number theory and dynamical systems
Diophantine Approximation and Dynamical Systems, La Trobe University, Melbourne, 6–8 January 2018
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 padic 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 muchdesired 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)
411457.
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) 29106
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:
D. Harari and E. Leichtnam "Extension du phenomene
de brisure spontanee de symetrie de BostConnes au cas des corps global
quelconques"
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
P. Cohen,
Dedekind zeta functions and quantum statistical mechanics
[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 BostConnes and HarariLeichtnam."
This is a recent development:
A. Connes
and M.
Marcolli, "From Physics to Number
Theory via Noncommutative Geometry. Part I: Quantum Statistical
Mechanics of Qlattices" (preprint 04/04)
[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 Qlattices 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 Lfunctions 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, "BostConnesMarcolli systems for Shimura
varieties" (preprint 03/05)
[abstract:] "We construct a Quantum Statistical Mechanical system $(A,\sigma_t)$ analogous
to the BostConnesMarcolli system...in the case of Shimura varieties. Along the way, we
define a new BostConnes 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, "Qlattices: 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 NeukirchUchida 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 normpreserving 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 BerryKeating 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 WienerKhintchine 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 RiemannWeil explicit formula
as Lefschetztype 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."
C. Deninger, "Some ideas on dynamical systems and
the Riemann zeta function"
"...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) 163186.
"In this article we describe what a cohomology theory related to zeta
and Lfunctions 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)
115.
C. Deninger, "A note on arithmetic
topology and dynamical systems"
[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."
C. Deninger, "On the nature of the
'explicit formulas' in analytic number theory  a simple example"
[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
3dimensional laminated space."
C. Deninger, "Number theory and dynamical
systems on foliated spaces"
[Abstract:] " We discuss analogies between number theory and the theory of dynamical systems
on spaces with a onecodimensional 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 ArakelovEuler characteristic and an Euler
characteristic in the sense of Connes. Finally the role of generalized solenoids as phase
spaces in our picture is explained."
C. Deninger's
homepage
partial bibliography
S. Kolyada, et al. (eds.), Dynamical Numbers: Interplay between Dynamical Systems and Number Theory (AMS, 2010)
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 twovolume 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) 3572.
"This paper describes the occurrence of numbertheoretic 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]
work by Andreas Knauf, et. al.
on number theoretical spin chains
J. Fiala and P. Kleban, "Generalized number theoretic spin
chainconnections to dynamical systems and expectation values", J. of Stat. Physics 121
(2005) 553577
[abstract:] "We generalize the number theoretic spin chain, a onedimensional
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 threeterm equation, which is useful in the study of the
Selberg zetafunction. 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) 4152.
[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 weakmixing 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) 573591.
[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 WienerIkehara 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 ergodictheoretic 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 GreenTao
Theorem on arithmetic progressions in the primes: an ergodic point of view", Bull. Amer. Math. Soc. 43
(2006), 323
[abstract:] "A longstanding 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) 198
[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 ergodictheoritical 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) 231240
[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) 555598
B. Bagchi, Recurrence in topological dynamics
and the Riemann hypothesis", Acta Mathematica Hungarica 50 (1987) 227240
V.M. Popov, "On stability properties which are equivalent to Riemann hypothesis", Libertas Math.
5 (1985) 5561
[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 differentialdelay systems is
introduced whose stability is equivalent to the Riemann hypothesis (RH). The systems
considered have a linear main part in which a zetalike term and the EulerMascheroni
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 codimensionone 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) 146
[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 nonvanishing of a certain twisted Lfunction 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, nondifferentiability of pressure,
and the critical exponent of transition", Advances in Mathematics 101
(1993) 133165
[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 nondifferentiablity, 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 noncommutative geometry" (Selecta
Mathematica (New Series) 8 no. 3 (2002) 475520.
[abstract:] "Using techniques introduced by D. Mayer, we prove an
extension of the classical GaussKuzmin 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
noncommutative modular curve, and show that the modular complex can
be seen as a sequence of K_{0}groups of the related
crossedproduct 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 noncommutative 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) 348376.
"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 GaussKuzmin operator, the
construction of certain nontrivial homology classes associated to
nonclosed geodesics on modular curves, certain Selberg zeta functions
and C* algebras related to shift invariant sets."
C. Consani and M. Marcolli, "Noncommutative geometry,
dynamics, and infinityadic 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) 779784
[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)
19451972.
[abstract:] "We construct spectral triples associated to
SchottkyMumford 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 ksplit 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 CuntzKrieger C*algebra."
Marcolli's current CV
also mentions that she has an intriguinglytitled 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
selfsimilar flows and Diophantine approximation", Contemporary
Mathematics volume 290 (AMS 2001) 113138.
"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
selfsimilar 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 Lfunctions. 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) 16631681.
[Abstract:] "We study the dynamical zeta function Z_{D}(s) related to
the periodic trajectories of the billiard flow for several disjoint strictly convex bodies in
R^{3}. We show that the analytic properties of Z_{D}(s)
close to the line of absolute convergence Re[s] = s_{0} 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 Lfunctions" (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)
253298
[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) 663668
[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. GarciaPerez, 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 nonoverlapping 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 parameterfree nonMarkovian 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) 107118
[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 T_{z} 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 T_{z} [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 T_{2}. 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) 1723
[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
(nondiscrete) periods proportional to very complicate noninteger 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
modelocked. this is most readily attained by tuning the ratio to
the 'golden mean' (5^{1/2}  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 numbertheoretic 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 D_{H} = 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 numbertheoretic
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 numbertheoretically 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 numbertheoretic 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
nonnegative 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
pparts 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
EulerFermat 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) 396408
[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) 105130.
"The orbits of the generalized ArnoldSinai cat maps, or hyperbolic automorphisms of the
twodimensional 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."
Z. Rudnick,
"Value distribution for eigenfunctions of desymmetrized quantum maps"
"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 semicircle 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) 117132.
A. Khrennikov and M. Nilsson, "On
the number of cycles of padic dynamical systems",
Journal of Number Theory 90 (2001) 255264
[abstract:] "We found the asymptotics, p, for the number of
cycles for iteration of monomial functions in the fields of
padic 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 padic numbers and a model of noisy cognitive
processes", Physica D 130 (1999) 112
A. Khrennikov, "padic dynamical systems: description of concurrent
struggle in biological population with limited growth", Dokl. Akad.
Nauk 361 no. 6 (1998) 752754.
S. De Smedt, A. Khrennikov, "A padic behaviour of dynamical systems",
Rev. Mat. Comput. 12 (1999) 301323
S. Matsutani, "padic differencedifference LotkaVolterra equation
and ultradiscrete limit", Int. J. Math. and Math. Sci. 27 (2001) 251260
[abstract:] "We study the differencedifference LotkaVolterra equations in padic number space and its
padic valuation version. We point out that the structure of the space given by taking the ultradiscrete limit is
the same as that of the padic valuation space. Since ultradiscrete 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, "LotkaVolterra equation over a finite ring $\mathbb{Z}/p^N
\mathbb{Z}$", J. Phys. A 34 (2001) 1073710744
[abstract:] "The discrete LotkaVolterra equation over $p$adic space was constructed since $p$adic space is a
prototype of spaces with nonArchimedean valuations and the space given by taking the ultradiscrete limit studied in
soliton theory should be regarded as a space with the nonArchimedean 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 selfcontained 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,
Noncommutative 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."
M. Nevins and
D. Rogers, "Quadratic maps as dynamical systems on
the padic numbers"
[abstract:] "We describe the trajectories of the successive iterates of the square map
and its perturbations on the field of padic numbers. We show that the cycles of the
square map on Q_{p} arise from cycles of the square map on
F_{p}, and that all nonperiodic trajectories in the unit disk densely
define a compact open subset. We find that the perturbed maps x > x^{2} + a, with a inside the unit circle, have similar dynamics to x > x^{2}, but that each fundamental cycle arising from F_{p} can further admit harmonic cycles, for different choices of p and a. In contrast, the cycles of the maps x > x^{2} + 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,
"pAdic dynamics and Sullivan's no wandering domain theorem",
Compositio Mathematica 122 (2000) 281298
[abstract:] "In this paper we study dynamics on the Fatou set of a rational function
f(z) defined over a finite extension Q_{p}, the field of
padic rationals. Using a notion of 'components' of the Fatou set defined in "Hyperbolic
maps in padic dynamics", we state and prove an analogue of Sullivan's No Wandering Domains Theorem for
padic rational functions which have no wild recurrent Julia critical points."
R. Benedetto, "Examples of wandering domains in padic polynomial dynamics"
(Comptes Rendus Mathématique. Académie des Sciences. Paris , 335 (2002), 615620.
[abstract:] "For any prime p > 0, we construct padic polynomial
functions in C_{p}[z] whose Fatou sets have wandering domains."
R. Benedetto, "Nonarchimedean holomorphic maps
and the Ahlfors Islands Theorem" (American Journal of Mathematics, accepted.) [DVI format]
[abstract:] "We present a padic and nonarchimedean 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 nonarchimedean 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 nonarchimedean 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 nonarchimedean dynamics"
Proceedings of the London Mathematical Society 84 no. 3 (2002) 231256
[abstract:] "We expand the notion of nonarchimedean connected components introduced
in "Hyperbolic maps in padic 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 nonarchimedean 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 "pAdic 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 padic maps
with interesting or pathological dynamics."
R. Bendetto, "Hyperbolic maps in padic dynamics",
Ergodic Theory and Dynamical Systems 21 (2001) 111
[abstract:] "In this paper we study the dynamics of a rational function f(z)
defined over a finite extension Q_{p}, the field of padic
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
padic 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."
D. Chistyakov, "Fractal
geometry for images of continuous map of padic numbers
and padic solenoids into Euclidean spaces"
[abstract:] "Explicit formulas are obtained for a family of continuous mappings of
padic 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 padic solenoid is an invariant set of fractional
dimension for a dynamic system. Computer drawings of some fractal images are presented."
D. Chistyakov, "Fractal
measures, padic numbers and continuous transition between dimensions"
[abstract:] "Fractal measures of images of continuous maps from the set of padic
numbers Q_{p} into complex plane C are analyzed. Examples of
'anomalous' fractals, i.e. the sets where the Ddimensional Hausdorff measures (HM)
are trivial, i.e. either zero, or sigmainfinite (D is the Hausdorff dimension (HD)
of this set) are presented. Using the Caratheodory construction, the generalized
scalecovariant HM (GHM) being nontrivial on such fractals are constructed. In particular,
we present an example of 0fractal, the continuum with HD=0 and nontrivial GHM invariant
w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains
in R^{n}, the formula for the change of variables
for GHM is obtained. The family of continuous maps Q_{p} 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>Q_{p} is C
(real axis in C.); 2) the fractal measures coincide with the images of the
Haar measure in Q_{p}, 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 Q_{p} are holomorphic in d,
similarly to the dimensional regularization method in QFT."
V. Anashin, "Uniformly
distributed sequences of padic integers, II"
[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 padic
integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1.
Equiprobable (in particular, measurepreserving) 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 padic unit disk."
E. Thiran, D. Verstegen and J. Weyers, "padic dynamics", Journal of Statistical
Physics 54 nos. 34 (1989) 893913
D. Verstegen, "padic dynamical systems" from Number
Theory and Physics (J.M. Luck, P. Moussa and M. Waldschmidt, eds.), Springer
Proceedings in Physics 47 (Springer, 1990) 235242
L. Hsia, "A
weak Néron model with applications to padic dynamical systems",
Composito Math. 100 (1996) 277304
HuaChieh Li, "padic periodic points and Sen's theorem", J. Number Theory
56 no. 2 (1996) 309318
J. Lubin, "Nonarchimedean
dynamical systems", Compositio Math. 94 no,. 3 (1994) 321346
J. Lubin, "Formal flows on the nonarchimedean open unit disk", Compositio Math.
124 (2000) 123136
S. BenMenahem, "pAdic iterations", preprint, TelAviv UP (1988) 162788
C. Chicchiero,
Notes on symbolic dynamics, entropy, and prime
numbers
Gamba,
Hernando and Romanelli's
calculation of Liapunov exponents for the distribution
of primes
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 913, 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
PostDocs with some background in ergodic theory and number theory.
Ergodic Prime Number Theory blog, Beijing (largely
in Chinese)
N. Guffey and N. Petulante,
"The Newtonian orbits of the Riemann zeta
function: a step towards a proof of the Riemann Hypothesis"
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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