dynamical systems. We show that the artinmazur zeta function can be extended to a meromorphic function via an expansion of 1 defined by using the transformation. How we measure reads a read is counted each time someone views a publication summary such. From what weve said before, this series encodes the whole sequence jfixtnj. Perhaps foolishly, i will still use the same notation e for the now. The artinmazur zeta function of a dynamically affine rational map in positive characteristic by andrew bridy get pdf 251 kb. Artins parents left germany in 1937, because michael artins maternal grandfather was jewish. Then we construct a zeta function in the usual way.
The zeta function of the beta transformation ergodic theory. We present and study certain hypotheses that imply a dichotomy for the artinmazur zeta function of the dynamical system. While the artinmazur zeta function counts the periodic points geometrically, the lefschetz zeta function counts them homologically. Emil artin relies on coneptual approaches to illuminating the gamma functions features instead of tedioius calculations. Milne niranjan ramachandran november, 20 abstract beginning with the conjecture of artin and tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function. His parents were natalia naumovna jasny natascha and emil artin, preeminent algebraist of the 20th century. In that case, the artinmazur zeta function simplifies to, where a is the adjacency matrix. The mazur stein book has probably the best street cred among the rh books for the general audience. We study the rationality of the artinmazur zeta function of a dynamical system defined by a polynomial selfmap of. The mazurstein book has probably the best street cred among the rh books for the general audience. The artinmazur zeta function contains a single variablezand was constructed by analogy with thegeometric theoryzeta function above. Motivic complexes and special values of zeta functions.
The proof relies on the meromorphic extension of its artinmazur zeta function. The artinmazur zeta function of a dynamically affine. The riemann hypothesis conjectures a characterization of the roots of this zeta function and is regarded as one of the outstanding problems in mathematics. The artin mazur zeta function was adopted later by milnor and thurston 67 to count periodic points for a piecewise monotone map of the interval.
Furthermore, the zeta function can be expressed in terms of closed orbits as. The nielsen and the reidemeister zeta functions of maps on. We show that the artinmazur zeta function can be extended to a meromorphic function via an expansion of 1 defined by. Riemann and his zeta function 3 logz lnz the natural logarithm of a complex number z. Mazur s name is on one of the grand generalizations of the riemann zeta functions, the artin mazur zeta function which has exploded into a major tool under the lead. Artin mazur zeta functions in arithmetic dynamics 3 4 but the sequence d ncannot be eventually periodic, contradiction. We determine the rationality or transcendence of the artinmazur zeta function of a dynamically affine selfmap of. Iharas zetafunktion av en graf kan ses som ett exempel pa artinmazurs zetafunktion. This zeta function is a reduced version of the smale zeta function for flows and is the natural analogue of the artinmazur zeta function for diffeomorphisms, whereas smales was inspired by selbergs. The artin mazur zeta function is formally similar to the local zeta function, when a diffeomorphism on a compact manifold replaces the frobenius mapping for an algebraic variety over a finite field. The burnside ring and the universal zeta function of finite.
Sections 9 and i0 state several versions of our main theorem which computes this artinmazur zeta function in terms of the kneading determinant. Inspired by the hasseweil zeta function of an algebraic variety over a. Pdf artinmazur zeta function on trees with infinite edges. A good knowledge of the eigenvalues of the transfer operator is needed, e. The prime number theorem, hardys theorem on the zeros of. Artinmazur zeta function barnes zeta function bessel zeta function beurling zeta function cotangent zeta function dedekind zeta function epstein zeta function fibonacci zeta function goss zeta function hasseweil zeta function height zeta function igusa zeta function ihara zeta function lefschetz zeta function lerch zeta function local zeta. Dynamical zeta functions, nielsen theory alexander felshtyn. This zeta function is a reduced version of the smale zeta function for flows and is the natural analogue of the artin mazur zeta function for diffeomorphisms, whereas smales was inspired by selbergs. The artin l function, is then the infinite product over all prime ideals of these factors. Orbit growth of periodicfinitetype shifts via artin. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change.
Artin s parents left germany in 1937, because michael artin s maternal grandfather was jewish. Motivic complexes and special values of zeta functions james s. In that case, the artin mazur zeta function simplifies to, where a is the adjacency matrix. Artinmazur zetafunction ruelle zeta function ihara zetafunction of a graph airy zeta function, related to the zeros of the airy function. Artinmazur zeta function of a piecewise monotone in terv al map. The artinmazur zeta function to derive a formula for the dynamical zeta function, we observe that am eam sgneam. The ihara zeta function of a graph can be interpreted as an example of the artin mazur zeta function. Some remarks on random zeta functions 1033 we shall prove the. However, the formula 2 cannot be applied anymore if the real part. For the second point we assume additionally that the matrices are diagonal and that the law of the perturbation is smooth enough. In the case of a finite simple graph, there is a natural time evolution, the subshift of finite type defined by the graph. Suppose for every nthat tn has only a nite number of xed points.
1424 144 1032 463 73 1013 809 728 1339 1003 127 27 1453 691 917 104 1356 152 656 865 552 645 1299 582 845 7 323 1361 840 1242 566 1441 11 421 155 1269 6 285 942 1156 186 1308