## number theory and dynamical systems

Workshop on Algebraic, Number Theoretic and Graph Theoretic Aspects of Dynamical Systems, 2–6 February, 2015, UNSW, Sydney, Australia

Interactions between Dynamics of Group Actions and Number Theory, 9 June–4 July 2014, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK

Dynamics and Numbers, 1 June–31 July 2014, Max Planck Institute for Mathematics, Bonn

An indirect link between dynamical systems and number theory is provided by the theory of dynamical zeta functions. Many dynamical systems possess such zeta functions, which historically emerged in analogy with the Riemann zeta function.

Further down this page you will find a bibliography of other, more direct, connections of various kinds:

• application of dynamical systems thinking to number theoretical problems
• use of number theory results in dynamical systems problems
• analogies arising between results in dynamical systems theory and number theory
• 'abstract' dynamical systems defined in terms of the prime numbers
• dynamical systems in p-adic contexts

Most importantly, however, is the search for a dynamical system underlying (or "lurking behind" as N. Snaith writes in her Ph.D. thesis) the Riemann zeta function, and thereby underlying the integers and everything which follows from them!

(1) Berry and Keating's proposed Riemann dynamics would provide the much-desired spectral interpretation of the Riemann zeta zeros.

(2) A particular flow on a particular C*-algebra was introduced here:

J.-B. Bost and A. Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457.

This models a quantum statistical system whose partition function coincides with the Riemann zeta function. The only pole of zeta (at 1) is shown to correspond to a spontaneous breaking of symmetry in the system. Alain Connes went on to write

A. Connes, "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function", Selecta Mathematica (New Series) 5 (1999) 29-106

wherein the truth of the Riemann Hypothesis was reduced to the validity of a certain (dynamical) trace formula, and the zeta zeros given a spectral interpretation, but as an absorption rather than an emission spectrum. Also relevant here:

This generalises the result of Bost and Connes which interprets the Riemann zeta function as a partition function of a dynamical system (in the C*-algebra formalism) whereby the pole at s =1 is interpreted in terms of spontaneous symmetry breaking. The generalisation extends the result to general number fields, and is further improved in the following paper which generalises in such a way that the partition function becomes the appropriate Dedekind zeta function

[abstract] "In this article we construct a C*-dynamical system with partition function the Dedekind zeta function of a given number field and with a phase transition at the pole of this zeta function which detects a breaking of symmetry with respect to a natural symmetry group. This extends work of Bost-Connes and Harari-Leichtnam."

This is a recent development:

[abstract:] "This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field."

The following builds on the above work:

E. Ha and F. Paugam, "Bost-Connes-Marcolli systems for Shimura varieties" (preprint 03/05)

[abstract:] "We construct a Quantum Statistical Mechanical system $(A,\sigma_t)$ analogous to the Bost-Connes-Marcolli system...in the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the "correct" symmetries and "correct" partition function. We give a formalism that applies to general Shimura data (G,X). The object of this series of papers is to show that these systems have phase transitions and spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature."    [additional background information]

M. Marcolli, "Number Theory in Physics" (survey article, 07/05)

M. Marcolli and A. Connes, "Q-lattices: quantum statistical mechanics and Galois theory", Journal of Geometry and Physics 56 no. 1 (2006) 2–23

G. Cornelissen and M. Marcolli, "Quantum Statistical Mechanics, $L$-series and Anabelian Geometry" (preprint 09/2010)

[abstract:] "It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C*-algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck's "anabelian" program, much like the Neukirch--Uchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups. In the second part of the paper, we use these systems to prove the following. If there is an isomorphism of character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding $L$-series (not just the zeta function), then the number fields are isomorphic.This is also equivalent to the purely algebraic statement that there exists a topological group isomorphism as a above and a norm-preserving group isomorphism between the ideals of the fields that is compatible with the Artin maps via the other map."

G. Cornelissen, "Number theory and physics, an eternal rusty braid", Eidnhoven Mathematics Colloquiums, 9th November 2011

[abstract:] "I will describe joint work with Matilde Marcolli in which we apply ideas from quantum statistical mechanics and dynamical systems to solve the number theoretical analogue of the problem how to hear the shape of a drum".

Both the Berry-Keating and Connes, et.al. approaches are seen as promising leads in the quest to prove the Riemann hypothesis.

(3) J.-F. Burnol has proposed a renormalisation group flow on a space involving all universality domains of something like a system in statistical physics. The Wiener-Khintchine duality relation of statistical mechanics is related to the functional equation of the Riemann zeta function, and the zeta zeros to fixed points ("or another more subtle mechanism") of the flow.

(4) Finally, Christopher Deninger has been seeking a dynamical system which will provide a cohomological interpretation of the zeta zeros. The idea is to interpret the Riemann-Weil explicit formula as Lefschetz-type trace formula. There is some overlap with Connes work:

C. Deninger, "Arithmetic geometry and analysis on foliated spaces" (preprint 05/05)

[abstract:] "This report on the topics in the title was written for a lecture series at the Southwestern Center for Arithmetic Algebraic Geometry at the University of Arizona. It may serve as an introduction to certain conjectural relations between number theory and the theory of dynamical systems on foliated spaces. The material is based on streamlined and updated versions of earlier papers on this subject."

"...we explain how the theory of the Riemann zeta function naturally leads to the investigation of a class of dynamical systems on foliated spaces. The hope is that finding the right dynamical system will be an important step towards a better understanding of zeta(s)."

C. Deninger, "Some analogies between number theory and dynamical systems on foliated spaces", Documenta Mathematica, Extra Volume ICM I (1998) 163-186.

"In this article we describe what a cohomology theory related to zeta and L-functions for algebraic schemes over the integers should look like. We then point out some striking analogies with the leafwise reduced cohomology of certain foliated dynamical systems"

C. Deninger, "Lefschetz trace formulas and explicit formulas in analytic number theory", J. Reine Angew. Math. 441 (1993) 1-15.

[Abstract:] "We discuss analogies between the etale site of arithmetic schemes and the algebraic topology of dynamical systems. The emphasis is on Lefschetz numbers. We also discuss similarities between infinite primes in arithmetic and fixed points of dynamical systems."

[Abstract:] "We interpret the 'explicit formulas' in the sense of analytic number theory for the zeta function of an elliptic curve over a finite field as a transversal index theorem on a 3-dimensional laminated space."

[Abstract:] " We discuss analogies between number theory and the theory of dynamical systems on spaces with a one-codimensional foliation. The emphasis is on comparing the "explicit formulas" of analytic number theory with certain dynamical Lefschetz trace formulas. We also point out a possible relation between an Arakelov-Euler characteristic and an Euler characteristic in the sense of Connes. Finally the role of generalized solenoids as phase spaces in our picture is explained."

partial bibliography

J.H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics 241 (Springer, 2007)

M.M. Dodson and J.A.G. Vickers (editors), Number Theory and Dynamical Systems, LMS Lecture Notes 134 (Cambridge University Press, 1989)

D.W. Boyd, "Dynamical Thinking in Number Theory", a review of Ergodic Theory of Numbers by K. Dajani and C. Kraaikamp [Carus Monographs 29, MAA (2002)] [MAA review]

M. Einsiedler and T. Ward, Ergodic Theory: with a view towards Number Theory

[authors' description:] "This is the first part of a two-volume project that aims to develop enough of the basic machinery of ergodic theory to describe some of the recent applications of ergodic theory to number theory. Two specific emphases are to avoid reliance on background in Lie theory and to fully prove the material needed in measure theory which goes beyond the standard texts. This will be a rigorous introduction, developing the machinery of conditional measures and expectations, entropy, mixing and recurrence. Applications will include the ergodic proof of Szemeredi's theorem and the connection between the continued fraction map and the modular surface.

This web page contains some chapters in draft form of the book in pdf."

J. Lagarias, "Number Theory and Dynamical Systems" from The Unreasonable Effectiveness of Number Theory, (S.A. Burr, Ed.), Proceedings of Symposia in Applied Mathematics 46, (AMS, 1992) 35-72.

"This paper describes the occurrence of number-theoretic problems in dynamical systems. These include Hamiltonian dynamical systems, dissipative dynamical systems and discrete dynamical systems. Diophantine approximations play an important role."

H. Furstenburg, Recurrence in Ergodic Theory and Combinatorial Number Theory

book review by K. Petersen

A. Knauf, "Number theory, dynamical systems and statistical mechanics" (1998 lecture notes)

"In these lecture notes connections between the Riemann zeta function, motion in the modular domain and systems of statistical mechanics are presented."
[extensive survey article]

J. Fiala and P. Kleban, "Generalized number theoretic spin chain-connections to dynamical systems and expectation values", J. of Stat. Physics 121 (2005) 553-577

[abstract:] "We generalize the number theoretic spin chain, a one-dimensional statistical model based on the Farey fractions, by introducing a new parameter x > 0. This allows us to write recursion relations in the length of the chain. These relations are closely related to the Lewis three-term equation, which is useful in the study of the Selberg zeta-function. We then make use of these relations and spin orientation transformations. We find a simple connection with the transfer operator of a model of intermittency in dynamical systems. In addition, we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length. Some of these expectation values appear to be directly connected with the mechanism of the phase transition."

W. Parry, "An analogue of the prime number theorem for shifts of finite type and their suspensions", Israel Journal of Mathematics 45 (1983) 41-52.

[abstract:] "Following the classical procedure developed by Wiener and Ikehara for the proof of the prime number theorem we find an asymptotic formula for the number of closed orbits of a suspension of a shift of finite type when the suspended flow is topologically weak-mixing and when the suspending function is locally constant."

This work was extended in Mark Pollicott's Ph.D. thesis, supervised by Parry, and led to the following paper:

W. Parry and M. Pollicott, "An analogue of the prime number theorem for closed orbits of axiom A flows", Annals of Mathematics 118 (1983) 573-591.

[abstract:] "For an axiom A flow restricted to a basic set we extend the zeta function to an open set containing $\script{R}(s) \geq h$ where $h$ is the topological entropy. This enables us to give an asymptotic formula for the number of closed orbits by adapting the Wiener-Ikehara proof of the prime number theorem."

Chapter 2 ("An application of recurrence to arithmetic progressions") of Dynamical Systems and Ergodic Theory by M. Pollicott and M. Yuri (Cambridge Univ. Press, 1998)

B. Green and T. Tao, "The primes contain arbitrarily long arithmetic progressions" (preprint 04/04)

[abstract:] "We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemerédi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemerédi's theorem that any subset of a sufficiently pseudorandom set of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim. Using this, one may place the primes inside a pseudorandom set of 'almost primes' with positive relative density."

[from proof outline, p.4] "Perhaps surprisingly for a result about primes, our paper has at least as much in common with the ergodic-theoretic approach as it does with the harmonic analysis approach of Gowers. We will use a language which suggests this close connection, without actually relying explicitly on any ergodic theoretical concepts".

T. Tao, "Obstructions to uniformity, and arithmetic patterns in the primes" (preprint 05/05, submitted for special edition of Quarterly J. Pure Appl. Math. in honour of John Coates)

[abstract:] "In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving 'randomly', and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes."

B. Kra, "The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view", Bull. Amer. Math. Soc. 43 (2006), 3-23

[abstract:] "A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory."

B. Kra, "Ergodic methods in combinatorial number theory", Additive Combinatorics, March/April 2006

[abstract:] "Shortly after Szemeredi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof using ergodic theory. This lead to the field of ergodic Ramsey Theory, in which the problems are motivated by additive combinatorics and the proofs use ergodic theory. This has lead to new combinatorial results, some of which have yet to be obtained by other means, and to a deeper understanding of the structure of measure preserving systems. I will outline the ergodic theory background needed to understand these results, with an emphasis on recent developments in ergodic theory."

A. Arbieto, C. Moreira and C. Matheus, "The remarkable effectiveness of ergodic theory in number theory", Ensaios Matemáticos 17 (2009) 1-98

[abstract:]"The main goal of this survey is the description of the fruitful interaction between Ergodic Theory and Number Theory via the study of two beautiful results: the first one by Ben Green and Terence Tao (about long arithmetic progressions of primes) and the second one by Noam Elkies and Curtis McMullen (about the distribution of the sequence $\{\sqrt{n}\} mod 1$). More precisely, during the first part, we'll see how the ergodic-theoritical ideas of Furstenberg about the famous Szemerèdi theorem were greatly generalized by Green and Tao in order to solve the classical problem of finding arbitrarily long arithmetical progression of prime numbers, while the second part will focus on how Elkies and McMullen used the ideas of Ratner's theory (about the classification of ergodic measures related to unipotent dynamics) to compute explicitly the distribution of the sequence $\{\sqrt{n}\}$ on the unit circle."

R.C. Rhoades, "An application of ergodic theory to a classical problem in number theory (notes from the colloquium talk given by Jordan Ellenberg on December 9, 2005 at the University of Wisconsin)

"A famous theorem of Langrange asserts that every positive integer is the sum of four squares. One can ask many more general questions about representations of integers by quadratic forms, or quadratic forms by quadratic forms. We will describe a theorem "of Lagrange type" whose proof relies on ideas from ergodic theory. No prior knowledge of quadratic forms or ergodic theory will be assumed."

A. Bellow and H. Furstenburg, "An application of number theory to ergodic theory and the construction of uniquely ergodic models", Israel Journal of Mathematics 33 (1979) 231-240

[abstract:] "Using a combinatorial result of N. Hindman one can extend Jewett's method for proving that a weakly mixing measure preserving transformation has a uniquely ergodic model to the general ergodic case. We sketch a proof of this reviewing the main steps in Jewett's argument."

A.V. Malyshev, "Yu. V. Linnik's ergodic method in number theory", Acta Arithmetica 27 (1975) 555-598

B. Bagchi, Recurrence in topological dynamics and the Riemann hypothesis", Acta Mathematica Hungarica 50 (1987) 227-240

V.M. Popov, "On stability properties which are equivalent to Riemann hypothesis", Libertas Math. 5 (1985) 55-61

[abstract:] "As shown recently by the author, the location of zeros of Riemann's zeta function is related to the rate of growth of the solutions of some dynamical systems with applications in control theory. In this paper a new class of differential-delay systems is introduced whose stability is equivalent to the Riemann hypothesis (RH). The systems considered have a linear main part in which a zeta-like term and the Euler-Mascheroni constant are involved. The above result is of independent interest, being a new property which is equivalent to RH."

S.L. Cacciatori, M.A. Cardella, "Uniformization, unipotent flows and the Riemann hypothesis" (preprint 02/2011)

[abstract:] "We prove equidistribution of certain multidimensional unipotent flows in the moduli space of genus $g$ principally polarized abelian varieties (ppav). This is done by studying asymptotics of $\pmb{\Gamma}_{g} \sim Sp(2g,\mathbb{Z})$-automorphic forms averaged along unipotent flows, toward the codimension-one component of the boundary of the ppav moduli space. We prove a link between the error estimate and the Riemann hypothesis. Further, we prove $\pmb{\Gamma}_{g - r}$ modularity of the function obtained by iterating the unipotent average process $r$ times. This shows uniformization of modular integrals of automorphic functions via unipotent flows."

V. Vatsal, "Uniform distribution of Heegner points", Inventiones Mathematicae 148 (2002) 1-46

[uses ergodic results of Ratner "to show that Heegner points are uniformly distributed along certain curves, and hence to establish a conjecture of Mazur concerning the non-vanishing of a certain twisted L-function of an elliptic curve over an imaginary quadratic field....a result that seems very far from ergodic theory (D.W. Boyd)]

A.O. Lopes, "The zeta function, non-differentiability of pressure, and the critical exponent of transition", Advances in Mathematics 101 (1993) 133-165

[excerpt from abstract:] "The main purpose of this paper is to analyze the lack of differentiablility of the pressure and, from the behaviour of the pressure around the point of non-differentiablity, to derive an asymptotic formula for the number of periodic orbits (under certain restrictions related to the norm of the periodic orbit) of a dynamical system. This kind of result is analogous to the well known Theorem of Distribution of Primes...We introduce a functional equation relating the pressure and the Riemann zeta function, and this equation plays an essential role in the proof of our results."

Yu. I. Manin and M. Marcolli, "Continued fractions, modular symbols, and non-commutative geometry" (Selecta Mathematica (New Series) 8 no. 3 (2002) 475-520.

[abstract:] "Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient $PGL(2,\bold{Z})\setminus\bold{P}^1(\bold{R})$ should be considered as non-commutative modular curve, and show that the modular complex can be seen as a sequence of K0-groups of the related crossed-product C*-algebras.

This paper is an expanded version of the previous "On the distribution of continued fractions and modular symbols". The main new features are Section 4 on non-commutative geometry and the modular complex and Section 1.2.2 on the Mixmaster Universe."

M. Marcolli, "Limiting modular symbols and the Lyapunov spectrum" (Journal of Number Theory 98 No. 2 (2003) 348-376.

"This paper consists of variations upon the theme of limiting modular symbols. Topics covered are: an expression of limiting modular symbols as Birkhoff averages on level sets of the Lyapunov exponent of the shift of the continued fraction, a vanishing theorem depending on the spectral properties of a generalized Gauss-Kuzmin operator, the construction of certain non-trivial homology classes associated to non-closed geodesics on modular curves, certain Selberg zeta functions and C* algebras related to shift invariant sets."

C. Consani and M. Marcolli, "Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry" (to appear in Selecta Mathematica)

[abstract:] "In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the 'closed fibers at infinity'. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g > 2. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper."

K. Consani and M. Marcolli, "Triplets spectraux en geometrie d'Arakelov" Comptes Rendus Acad. Sci. Paris Ser. I 335 (2002) 779-784

[abstract:] "This note is a brief overview of the results of math.AG/0205306. We use Connes' theory of spectral triples to provide a connection between Manin's model of the dual graph of the fiber at infinity of an Arakelov surface and the cohomology of the mapping cone of the local monodromy."

C. Consani and M. Marcolli, "Spectral triples from Mumford curves", International Math. Research Notices 36 (2003) 1945-1972.

[abstract:] "We construct spectral triples associated to Schottky-Mumford curves, in such a way that the local Euler factor can be recovered from the zeta functions of such spectral triples. We propose a way of extending this construction to the case where the curve is not k-split degenerate."

M. Marcolli, "Modular curves, C* algebras, and chaotic cosmology" (preprint 12/03)

[abstract:] "We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. In particular we consider a class of solutions with bounded number of cycles in each Kasner era and describe their dynamical properties, which we relate to the noncommutative geometry of a moduli space of such solutions, given by a Cuntz-Krieger C*-algebra."

Marcolli's current CV also mentions that she has an intriguingly-titled article in preparation with N. Ramachandran: "Tower power: KMS states and complex multiplication", and two books in preparation - Noncommutative Geometry with Alain Connes and Arithmetic Noncommutative Geometry (notes of lectures given at the University of Toronto and the University of Nottingham).

M.L. Lapidus and M. van Frankenhuysen, "A prime orbit theorem for self-similar flows and Diophantine approximation", Contemporary Mathematics volume 290 (AMS 2001) 113-138.

"Assuming some regularity of the dynamical zeta function, we establish an explicit formula with an error term for the prime orbit counting function of a suspended flow. We define the subclass of self-similar flows, for which we give an extensive analysis of the error term in the corresponding prime orbit theorem...The precise order of the error term depends on the 'dimension free' region of the dynamical zeta function, as in the classical Prime Number Theorem. This region in turn depends on properties of Diophantine approximation of the weights of the flow."

G. Chalmers, "Computational derivation to zeta zeros and prime numbers" (preprint 03/05)

[abstract:] "A route to the derivation of the numbers $s$ to the transcendental equation $\zeta(s)=0$ is presented. The solutions to this equation require the solving of a geodesic flow in an infinite dimensional manifold. These solutions enable one approach to a formula generating the prime numbers."

K.A. Broughan and A.R. Barnett, "The holomorphic flow of the Riemann zeta function" (preprint, 05/00)

[abstract:] "The flow of the Riemann zeta function is considered and phase portraits presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases, the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest using the Chebyshev method and using Mathematica.

The phase diagrams suggest new analytic properties of zeta, a number of which are proved and a number of which are given in the form of conjectures."

K. Broughan's research page includes further connections between dynamical systems and the zeta and xi functions and more phase portraits of the zeros of these functions

L.A. Bunimovich and C.P. Dettmann, "Open circular billiards and the Riemann hypothesis", Phys. Rev. Lett. 94 (2005) 100201

[abstract:] "We obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeroes of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and major unsolved problems in number theory."

L.A. Bunimovich and C.P. Dettmann, "Escape from a circle and Riemann hypotheses" (preprint 03/2006)

[abstract:] "We consider open circular billiards with one and with two holes. The exact formulas for escape are obtained which involve the Riemann zeta function and Dirichlet L-functions. It is shown that the problem of finding the exact asymptotics in the small hole limit for escape in some of these billiards is equivalent to the Riemann hypothesis."

C.P. Dettman, "New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis" (preprint 03/2011)

[abstract:] "The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor $\sqrt{t}$ are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution, with an explicit formula for the variance, is recovered when scaling by $\sqrt {t\ln t}$, as shown by Sz\'asz and Varj\'u in 2007. Here we explore the higher dimensional infinite horizon case, giving an explicit formula for the variance in the superdiffusive case, connections with the Riemann Hypothesis in the small scatterer limit, and evidence for a critical dimension $d=6$ beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions."

V. Petkov, "Analytic singularities of the dynamical zeta function", Nonlinearity 12 (1999) 1663-1681.

[Abstract:] "We study the dynamical zeta function ZD(s) related to the periodic trajectories of the billiard flow for several disjoint strictly convex bodies in R3. We show that the analytic properties of ZD(s) close to the line of absolute convergence Re[s] = s0 are similar to the behaviour close to the line Re[s] = 1 of the inverse Q(s) = 1/R(s) of the classical Riemann zeta function R(s)."

P. Vishe, "Dynamical methods for rapid computations of L-functions" (2011 PhD dissertation)

[abstract:] "Let $f$ be a holomorphic or Maass cusp form on the upper half plane. We use the slow divergence of the horocycle flow on the upper half plane to get an algorithm to compute $L(f,1/2+iT)$ up to a maximum error $O(T^{-\gamma})$ using $O(T^{7/8+\eta})$ operations. Here $\gamma$ and $\eta$ are any positive numbers and the constants in $O$ are independent of $T$. We thus improve the current approximate functional equation based algorithms which have complexity $O(T^{1+\eta})$."

A. Verjovsky, "Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold", Dyamical Systems. Proceedings of the 3rd international school of dynamical systems, Santiago de Chile, 1990 (R. Bamon, et.al., eds.) Longman Scientific and Tehcnical Pitman Res. Notes Math. Ser 285 (1993) 253-298

[abstract:] "Let $M={\rm SL}_2( Z)\backslash {\rm SL}_2( R)$. A theorem of Dani gives on this space, for each $y>0$, ergodic measures supported on closed orbits of period $1/y$ of the horocyclic flow. Let $m(y)$ denote the measures and $m$ Haar measure. For smooth functions $f$ on $M$ satisfying $\int_Mf\,dm=0$, Zagier has shown $\int_Mf\,dm_y=o(y^{1/2})$ as $y\to 0$ and that the Riemann hypothesis is equivalent to $(*)$ $\int_M f\,dm_y=o(y^{3/4-\epsilon})$ as $y\to 0$. The article shows that $(*)$ does not hold for the characteristic function of certain "boxes" in $M$. This of course does not disprove the Riemann hypothesis. The proof is based on reducing the problem to a lattice point counting problem. The impossibility of improving the error is related to an analogous situation in the circle problem related to the slow decay of Fourier transforms of characteristic functions."

Wang Liang and Huang Yan, "A kind of dynamic model of prime numbers" (preprint 01/06)

[abstract:] "A dynamic sieve method is designed according to the basic sieve method. It mainly refers to the symbolic dynamics theory. By this method, we could connect the prime system with familiar 'Logistic Mapping'. An interesting discovery is that the pattern of primes could be depicted by a series of orbits of this mapping. Some heuristic proofs for open problems like twin primes are obtained through this relation. This research gives a new viewpoint for the distribution of prime numbers."

A. Costé, "Un système dynamique lié à la suite des nombres premiers", Comptes Rendus de l'Académie des Sciences (Series I) 333 (2001) 663-668

[abstract (translation):] "We study the dynamical system defined by the map $\Phi: ]0,1] \rightarrow ]0,1]$, where $\Phi(x)= px - 1$ on ]1/p,1/q] if q and p are consecutive prime numbers. We relate the existence of an absolutely continuous invariant measure to ergodicity of a Markov chain P on the union of orbits stemming from numbers 1/p (p prime). We prove that ergodicity of P implies ergodicity of $\Phi$ . We establish a link between transfer probabilities of order n and some sets of sequences of the symbolic dynamic. This leads to a way of computing these coefficients using Monte Carlo method. We propose an algorithm which leads to a density indicating a good experimental fit with a typical orbit."

G. Garcia-Perez, M. Angeles Serrano and M. Boguna, "The complex architecture of primes and natural numbers" (preprint 02/2014)

[abstract:] "Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cram\'er's conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cram\'er's version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help not only to conjecture but also to prove results about primes and the complex architecture of natural numbers."

C. Bonanno and M.S. Mega, "Toward a dynamical model for prime numbers", Chaos, Solitons and Fractals 20 (2004) 107-118

[abstract:] "We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between the prime numbers and intermittency."

The main idea here is that the Manneville map Tz exhibits a phase transition at z = 2, at which point the mean Algorithmic Information Content of the associated symbolic dynamics is n/log n. n is a kind of iteration number. For this to work, the domain of Tz [0,1] must be partitioned as [0,0.618...] U [0.618...,1] where 1.618... is the golden mean.

The authors attempt to exploit the resemblance to the approximating function in the Prime Number Theorem, and in some sense model the distribution of primes in dynamical terms, i.e. relate the prime number series (as a binary string) to the orbits of the Manneville map T2. Certain refinements of this are then explored.

"We remark that this approach to study prime numbers is similar to the probabilistic approach introduced by Cramér...that is we assume that the [binary] string [generated by the sequence of primes]...is one of a family of strings on which there is a probability measure..."

J.A.C. Gallas, "On the origin of periodicity in dynamical systems", Physica A: Statistical Mechanics and its Applications 283 (2000) 17-23

[abstract:] "We prove a theorem establishing a direct link between macroscopically observed periodic motions and certain subsets of intrinsically discrete orbits which are selected naturally by the dynamics from the skeleton of unstable periodic orbits (UPOs) underlying classical and quantum dynamics. As a simple illustration, an infinite set of UPOs of the quadratic (logistic) map is used to build ab initio the familiar trigonometric and hyperbolic functions and to show that they are just the first members of an infinite hierarchy of functions supported by the UPOs. Although all microscopic periodicities of the skeleton involve integer (discrete) periods only, the macroscopic functions resulting from them have real (non-discrete) periods proportional to very complicate non-integer numbers, e.g. $2\pi$ and $2\pi i$, where i =(-1)1/2."

P. Cvitanovic, "Circle Maps: Irrationally Winding" from Number Theory and Physics, eds. C. Itzykson, et. al. (Springer, 1992)

"We shall start by briefly summarizing the results of the "local" renormalization theory for transitions from quasiperiodicity to chaos. In experimental tests of this theory one adjusts the external frequency to make the frequency ratio as far as possible from being mode-locked. this is most readily attained by tuning the ratio to the 'golden mean' (51/2 - 1)/2. The choice of the golden mean is dictated by number theory: the golden mean is the irrational number for which it is hardest to give good rational approximants. As experimental measurements have limited accuracy, physicists usually do not expect that such number-theoretic subtleties as how irrational a number is should be of any physical interest. However, in the dynamical systems theory to chaos the starting point is the enumeration of asymptotic motions of a dynamical system, and through this enumeration number theory enters and comes to play a central role.

Number theory comes in full strength in the 'global' theory of circle maps, the study of universal properties of the entire irrational winding set - the main topic of these lectures. We shall concentrate here on the example of a global property of the irrational winding set discovered by Jensen, Bak and Bohr: the set of irrational windings for critical circle maps with cubic inflection has the Hausdoff dimension DH = 0.870..., and the numerical work indicates that this dimension is universal. The universality (or even existence) of this dimension has not yet been rigorously extablished. We shall offer here a rather pretty explanation of this universality (or even existence) of this dimension has not yet been rigorously established. We shall offer here a rather pretty explanation of this universality in form of the explicit formula (39) which expresses this Hausdorff dimension as an average over the Shenker universal scaling numbers. The renormalization theory of critical circle maps demands at present rather tedious numerical computations, and our intuition is much facilitated by approximating circle maps by number-theoretic models. The model that we shall use here to illustrate the basic concepts might at first glance appear trivial, but we find it very instructive, as much that is obscured by numerical work required by the critical maps is here readily number-theoretically accessible. Indicative of the depth of mathematics lurking behind physicists' conjectures is that fact that the properties that one would like to establish about the renormalization theory of critical circle maps might turn out to be related to number-theoretic abysses such as the Riemann conjecture, already in the context of the 'trivial' models."

D.H. Lenz, "Substitution dynamical systems: characterization of linear repetitivity and applications" (preprint 02/03)

"We consider dynamical systems arising from substitutions over a finite alphabet. We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive substitutions to minimal substitutions. This includes applications to random Schrödinger operators and to number theory."

G. Everest, Y. Puri and T. Ward, "Integer sequences counting periodic points"

[Abstract:] "An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This is applied to study linear recurrence sequences which count periodic points. Instances where the p-parts of an integer sequence themselves count periodic points are studied. The Mersenne sequence provides one example, and the denominators of the Bernoulli numbers provide another. The methods give a dynamical interpretation of many classical congruences such as Euler-Fermat for matrices, and suggest the same for the classical Kummer congruences satisfied by the Bernoulli numbers."

F. Beukers, J.A. Sanders and J.P. Wang, "On integrability of systems of evolution equations", Journal of Differential Equations 172 (2001) 396-408

[abstract:] "We prove the conjecture...that almost all systems [a particular] family...have at most finitely many symmetries by using number theory. We list the nine exceptional cases when the systems do have infinitively many symmetries. For such systems, we give the recursive operators to generate their symmetries. We treat both the commutative and the noncommutative (or quantum) cases. This is the first example of a class of equations where such a classification has been possible."

I. Percival and F. Vivaldi, "Arithmetical properties of strongly chaotic motions", Physica D 25 (1987) 105-130.

"The orbits of the generalized Arnold-Sinai cat maps, or hyperbolic automorphisms of the two-dimensional torus, typify purely chaotic, Anosov motion. We transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers, classify all periodic orbits, and show how to determine their periods and initial conditions. The methods are based on ideal theory in quadratic fields, which is reviewed. It is shown that the structure of orbits rests upon some basic arithmetical notions, such as unique factorization into prime ideals."

"We study the value distribution and extreme values of eigenfunctions for the "quantized cat map". This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map - a commutative group of unitary operators which commute with the map, which we called "Hecke operators". The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call "Hecke eigenfunctions". In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the "split primes" for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weil's theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions."

S. De Smedt and A. Khrennikov, "Dynamical systems and theory of numbers", Comment. Math. Univ. St. Pauli 46 no. 2 (1997) 117-132.

A. Khrennikov and M. Nilsson, "On the number of cycles of p-adic dynamical systems", Journal of Number Theory 90 (2001) 255-264

[abstract:] "We found the asymptotics, p, for the number of cycles for iteration of monomial functions in the fields of p-adic numbers. This asymptotics is closely connected with classical results on the distribution of prime numbers."

D. Dubischar, V.M. Gundlach, O. Steinkamp, and A. Khrennikov, "Attractors of random dynamical systems over p-adic numbers and a model of noisy cognitive processes", Physica D 130 (1999) 1-12

A. Khrennikov, "p-adic dynamical systems: description of concurrent struggle in biological population with limited growth", Dokl. Akad. Nauk 361 no. 6 (1998) 752-754.

S. De Smedt, A. Khrennikov, "A p-adic behaviour of dynamical systems", Rev. Mat. Comput. 12 (1999) 301-323

S. Matsutani, "p-adic difference-difference Lotka-Volterra equation and ultra-discrete limit", Int. J. Math. and Math. Sci. 27 (2001) 251-260

[abstract:] "We study the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We point out that the structure of the space given by taking the ultra-discrete limit is the same as that of the p-adic valuation space. Since ultra-discrete limit can be regarded as a classical limit of a quantum object, it implies that a correspondence between classical and quantum objects might be associated with valuation theory."

S. Matsutani, "Lotka-Volterra equation over a finite ring $\mathbb{Z}/p^N \mathbb{Z}$", J. Phys. A 34 (2001) 10737-10744

[abstract:] "The discrete Lotka-Volterra equation over $p$-adic space was constructed since $p$-adic space is a prototype of spaces with non-Archimedean valuations and the space given by taking the ultra-discrete limit studied in soliton theory should be regarded as a space with the non-Archimedean valuations given in my previous paper (Matsutani, S 2001 Int. J. Math. Math. Sci.). In this paper, using the natural projection from a $p$-adic integer to a ring $\mathbb{Z}/p^N \mathbb{Z}$, a soliton equation is defined over the ring. Numerical computations show that it behaves regularly."

P.E. Cartier, B. Julia, P. Moussa and P. Vanhove (eds.), Frontiers in Number Theory, Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, due March 2006)

[publisher's description:] "This book presents pedagogical contributions on selected topics relating Number Theory, Theoretical Physics and Geometry. The parts are composed of long self-contained pedagogical lectures followed by shorter contributions on specific subjects organized by theme. Most courses and short contributions go up to the recent developments in the fields; some of them follow their author's original viewpoints. There are contributions on Random Matrix Theory, Quantum Chaos, Non-commutative Geometry, Zeta functions, and Dynamical Systems. The chapters of this book are extended versions of lectures given at a meeting entitled Number Theory, Physics and Geometry, held at Les Houches in March 2003, which gathered mathematicians and physicists."

[abstract:] "We describe the trajectories of the successive iterates of the square map and its perturbations on the field of p-adic numbers. We show that the cycles of the square map on Qp arise from cycles of the square map on Fp, and that all nonperiodic trajectories in the unit disk densely define a compact open subset. We find that the perturbed maps x |-> x2 + a, with a inside the unit circle, have similar dynamics to x |-> x2, but that each fundamental cycle arising from Fp can further admit harmonic cycles, for different choices of p and a. In contrast, the cycles of the maps x |-> x2 + a, with a on the boundary of the unit circle, are no longer tied to those of the square map itself. In all cases we give a refined algorithm for computing the finitely many periodic points of the map."

R. Benedetto, "p-Adic dynamics and Sullivan's no wandering domain theorem", Compositio Mathematica 122 (2000) 281-298

[abstract:] "In this paper we study dynamics on the Fatou set of a rational function f(z) defined over a finite extension Qp, the field of p-adic rationals. Using a notion of 'components' of the Fatou set defined in "Hyperbolic maps in p-adic dynamics", we state and prove an analogue of Sullivan's No Wandering Domains Theorem for p-adic rational functions which have no wild recurrent Julia critical points."

R. Benedetto, "Examples of wandering domains in p-adic polynomial dynamics" (Comptes Rendus Mathématique. Académie des Sciences. Paris , 335 (2002), 615--620.

[abstract:] "For any prime p > 0, we construct p-adic polynomial functions in Cp[z] whose Fatou sets have wandering domains."

R. Benedetto, "Non-archimedean holomorphic maps and the Ahlfors Islands Theorem" (American Journal of Mathematics, accepted.) [DVI format]

[abstract:] "We present a p-adic and non-archimedean version of some classical complex holomorphic function theory. Our main result is an analogue of the Five Islands Theorem from Ahlfors' theory of covering surfaces. For non-archimedean holomorphic maps, our theorem requires only two islands, with explicit and nearly sharp constants, as opposed to the three islands without explicit constants in the complex holomorphic theory. We also present non-archimedean analogues of other results from the complex theory, including theorems of Koebe, Bloch, and Landau, with sharp constants."

R. Benedetto, "Components and periodic points in non-archimedean dynamics" Proceedings of the London Mathematical Society 84 no. 3 (2002) 231-256

[abstract:] "We expand the notion of non-archimedean connected components introduced in "Hyperbolic maps in p-adic dynamics". We define two types of components and discuss their uses and applications in the study of dynamics of a rational function f in K(z) defined over a non-archimedean field K. Using this theory, we derive several results on the geometry of such components and the existence of periodic points within them. Furthermore, we demonstrate that for appropriate fields of definition, the conjectures stated in "p-Adic dynamics and Sullivan's No Wandering Domains Theorem", including the No Wandering Domains conjecture, are equivalent regardless of which definition of 'component' is used. We also give a number of examples of p-adic maps with interesting or pathological dynamics."

R. Bendetto, "Hyperbolic maps in p-adic dynamics", Ergodic Theory and Dynamical Systems 21 (2001) 1-11

[abstract:] "In this paper we study the dynamics of a rational function f(z) defined over a finite extension Qp, the field of p-adic rationals. After proving some basic results, we define a notion of 'components' of the Fatou set, analogous to the topological components of a complex Fatou set. We define hyperbolic p-adic maps and, in our main theorem, characterize hyperbolicity by the location of the critical set. We use this theorem and our notion of components to state and prove an analogue of Sullivan's No Wandering Domains Theorem for hyperbolic maps."

[abstract:] "Explicit formulas are obtained for a family of continuous mappings of p-adic numbers $\Qp$ and solenoids $\Tp$ into the complex plane $\sC$ and the space \~$\Rs ^{3}$, respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpinski triangle are images of the unit balls in $\Qn{2}$ and $\Qn{3}$. In each of the families, the subset of the embeddings is found. For these embeddings, the Hausdorff dimensions are calculated and it is shown that the fractal measure on the image of $\Qp$ coincides with the Haar measure on $\Qp$. It is proved that under certain conditions, the image of the p-adic solenoid is an invariant set of fractional dimension for a dynamic system. Computer drawings of some fractal images are presented."

[abstract:] "Fractal measures of images of continuous maps from the set of p-adic numbers Qp into complex plane C are analyzed. Examples of 'anomalous' fractals, i.e. the sets where the D-dimensional Hausdorff measures (HM) are trivial, i.e. either zero, or sigma-infinite (D is the Hausdorff dimension (HD) of this set) are presented. Using the Caratheodory construction, the generalized scale-covariant HM (GHM) being non-trivial on such fractals are constructed. In particular, we present an example of 0-fractal, the continuum with HD=0 and nontrivial GHM invariant w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains in Rn, the formula for the change of variables for GHM is obtained. The family of continuous maps Qp in C continuously dependent on "complex dimension" d in C is obtained. This family is such that: 1) if d = 2(1), then the image of b>Qp is C (real axis in C.); 2) the fractal measures coincide with the images of the Haar measure in Qp, and at d = 2(1) they also coincide with the flat (linear) Lebesgue measure; 3) integrals of entire functions over the fractal measures of images for any compact set in Qp are holomorphic in d, similarly to the dimensional regularization method in QFT."

[abstract:] "The paper describes ergodic (with respect to the Haar measure) functions in the class of all functions, which are defined on (and take values in) the ring of p-adic integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1. Equiprobable (in particular, measure-preserving) functions of this class are described also. In some cases (and especially for p = 2) the descriptions are given by explicit formulae. Some of the results may be viewed as descriptions of ergodic isometric dynamical systems on p-adic unit disk."

E. Thiran, D. Verstegen and J. Weyers, "p-adic dynamics", Journal of Statistical Physics 54 nos. 3-4 (1989) 893-913

D. Verstegen, "p-adic dynamical systems" from Number Theory and Physics (J.-M. Luck, P. Moussa and M. Waldschmidt, eds.), Springer Proceedings in Physics 47 (Springer, 1990) 235-242

L. Hsia, "A weak Néron model with applications to p-adic dynamical systems", Composito Math. 100 (1996) 277-304

Hua-Chieh Li, "p-adic periodic points and Sen's theorem", J. Number Theory 56 no. 2 (1996) 309-318

J. Lubin, "Nonarchimedean dynamical systems", Compositio Math. 94 no,. 3 (1994) 321-346

J. Lubin, "Formal flows on the nonarchimedean open unit disk", Compositio Math. 124 (2000) 123-136

S. Ben-Menahem, "p-Adic iterations", preprint, Tel-Aviv UP (1988) 1627-88

Y. Bugeaud (ed.), Dynamical Systems and Diophantine Approximation (Société Mathématique de France, 2011)

[publisher's description:] "On June 7–9, 2004, a conference on Dynamical Systems and Diophantine Approximation was held at the Institut Henri Poincaré. One of the aims of this conference was to give a survey of research tools at the interface between these two domains. The editors' goal was also to highlight methods and open questions. The proceedings the editors are presenting in this volume reflect the spirit of this conference. The reader will find surveys and articles on the convergence or divergence points between dynamical systems and Diophantine approximation. All the papers are accessible to a wide audience."

Summer School on Dynamical Systems and Number Theory, Graz, July 9-13, 2007

This summer school is organised as a part of the National Research Network "Analytic Combinatorics and Probabilistic Number Theory" supported by the Austrian Science Foundation.

The purpose of the summer school is to introduce and enlighten the powerful interplay between dynamical systems and number theory. The four courses focus on different recent research developments in that direction. The summer school is therefore designed for PhD students and young Post-Docs with some background in ergodic theory and number theory.

Ergodic Prime Number Theory blog, Beijing (largely in Chinese)

This document is based on a talk given at a MAA meeting in 1999 when the author was a student at Bowie State University (Maryland, USA) under the supervision of N. Petulante.

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