Markov Constant and Quantum Instabilities
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Growth and Integrability in Multi-Valued Dynamics
Growth and Integrability in Multi-Valued Dynamics Kathryn Spalding A Doctoral Thesis Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University January 2018 Certificate of Originality I certify that I am responsible for the work submitted in this thesis, that the original work is my own except as specified in acknowledgements or footnotes, and that neither the thesis nor the original work contained therein has been submitted to this or any other institution for a degree. Signed: ........................................................................ Date: ........................................................................ 1 Abstract This thesis is focused on the problem of growth and integrability in multi-valued dynamics generated by SL2(Z) actions. An important example is given by Markov dynamics on the cubic surface x2 + y2 + z2 = 3xyz; generating all the integer solutions of this celebrated Diophantine equation, known as Markov triples. To study the growth problem of Markov numbers we use the binary tree representation. This allows us to define the Lyapunov exponents Λ(x) as the function of the paths on this 1 tree, labelled by x 2 RP . We prove that Λ(x) is a P GL2(Z)-invariant function, which is zero almost everywhere but takes all values in [0; ln '] (where ' denotes the golden ratio). We also show that this function is monotonic, and that its restriction to the Markov-Hurwitz set of most irrational numbers is convex in the Farey parametrisation. We also study the growth problem for integer binary quadratic forms using Conway's to- pograph representation. It is proven that the corresponding Lyapunov exponent ΛQ(x) = 2Λ(x) except for the paths along the Conway river. -
A Comparison of Dispersion and Markov Constants
ACTA ARITHMETICA LXIII.3 (1993) A comparison of dispersion and Markov constants by Amitabha Tripathi (Fairmont, W.Va.) 0. Introduction. Let {xn} be a sequence of numbers, 0 ≤ xn ≤ 1. In [3], H. Niederreiter introduced a measure of denseness of such a sequence as follows: For each N ≥ 1, let dN = sup min |x − xn| 0≤x≤1 1≤n≤N and define D({xn}) = lim sup NdN . N→∞ In particular, for irrational α, the dispersion constant D(α) is defined by D({nα mod 1}). It is well known that, for irrational α, the Markov constant M(α) is defined by M(α)−1 = lim inf nknαk , n→∞ where kxk denotes the distance from x to the nearest integer. In [3], Niederreiter asks if M(α)<M(β) implies D(α)<D(β). V. Drobot [1] has shown this to be false by producing a counterexample of two quadratic irrationals, both with continued fraction expansion with period length nine. In this paper, we classify some infinite families of pairs (α, β) of irrational numbers that satisfy M(α) < M(β) and D(α) > D(β). We first outline the method of V. Drobot [1] to compute D(α) for quadratic irrationals α. If α is a real irrational number with continued fraction expansion α = [a0; a1, a2,...], let λi = [0; ai, ai−1, . , a1],Λi = [ai+1; ai+2,...],Mi = λi + Λi . We define −1 2 ψi(x) = Mi [−x + (Λi − λi − 1)x + Λi(1 + λi)] , xi = (Λi − λi − 1)/2 , ni is the integer closest to xi . -
Metric Diophantine Approximation and Dynamical Systems Dmitry Kleinbock
Metric Diophantine approximation and dynamical systems Dmitry Kleinbock Brandeis University, Waltham MA 02454-9110 E-mail address: [email protected] This is a rough draft for the Spring 2007 course taught at Brandeis. Comments/corrections/suggestions welcome. Last modified: March 29, 2007. Here is a short course description: The main theme is studying the way the set of rational numbers Q sits inside real numbers R. This seemingly simple set-up happens to lead to quite intricate problems. The word ‘metric’ comes from ‘measure’ and implies that the emphasis will be put on properties of ‘almost all’ numbers; more precisely, for a certain approximation property of real numbers we will be interested in the magnitude (for example, in terms of Lebesgue measure or Hausdorff dimension) of the set of numbers having this property. The n situation becomes even more interesting when one similarly considers Q n sitting in R ; this has been the area of several exciting recent developments, which we may talk about in the second part of the course. The introductory part will feature: rate of approximation of real num- bers by rationals; theorems of Kronecker, Dirichlet, Liouville, Borel-Cantelli, Khintchine; connections with dynamical systems: circle rotations, hyper- bolic flow in the space of lattices, geodesic flow on the modular survace, Gauss map (continued fractions). Other topics could be (choice based upon preferences of the audience): • Hausdorff measures and dimension, fractal sets and measures they support • W. Schmidt’s (α, β)-games, winning sets, badly approximable num- bers • ubiquitious systems, Jarnik-Besicovitch Theorem • inhomogeneous approximation, shrinking target properties for cir- cle rotations • multidimensional theory, Diophantine properties of measures on n R , a connection with flows on SLn+1(R)/SLn+1(Z) References: • W. -
Yang-Baxter and Braid Dynamics
Yang-Baxter and braid dynamics Alexander Veselov, Loughborough University GADUDIS, Glasgow, April 1, 2009 Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics I Diophantine analysis and Markov equation I Yang-Baxter versus braid relations I Braid group and modular group Plan I Dynamics and groups Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics I Diophantine analysis and Markov equation I Yang-Baxter versus braid relations Plan I Dynamics and groups I Braid group and modular group Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics I Diophantine analysis and Markov equation Plan I Dynamics and groups I Braid group and modular group I Yang-Baxter versus braid relations Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics Plan I Dynamics and groups I Braid group and modular group I Yang-Baxter versus braid relations I Diophantine analysis and Markov equation Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. -
Arxiv:1603.08360V5 [Math.DS] 5 May 2020 the Growth of Markov Numbers M = 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325
LYAPUNOV SPECTRUM OF MARKOV AND EUCLID TREES K. SPALDING AND A.P. VESELOV Abstract. We study the Lyapunov exponents Λ(x) for Markov dy- 1 namics as a function of path determined by x 2 RP on a binary planar tree, describing the Markov triples and their \tropical" version - Euclid triples. We show that the corresponding Lyapunov spectrum is [0; ln '], where ' is the golden ratio, and prove that on the Markov-Hurwitz set X of the most irrational numbers the corresponding function ΛX is monotonically increasing and in the Farey parametrization is convex. 1. Introduction In 1880 Andrei A. Markov, a 24-year old student from St Petersburg, discovered in his master's thesis [31] a remarkable connection between Dio- phantine analysis and the following Diophantine equation x2 + y2 + z2 = 3xyz; (1) known nowadays as the Markov equation. The solutions of this celebrated equation are known as Markov triples and can be found from the obvious one (1; 1; 1) by compositions of Vieta involutions (x; y; z) ! (x; y; 3xy − z) (2) and permutations of x; y; z: The numbers, which appear in Markov triples, are called Markov numbers, the set of which we denote by M. Their arithmetic was studied by Frobenius [17], see recent development in [5]. For more history and details we refer to the very nicely written book [1] by Aigner. arXiv:1603.08360v5 [math.DS] 5 May 2020 The growth of Markov numbers m = 1; 2; 5; 13; 29; 34; 89; 169; 194; 233; 433; 610; 985; 1325;::: on the real line was studied by Zagier [42] (see also McShane and Rivin [32]).