## Farey sequences, the Farey map and the Farey Tree also including continued fractions

Martin Huxley explains the possible relationship between Farey sequences (normally studied by number theorists) and physics:

"If you have a two-dimensional lattice model in which molecules can interact only if there is a clear line of sight between them, then the Farey sequence comes in. Make it the unit square lattice, and cut off points with x,y coordinates greater than Q. The lattice points visible from the origin have an eightfold symmetry; the basic repeating chunk is the lattice points with 0 < or = Q, 0 < y < x. For book-keeping you count the points (1,0) and (1,1) on the boundary with weight 1/2. The gradients y/x taken in increasing order form the Farey sequence of order Q.

At the Aarhus Conference on Number Theory and Spectral Theory at the beginning of december, Professor Knauf was talking about a model along these lines.

The condition that a lattice point is visible from the origin brings in the Möbius function and prime number theory. A nice way of stating the Riemann hypothesis is that the Farey sequence is distributed as uniformly in the interval 0 to 1 as it possibly can be, given that there must be a gap between the gradients 0 and 1/Q, and between 1/2 and either P/Q with P= (Q+1)/2 if Q is odd, or between 1/2 and P/(Q-1) with P = Q/2 if Q is even, and so on. This is in my 1972 lecture notes on the distribution of prime numbers (Oxford University Press). You can twist it to get the Riemann hypothesis for a Dirichlet L-function by taking different molecules in the lattice, in a repeating pattern that only depends on the x coordinate, so not a particularly physical model. Twist is a technical term for colouring the points by labelling them with directions in the plane or complex numbers from the unit circle;"

Francesco Amoroso provided the following Farey sequence bibliography:

Amoroso, F. "On the heights of a product of cyclotomic polynomials." Number theory, I (Rome, 1995). Rend. Sem. Mat. Univ. Politec. Torino 53 (1995), no. 3, 183--191.

Amoroso, F. "Algebraic numbers close to 1 and variants of Mahler's measure." J. Number Theory 60 (1996), no. 1, 80-96

Codeca, P. "Some properties of the local discrepancy of Farey sequences." Atti Accad. Sci. Istit. Bologna Cl. Sci. Fis. Rend (13) 8 (1980/81), no. 1-2, 163-173

Codeca, P. and Perelli, A. "On the uniform distribution (mod 1) of the Farey fractions and lp spaces." Math. Ann. 279 (1988), no.3, 413-422

Franel, J. "Les suites de Farey et les problemes des nombres premiers." Gottinger Nachrichten, 198-201 (1924)

Fujii, A. "A remark on the Riemann hypothesis." Comment. Math. Univ. St. Pauli 29 (1980), 195-201

Fujii, A. "Some explicit formulae in the theory of numbers. A remark on the Riemann Hypothesis." Proc. Japan Acad., Ser. A 57(1981), 326-330

Huxley, M.N. "The distribution of Farey points, I." Acta Arith. 18 (1971), 281-287

Kanemitsu, S. and Yoshimoto, M. "Farey series and the Riemann hypothesis." Acta Arith. 75 (1996), no. 4, 351-374

Kanemitsu, S. and Yoshimoto, M. "Farey series and the Riemann hypothesis. III." Ramanujan J. 1 (1997), no. 4, 363-378

Kopriva, J. "Contribution to the relation of the Farey series to the Riemann hypothesis on the zeros of the zeta function" (Czech), Casopis Pest. Mat. {\bf 78} (1953), 49-55

Kopriva, J. "Contribution to the relation of the Farey series to the Riemann hypothesis" (Czech), Casopis Pest. Mat. 79 (1954), 77-82

Landau, E. "Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel." Gottinger Nachrichten, 202-206 (1924);

Mikolas, M. "Sur l'hypothese de Riemann." C. R. Acad. Sci. Paris 228 (1949), 633-636

Mikolas, M. "Farey series and their connection with the prime number problem. I." Acta Univ. Szeged. Sect. Sci. Math.13 (1949), 93-117

Mikolas, M. "Farey series and their connection with the prime number problem. II." Acta Univ. Szeged. Sect. Sci. Math.14 (1951), 5-21

Mikolas, M. "On the asymptotic behaviour of Franel's sum and the Riemann hypothesis." Results Math 21(1992), no. 3-4, 368-378

Niederreiter, E. "The distribution of Farey points", Math. Ann. 201 (1973), 341-345

Yoshimoto, M. "Farey series and the Riemann hypothesis. II." Acta Math.
Hungar. 78 (1998), no. 4, 287-304

Zulauf, A. "The distribution of Farey Numbers." J. Reine Angew. Math.

An thorough account of the relationship between Farey sequences and the Riemann Hypothesis can be found in

H.M. Edwards, Riemann's Zeta Function (Academic Press, 1974) p.263-267

L. Vepstas, The Modular Group and Fractals - An exposition of the relationship between fractals, the Riemann zeta function, the modular group, the Farey fractions and the Minkowski Question Mark.

Ford circles provide a means to visualise Farey sequences

Keith Taylor's Farey Tree applet

The Phyllotaxis project's notes on the Farey Tree and the Golden Mean

The Farey Room (L. Vepstas's Farey map graphics)

B. Devaney, "The Mandelbrot Set and the Farey Tree", Amer. Math. Monthly 106 (1999) 289-302

S. Grynberg and M. Piacquadio, "Self-similarity of Farey staircases" (preprint 06/03)

[abstract:] "We study Cantor Staircases in physics that have the Farey-Brocot arrangement for the Q/P rational heights of stability intervals I(Q/P), and such that the length of I(Q/P) is a convex function of 1/P. Circle map staircases and the magnetization function fall in this category. We show that the fractal sets $\Omega$ underlying these staircases are connected with key sets in Number Theory via their $(\alpha, f(\alpha))$ multifractal decomposition spectra. It follows that such sets $\Omega$ are self similar when the usual (Euclidean) measure is replaced by the hyperbolic measure induced by the Farey-Brocot partition."

"In order to study Cantor staircases in physics – forced pendulums, magnetization, etc - showing the Farey-Brocot arrangement for intervals I(Q/P), a natural connection with Number Theory, precisely due to the ubiquitous presence of the Farey-Brocot partition. But when closely examining the behaviour of These staircases, we were forced to considerably refine the $J_{\beta}$ nested classes into the $G_{\beta}$ disjoint ones.

We are saying that problems in empirical Physics produced a refinement of key tools in Number theory

The properties of these $G_{\beta} allowed us to extract theoretical and practical information about the multifractal spectrum of such cantordusts$\Omega$underlying Cantor staircases in physics, and about the nature of the self-similarity of said staircases." P. Kleban, A. E. Özlük, "A Farey fraction spin chain" [abstract:] "We introduce a new number-theoretic spin chain and explore its thermodynamics and connections with number theory. The energy of each spin configuration is defined in a translation-invariant manner in terms of the Farey fractions, and is also expressed using Pauli matrices. We prove that the free energy exists and exhibits a unique phase transition at inverse temperature beta = 2. The free energy is the same as that of a related, non translation-invariant number-theoretic spin chain. Using a number-theoretic argument, the low-temperature (beta > 3) state is shown to be completely magnetized for long chains. The number of states of energy E = log(n) summed over chain length is expressed in terms of a restricted divisor problem. We conjecture that its asymptotic form is (n log n), consistent with the phase transition at beta = 2, and suggesting a possible connection with the Riemann zeta function. The spin interaction coefficients include all even many-body terms and are translation invariant. Computer results indicate that all the interaction coefficients, except the constant term, are ferromagnetic." J. Fiala, P. Kleban, A. Özlük, "The Phase Transition in Statistical Models Defined on Farey Fractions" (accepted for publication, J. Stat. Physics) [abstract:] "We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form [t (ln t)^2]^(-1). The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition." J. Fiala and P. Kleban, "Thermodynamics of the Farey Fraction Spin Chain", J. Stat. Physics 116 (2004) 1471-1490 [abstract:] "We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and renormalization group arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a "marginal" field is present) despite the unusual nature of the transition for h=0." 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." T. Prellberg, J. Fiala and P. Kleban, "Cluster approximation for the Farey fraction spin chain" (prepring 07/05) [abstract:] "We consider the Farey fraction spin chain in an external field h. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At h = 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy. For h <> 0, our results are largely consistent with those found previously using mean field theory and renormalization group arguments." M. Degli Esposti, S. Isola, A. Knauf, "Generalized Farey trees, transfer Operators and phase transitions" (preprint 06/2006) [abstract:] "We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter map is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral analysis of generalized transfer operators. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function." G.I. Watson, "Repulsive particles on a two-dimensional lattice" (preprint, 09/96) [abstract:] "The problem of finding the minimum-energy configuration of particles on a lattice, subject to a generic short-ranged repulsive interaction, is studied analytically. The study is relevant to charge ordered states of interacting fermions, as described by the spinless Falicov-Kimball model. For a range of particle density including the half-filled case, it is shown that the minimum-energy states coincide with the large-U neutral ground state ionic configurations of the Falicov-Kimball model, thus providing a characterization of the latter as 'most homogeneous' ionic arrangements. These obey hierarchical rules, leading to a sequence of phases described by the Farey tree. For lower densities, a new family of minimum-energy configurations is found, having the novel property that they are aperiodic even when the particle density is a rational number. In some cases there occurs local phase separation, resulting in an inherent sensitivity of the ground state to the detailed form of the interaction potential." [abstract:] "We prove that the Farey sequences can be express into equivalence classes labeled by a fractal parameter which looks like a Hausdorff dimension h defined within the interval 1 < h < 2. The classes h satisfy the same properties of the Farey series and for each value of h there exists an algebraic equation." [abstract:] "We consider Farey series of rational numbers in terms of fractal sets labeled by the Hausdorff dimension with values defined in the interval 1 < h < 2 and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that the universality classes of the quantum Hall transitions are classified in terms of h. The connection between Number Theory and Physics appears naturally in this context." [abstract:] "The concept of fractal index is introduced in connection with the idea of universal class$h$of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field theory (CFT)-quasiparticles taking into account the central charge$c[\nu]$and the particle-hole duality$\nu\longleftrightarrow\frac{1}{\nu}$, for integer-value$\nu$of the statistical parameter. The Hausdorff dimension$h$which labelled the universal classes of particles and the conformal anomaly are therefore related. We also establish a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension." W. da Cruz and R. de Oliveira, "Fractal index, central charge and fractons" [abstract:] "We introduce the notion of fractal index associated with the universal class$h$of particles or quasiparticles, termed fractons, which obey specific fractal statistics. A connection between fractons and conformal field theory (CFT)-quasiparticles is established taking into account the central charge$c[\nu]$and the particle-hole duality$\nu\longleftrightarrow\frac{1}{\nu}$, for integer-value$\nu$of the statistical parameter. In this way, we derive the Fermi velocity in terms of the central charge as$v\sim\frac{c[\nu]}{\nu+1}$. The Hausdorff dimension$h$which labelled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension." W. da Cruz, "Fractal sets of dual topological quantum numbers" (preprint, 06/03) [abstract:] "The universality classes of the quantum Hall transitions are considered in terms of fractal sets of dual topological quantum numbers filling factors, labelled by a fractal or Hausdorff dimension defined into the interval 1 < h < 2 and associated with fractal curves. We show that our approach to the fractional quantum Hall effect-FQHE is free of any empirical formula and this characteristic appears as a crucial insight for our understanding of the FQHE. According to our formulation, the FQHE gets a fractal structure from the connection between the filling factors and the Hausdoff dimension of the quantum paths of particles termed fractons which obey a fractal distribution function associated with a fractal von Neumann entropy. This way, the quantum Hall transitions satisfy some properties related to the Farey sequences of rational numbers and so our theoretical description of the FQHE establishes a connection between physics, fractal geometry and number theory. The FQHE as a convenient physical system for a possible prove of the Riemann hypothesis is suggested." M. Piacquadio Losada and E. Cesaratto, "Multifractal spectrum and thermodynamical formalism of the Farey tree" [abstract:] "The task of comparing the Hausdorff spectrum, the computational spectrum, and the Legendre spectrum of a fractal set endowed with a probability measure, was tackled by several authors - Cawley and Mauldin, Riedi and Mandelbrot, among others. For self-similar measures all three spectra coincide. We compare these spectra for the hyperbolic measure (inducing the Farey Tree partition), fundamentally different from the self-similar one." S. Rajesh and G. Ananthakrishna, "Incomplete approach to homoclinicity in a model with bent-slow manifold geometry" [abstract:] "The dynamics of a model, originally proposed for a type of instability in plastic flow, has been investigated in detail. The bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical behaviour, including period bubbling and period adding or Farey sequences. The complex bifurcation sequences, characterized by Mixed Mode Oscillations, exhibit partial features of Shilnikov and Gavrilov-Shilnikov scenario. Utilizing the fact that the model has disparate time scales of dynamics, we explain the origin of the relaxation oscillations using the geometrical structure of the bent-slow manifold. Based on a local analysis, we calculate the maximum number of small amplitude oscillations, s, in the periodic orbit of Ls type, for a given value of the control parameter. This further leads to a scaling relation for the small amplitude oscillations. The incomplete approach to homoclinicity is shown to be a result of the finite rate of 'softening' of the eigenvalues of the saddle focus fixed point. The latter is a consequence of the physically relevant constraint of the system which translates into the occurrence of back-to-back Hopf bifurcation." B. Barbanis, H. Varvoglis and Ch. L. Vozikis, "Imperfect fractal repellers and irregular families of periodic orbits in a 3-D model potential" [abstract:] "A model, plane symmetric, 3-D potential, which preserves some features of galactic problems,is used in order to examine the phase space structure through the study of the properties of orbits crossing perpendicularly the plane of symmetry. It is found that the lines formed by periodic orbits, belonging to Farey sequences, are not smooth neither continuous. Instead they are deformed and broken in regions characterised by high Lyapunov Characteristic Numbers (LCN's). It is suggested that these lines are an incomplete form of a fractal repeller, as discussed by Gaspard and Baras (1995), and are thus closely associated to the 'quasi-barriers' discussed by Varvoglis et al. (1997). There are numerical indications that the contour lines of constant LCN's possess fractal properties. Finally it is shown numerically that some of the periodic orbits -members of the lines- belong to true irregular families. It is argued that the fractal properties of the phase space should affect the transport of trajectories in phase or action space and,therefore, play a certain role in the chaotic motion of stars in more realistic galactic potentials." [abstract:] "It is well established that the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point. We present here for the first time a complete spectral analysis for an example of such an intermittent map, the Farey map. We give a simple proof that the transfer operator is self-adjoint on a suitably defined Hilbert space and show that its spectrum decomposes into a continuous part (the interval [0,1]) and isolated eigenvalues of finite multiplicity. Using a suitable first-return map, we present a highly efficient numerical method for the determination of all the eigenvalues, including the ones embedded in the continuous spectrum." B.P. Dolan, "Duality and the modular group in the quantum Hall effect", J. Phys. A 32 (1999) L243 [abstract:] "We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule |p1q2 - p2q1|=1 for quantum Hall transitions between filling factors nu1 = p1/q1 and nu2 = p2/q2; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus." B. Basu-Mallick, T. Bhattacharyya and D. Sen, "Novel multi-band quantum soliton states for a derivative nonlinear Schrödinger model" (preprint 07/03) [abstract:]"We show that localized N-body soliton states exist for a quantum integrable derivative nonlinear Schrödinger model for several non-overlapping ranges (called bands) of the coupling constant \eta. The number of such distinct bands is given by Euler's \phi-function which appears in the context of number theory. The ranges of \eta within each band can also be determined completely using concepts from number theory such as Farey sequences and continued fractions. We observe that N-body soliton states appearing within each band can have both positive and negative momentum. Moreover, for all bands lying in the region \eta > 0, soliton states with positive momentum have positive binding energy (called bound states), while the states with negative momentum have negative binding energy (anti-bound states)." B. Basu-Mallick, T. Bhattacharyya and D. Sen, "Multi-band structure of the quantum bound states for a generalized nonlinear Schrödinger model" (preprint 02/05) [abstract:] "By using the method of coordinate Bethe ansatz, we study N-body bound states of a generalized nonlinear Schrödinger model having two real coupling constants c and \eta. It is found that such bound states exist for all possible values of c and within several nonoverlapping ranges (called bands) of \eta. The ranges of \eta within each band can be determined completely using Farey sequences in number theory. We observe that N-body bound states appearing within each band can have both positive and negative values of the momentum and binding energy. N. Cornish and J. Levin, "The mixmaster universe: A chaotic Farey tale" [abstract:]"When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer dependent answers. Here we resolve the conflict by using observer independent, fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the 6-dimensional minisuperspace of the full Einstein equations, and in a 2-dimensional discretisation of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy and multifractal dimensions. As all of these quantities are coordinate, or gauge independent, there is no longer any ambiguity-the mixmaster universe is indeed chaotic." 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})$should be considered as non-commutative modular curve, and show that the modular complex can be seen as a sequence of K0-groups of the related crossed-product C*-algebras. This paper is an expanded version of the previous "On the distribution of continued fractions and modular symbols". The main new features are Section 4 on non-commutative geometry and the modular complex and Section 1.2.2 on the Mixmaster Universe." M. Marcolli, "Limiting modular symbols and the Lyapunov spectrum", Journal of Number Theory 98 No. 2 (2003) 348-376. [abstract:] "This paper consists of variations upon the theme of limiting modular symbols. Topics covered are: an expression of limiting modular symbols as Birkhoff averages on level sets of the Lyapunov exponent of the shift of the continued fraction, a vanishing theorem depending on the spectral properties of a generalized Gauss-Kuzmin operator, the construction of certain non-trivial homology classes associated to non-closed geodesics on modular curves, certain Selberg zeta functions and C* algebras related to shift invariant sets." H.-C. Kao, S.-C. Lee and W.-J. Tzeng, "Farey Tree and the Frenkel-Kontorova model" [abstract:]"We solved the Frenkel-Kontorova model with the potential$V(u)= -\frac{1}{2} |\lambda|(u-{\rm Int}[u]-\frac{1}{2})^2$exactly. For given$|\lambda|$, there exists a positive integer$q_c$such that for almost all values of the tensile force$\sigma$, the winding number$\omega$of the ground state configuration is a rational number in the$q_c$-th level Farey tree. For fixed$\omega=p/q$, there is a critical$\lambda_c$when a first order phase transition occurs. This phase transition can be understood as the dissociation of a large molecule into two smaller ones in a manner dictated by the Farey tree. A kind of 'commensurate-incommensurate' transition occurs at critical values of$\sigma\$ when two sizes of molecules co-exist. 'Soliton' in the usual sense does not exist but induces a transformation of one size of molecules into the other."

C. Gruber, D. Ueltschi and J. Jedrzejewski, "Molecule formation and the Farey Tree in the one-dimensional Falicov-Kimball model"

[abstract:]"The ground state configurations of the one--dimensional Falicov-Kimball model are studied exactly with numerical calculations revealing unexpected effects for small interaction strength. In neutral systems we observe molecular formation, phase separation and changes in the conducting properties; while in non--neutral systems the phase diagram exhibits Farey tree order (Aubry sequence) and a devil's staircase structure. Conjectures are presented for the boundary of the segregated domain and the general structure of the ground states."

B. Basu-Mallick, T. Bhattacharyya and D. Sen, "Clusters of bound particles in the derivative delta-function Bose gas" (preprint 10/2001)

[abstract:] "In this paper we discuss a novel procedure for constructing clusters of bound particles in the case of a quantum integrable derivative delta-function Bose gas in one dimension. It is shown that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasi-momenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. We also establish a connection between these special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles associated with the derivative delta-function Bose gas and allows us to study various properties of these clusters like their size and their stability under the variation of the coupling constant."

S. Perrine, "Recherches autour de la theorie de Markoff" (compact version of a text published as La Theorie de Markoff et ses Developements (Tessier & Ashpool, 2002))

[abstract:] "The text deals with generalizations of the Markoff equation in number theory, arising from continued fractions. It gives the method for the complete resolution of such new equations, and their interpretation in algebra and algebraic geometry. This algebraic approach is completed with an analytical development concerning fuchsian groups. The link with the Teichmuller theory for punctured toruses is completely described, giving their classification with a reduction theory. More general considerations about Riemann surfaces, geodesics and their hamiltonian study are quoted, together with applications in physics, 1/f noise and zeta function. Ideas about important conjectures are presented. Reasons why the Markoff theory appears in different geometrical contexts are given, thanks to decomposition results in the group GL(2,Z)."

[personal note from author:] "My text deals in fact with trees which are similar to the Farey tree. With my trees I build generalizations of the Markoff equation (my generalizations) which are in fact trace formulas (I explained how). These equations are known to have interpretations on some bundles (that is to say sometimes particles...). With Michel [Planat], we tried to understand the link that he discovered with his oscillators, hence with 1/f noise, the idea being that it could be of arithmetical origin."

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