# Applications of number theory in dynamical systems

I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics. While there may be overlap with arithmetic dynamics (see, for instance Current Trends and Open Problems in Arithmetic Dynamics by Benedetto, DeMarco, Ingram, Jones, Manes, Silverman & Tucker), I would like examples leaning more towards traditional dynamical systems, in other words differential equations or discrete dynamical systems over the reals or complex numbers.

Note that Lagarias writes in The Unreasonable Effectiveness of Number Theory, Proceedings of Symposia in Applied Mathematics, Volume 46, 1992, American Mathematical Society in the chapter Number Theory and Dynamical Systems:

Number theoretic problems have occurred repeatedly in dynamical systems. This initially seems surprising, since number theory deals with discrete objects.

A citation search using this reference was unrewarding. Other nonrecent papers which might yield a successful citation search would be welcome. Also, useful search terms would be appreciated.

• KAM theory must be a major part of an answer here. These are questions about perturbation of Hamiltonian systems and persistence of periodic orbits that have nothing on the face of it to do with number theory. It emerges that Diophantine approximation of rotation numbers plays a key role in the theory. Aug 26, 2019 at 18:14
• You could also look at subsequence ergodic theorems: you make measurements of an observable defined on a measure-preserving transformation at times $n^2$ or $p_n$ (the $n$th prime) and ask about convergence similar to Birkhoff's or von Neumann's ergodic theorems. This is maybe weaker as an example because one is starting here with a number-theoretic question about dynamical systems, rather than having the number theory arise naturally. Aug 26, 2019 at 18:17
• The dynamical degree of a rational map $f:\mathbb C\mathbb P^N\to\mathbb C\mathbb P^N$ is the quantity $\delta(f):=\lim_{n\to\infty} (\deg f^n)^{1/n}$. Sometimes $\log\delta(f)$ is called the algebraic entropy. It had been conjectured by Bellon and Vialet that $\delta(f)$ is always an algebraic number. A recent paper: A transcendental dynamical degree, Jason P. Bell, Jeffrey Diller, Mattias Jonsson, arxiv.org/abs/1907.00675 provides a counterexample. Aug 26, 2019 at 18:30
• I just thought of another extremely nice example. Margulis; and later Parry and Pollicott used ideas from number theory (zeta functions, Tauberian theorems etc) to study the growth rate of the number of closed geodesics of m length up to $L$ in the geodesic flow on a compact surface of negative curvature. Aug 27, 2019 at 0:33
• @AnthonyQuas - Following up to note that Artin-Mazur zeta functions are used in a similar vein in symbolic dynamics. Of course, there is a close relationship here with Anosov flows. Sep 5, 2019 at 13:24

From Wikipedia: "Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers."

First see examples in the MO-Q "Where is number theory used in the rest of mathematics?"

I commented there, "Goldman in his book The Queen of Mathematics alludes to an article by Weinberg where the partition function of number theory is related to the states of a vibrating string."

Edit Jan, 10, 2021: (Start)

Thanks to a comment by Terry Tao on the adeles and the functional symmetry equation for the Riemann zeta function, I found these interesting series of papers

1. "Linear Fractional p-Adic and Adelic Dynamical Systems" by Dragovich, Khrennikov, and Mihajlovic

2. "Adeles in mathematical physics" by Dragovich, in which the author states "It is remarkable that such a simple physical system as a harmonic oscillator is related to so significant a mathematical object as the Riemann zeta function."

This is only surprising if one focuses only on the standard number theorist’s narrative on the algebra and calculus and does not note that, through the Mellin transform, the e.g.f. of the Bernoulli polynomials underpin the Riemann, Hurwitz, and Lerch zeta functions and that of the Hermite numbers underlie the corresponding Landau-Riemann xi function via the associated Jacobi theta functions. Both the Bernoulli and the (appropriate) Hermite polynomials are Appell sequences with lowering/destruction/annihilation and raising/creation ops, or ladder ops. A family of Hermite polynomials pops up in eigenfunctions for quantum harmonic oscillators and therefore in QFT; as early as Schwinger, the Hurwitz zeta function was related to a particular model in QED; and later zeta function regularization was introduced into physics by Hawking. This is not surprising since the Sheffer Appell polynomial formalism parallels that of symmetric polynomials/functions with associated trace formulas (eigenvalue/zero/pole sums), volumes associated to determinants, log/exp mappings (Newton identities), etc. (Langlands-program, GUE and random matrix integration, and all that stuff). What would be surprising is if there were no connections.

1. Adelic harmonic oscillator” by Dragovich, in which he states, “The Mellin transform of a simplest vacuum state leads to the well known functional relation for the Riemann zeta function. “

2. From p-adic to zeta strings” by Dragovich, in which he “briefly discusses some properties of ... Lagrangians, related potentials, equations of motion, mass spectra and possible connections with ordinary strings. This is a review of published papers with some new views.”

3. p-Adic Mathematical Physics: The First 50 Years” by Dragovich, Khrennikov, Kozyrev, Volovich, and Zelenov--a slightly earlier survey article to the previous one, which relates p-adics to diverse fields of research.

An article with a similar flavor is “A Correspondence Principle” by. Hughe and Ninham, where parallels are drawn among the Poisson summation formula, Dirac combs, and functional symmetry equations, and these are associated with the properties of solutions of various differential equations of mathematical physics.

(End)

The Dedekind eta function enters into the dynamics of modular flows and the Lorenz equations through knot theory. (It also has applications in statistical mechanics and string theory.) See refs to Ghys work on knots and dynamics in MO-Q "The Dedekind eta function in physics" and answers to the duplicate question on PhysicsOverflow.

With a more combinatorial flavor:

Solutions to the inviscid Burgers' the KdV, and the KP equations of hydrodynamics and to the general evolution equations for flow fields generated by tangent vectors are related to classic integer arrays. (Not so surprising since the iterated operators $$(x^{m+1}d/dx)^n$$ are related to classic integer arrays and combinatorics.)

The integers relate to solutions of the Burgers' equation through the combinatorics of the associahedra and its relations to compositional inversion through OEIS A133437 as sketched in the answer to MO-Q "Why is there a connection between enumerative geometry and nonlinear waves?"

A bivariate e.g.f. for the Eulerian numbers (A008292/A123125) with its associated quadratic ($$sl_2$$) infinigen provides a soliton solution of the 1-D KdV equation. (The Eulerians are rife with ($$A_n$$ and $$B_n$$) connnections to enumerative algebraic geometry, as discussed by Hirzebruch, Losev and Manin, Batryev and Blume, Cohen, et al.)

Lauren Williams in "Enumeration of totally positive Grassmann cells" (see this recent layman's intro) develops a polynomial generating function $$A_{k,n}(q)$$ whose $$q^d$$ coefficient is the number A046802 of totally positive cells in $$G^+(k,n)$$ that have dimension $$d$$ and goes on to show that for the binomial transform $$\hat{E}_{k,n}(q)=q^{k-n}\sum^n_{i=0} (-1)^i \binom{n}{i} A_{k,n-i}(q)$$ that $$\hat{E}_{k,n(}(1)=E_{k,n}$$, the Eulerian numbers A008292, and $$\hat{E}_{k,n}(0)=N_{k,n}$$, the Narayana numbers A001263. She reiterates this in her presentation "The Positive Grassmannian (a mathematician's perspective)" and relates $$G^+$$ to soliton shallow-water-wave solutions of a KP equation, noting also the roles of $$G^+$$ in computing scattering amplitudes in string theory, a relation to free probability, and the occurrence of the Eulerians and Narayanaians in the BCFW recurrence and twistor string theory. (See links to Williams' papers in A046802.)

The refined Eulerian numbers A145271 arise in the series expansion for flow fields generated by exponentiation of tangent vectors.

The conservation equations associated to the Burgers' and KdV equations also have connections to classic integer sequences. See also "Set partitions and integrable hierarchies" by V.E. Adler.

See also some refs and comments on the relation of the cycle index polynomials for the symmetric group $$S_n$$ A036039 (refined Stirling polynomials of the first kind, related to the elementary Schur polynomials and Faber polynomials) to tau functions and integrable hierarchies (and zeta functions). The Faber polynomials A263916 are also related to integrable systems, evolution equations, and number theoretic relations.