# Continuous dynamical systems and analytic number theory: connections

Are there any connections between continuous dynamical systems and (analytic) number theory?

• Continued fractions encode the geodesic flow on the space $PSL_{2}(\mathbb{R})/PSL_{2}(\mathbb{Z})$, so yes.
– Asaf
Commented Oct 2, 2023 at 12:21
• It seems some of the answers given are discrete-time, not continuous. Commented Oct 2, 2023 at 22:36
• @Jesse Elliott. There's some ambiguity with regard to the modifier "continuous". A discrete-time dynamical system f^(n) might be regarded as "continuous" if the map f is a continuous self-map of a topological space. Commented Oct 3, 2023 at 19:29
• To further Asaf's comment: See work of Einsiedler, Katok and Lindenstrauss on Littlewood's conjecture. en.wikipedia.org/wiki/Littlewood_conjecture Commented Oct 3, 2023 at 19:34

There is a whole area which studies such connections, e.g., revolving around the Sarnak Conjecture: a continuous discrete-time dynamical system $$(X,T)$$ of zero topological entropy, where $$X$$ is a compact metric space and $$T$$ is a homeomorphism, must be Möbius disjoint, i.e., orthogonal to the Möbius function from number theory. This means $$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n\le N}f(T^nx) \mu(n)=0$$ for all $$x\in X$$ and all real continuous functions $$f$$ on $$X$$.

For a review, as of 2017, see "Sarnak's Conjecture -- what's new".

As usual in those matters, this is more a long comment than a full comprehensive answer.
I am not aware of works applying directly the theory of dynamical systems to the analytic theory of numbers. Nevertheless I know that the Siegel-Shidlovskii theory of transcendental numbers relies on the theory of homogeneous linear ordinary differential equations of polynomial, or more generally rational, coefficients which is undoubtedly a chapter of this theory.
It seems that the roots of this theory lie in an ancient observation by A. M. Legendre (according to [1], chapter 1, §4, p. 7 and [2], §4, p. 39): by considering the power series $$f_\alpha(x)=\sum_{n=0}^\infty \frac{x^n}{n! \alpha(\alpha+1)\cdots(\alpha+n-1)},\quad \alpha\neq 0, -1, -2, \ldots,$$ which is an entire solution of the following linear ODE with polynomial coefficient $$x f^{\prime\prime}(x)+\alpha f^{\prime}(x)= f(x),$$ Legendre noted that the ratio $$f_\alpha(x)/ f^{\prime}_\alpha(x)$$ is irrational for any rational $$x\neq 0$$ and $$\alpha$$ satisfying the above requirements. Later Erik Stridsberg proved the irrationality of the dividend $$f_\alpha(x)$$ and of the divisor $$f^{\prime}_\alpha(x)$$ for the same values of $$x$$ and $$\alpha$$, but decisive step in this approach was done by Carl Ludwig Siegel in 1949.
Siegel defined the class of $$E$$-functions by using the following linear system of ODEs with rational coefficients $$\frac{\mathrm{d}}{\mathrm{d} x} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix} = \begin{pmatrix} Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\ Q_{2,1}(x) & Q_{2,2}(x) & \ldots &Q_{2,m}(x) \\ \vdots & \vdots & \ddots & \vdots \\ Q_{m,1}(x) & Q_{m,2}(x) & \ldots &Q_{m,m}(x) \end{pmatrix} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix},$$ and was able to prove, assuming suitable hypotheses on the coefficients and on its solutions $$\big(f_1(x), \ldots, f_m(x)\big)$$ are verified, that all the $$m$$ numbers $$f_1(\alpha), \ldots, f_m(\alpha)$$ where $$\alpha$$ is and algebraic number which is not $$0$$ nor it is a pole of the functions $$\{Q_{i,j}\}_{1\le i,j\le m}$$ are algebraically independent. Later on Andrei Borisovich Shidlovskii, was able to simplify in a meaningful way the hypothesis under which the results of Siegel holds, and thus develop further Siegel's original approach. The two references cited below should be more than sufficient to give a sketch of the theory.

References

[1] Andrei Borisovich Shidlovskii, Transcendental numbers. With a foreword by W. Dale Brownawell. Translated from the Russian by Neal Koblitz. (English) De Gruyter Studies in Mathematics, 12. Berlin-New York: Walter de Gruyter, pp. xix+466 (1989), ISBN:3-11-011568-9, MR1033015, Zbl 0689.10043.

[2] Naum Il’ich Fel’dman and Andrei Borisovich Shidlovskii, "The development and present state of the theory of transcendental numbers" (English. Russian original) Russian Mathematical Surveys 22, No. 3, 1-79 (1967); translation from Uspekhi Matematicheskikh Nauk [N. S.] 22, No. 3(135), 3-81 (1967), MR214551, Zbl 0178.04801.

• Very interesting!
– M.G.
Commented Oct 2, 2023 at 13:46

A number of people including my collaborator, Máté Wierdl, have worked extensively on subsequence ergodic theorems. The idea is that one has an integer sequence $$t_1 and one forms the subsequence ergodic averages $$\frac 1N \Big(f(T^{t_1}x)+\ldots+f(T^{t_n}x)\Big).$$

If the transformation $$T$$ is an ergodic measure preserving transformation of a probability space $$(X,\mu)$$ and $$(t_n)$$ is just the sequence of natural numbers, the Birkhoff ergodic theorem shows that for $$L^1$$ functions $$f$$, the above averages converge, almost surely, to the integral of $$f$$.

A result of Bourgain established convergence for the sequence of squares for $$f\in L^2$$. Wierdl showed convergence for $$f\in L^p$$ ($$p>1$$) for the sequence of primes. A paper of Boshernitzan, Kolesnik, Wierdl and myself showed that if $$t_n=\lfloor g(n)\rfloor$$ where $$g(n)$$ grows at a roughly polynomial rate, but is sufficiently far from rational polynomials, then again there is convergence for $$f\in L^p$$ ($$p>1$$).

All of these results rely critically on controlling exponential sums, and therefore analytic number theory.

See Cellarosi and Marklof's paper https://arxiv.org/abs/1501.07661

They are interested in a limit theorem of the expression

$$X_N(t):=\frac{1}{\sqrt N}\sum_{n=1}^{\lfloor Nt\rfloor }e\left(\left(\frac12n^2+\beta n\right)x+\alpha n\right)$$ where $$x$$ is random and $$(\alpha, \beta)\in \mathbb R^2\setminus \mathbb Q^2$$. They show that independent of how $$x$$ is distributed, $$X_N$$ converges to $$\sqrt{t}\Theta_\chi(g\Phi^{2 \log t})$$ where $$\Phi$$ is the geodesic flow on $$\widetilde {SL(2,\mathbb R)}\ltimes \mathbb H$$ and $$g$$ is drawn wrt Haar measure.

• Variable $t$ on the left, $x$ on the right side of the equation? And where does $\chi$ come from? Commented Oct 3, 2023 at 1:08
• @GerryMyerson Fixed the first one and $\Theta_\chi$ is a complicated automorphic function. Commented Oct 3, 2023 at 1:14