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doi: 10.3934/dcds.2021084

On trigonometric skew-products over irrational circle-rotations

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA

Received  May 2020 Revised  March 2021 Published  May 2021

We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach.

Citation: Hans Koch. On trigonometric skew-products over irrational circle-rotations. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021084
References:
[1]

C. AistleitnerG. LarcherF. PillichshammerS. S. Eddin and R. F. Tichy, On Weyl products and uniform distribution modulo one, Monatshefte für Math., 185 (2018), 365-395.  doi: 10.1007/s00605-017-1100-8.  Google Scholar

[2]

P. ArnouxS. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith., 87 (1999), 209-217.  doi: 10.4064/aa-87-3-209-217.  Google Scholar

[3]

A. Avila and S. Jitomirskaya, The ten martini problem, Ann. Math., 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.  Google Scholar

[4]

A. Avila and R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. Math., 164 (2006), 911-940.  doi: 10.4007/annals.2006.164.911.  Google Scholar

[5]

J. Beck, The modulus of polynomials with zeros on the unit circle: a problem of Erdos, Ann. Math., 134 (1991), 609-651.  doi: 10.2307/2944358.  Google Scholar

[6]

V. Berthé, S. Ferenczi and L. Q. Zamboni, Interactions between dynamics, arithmetic and combinatorics: The good, the bad, and the ugly, Algebraic and Topological Dynamics, Contemp. Math., 385 (2005), 333–364; Amer. Math. Soc., Providence.  Google Scholar

[7]

S. Grepstad and M. Neumüller, Asymptotic behaviour of the Sudler product of sines for quadratic irrationals, J. Math. Anal. Appl., 465 (2018), 928-960.  doi: 10.1016/j.jmaa.2018.05.045.  Google Scholar

[8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford Univ. Press, New York, 2008.   Google Scholar
[9]

P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. Lond. A, 68 (1955), 874-892.   Google Scholar

[10]

Y. HatsugaiM. Kohmoto and Y.-S. Wu, Quantum group, Bethe ansatz equations, and Bloch wave functions in magnetic fields, Phys. Rev. B, 53 (1996), 9697-9712.   Google Scholar

[11]

D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B, 14 (1976), 2239-2249.   Google Scholar

[12]

O. Knill and F. Tangerman, Self-similarity and growth in Birkhoff sums for the golden rotation, Nonlinearity, 24 (2011), 3115-3127.  doi: 10.1088/0951-7715/24/11/006.  Google Scholar

[13]

H. Koch, Golden mean renormalization for the almost Mathieu operator and related skew products, J. Math. Phys., 62 (2021), 042702, 12 pp. doi: 10.1063/5.0005429.  Google Scholar

[14]

H. Koch and S. Kocić, Renormalization and universality of the Hofstadter spectrum, Nonlinearity, 33 (2020), 4381-4389.  doi: 10.1088/1361-6544/ab8693.  Google Scholar

[15]

H. Koch, Asymptotic scaling and universality for skew products with factors in ${{{\rm SL}}}(2, {\mathbb{R}})$, Preprint 2021, Available e.g. at http://web.ma.utexas.edu/users/koch/papers/skewunivers/ Google Scholar

[16]

Y. Last, Zero measure spectrum for the almost Mathieu operator, Comm. Math. Phys., 164 (1994), 421-432.   Google Scholar

[17]

M. Lerch, Question 1547, L'Intermédiaire des Mathématiciens, 11 (1904), 144-145.   Google Scholar

[18]

D. S. Lubinsky, The size of $(q; q)_n$ for $q$ on the unit circle, J. Number Theory, 76 (1999), 217-247.  doi: 10.1006/jnth.1998.2365.  Google Scholar

[19]

A. M. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Semin. Univ. Hamburg, 1 (1922), 77-98.  doi: 10.1007/BF02940581.  Google Scholar

[20]

J. M. Thuswaldner, $S$-adic sequences. A bridge between dynamics, arithmetic, and geometry, Preprint 2019. Google Scholar

[21]

V. T. Sós, On the distribution $ \rm mod \; 1 $ of the sequence $n\alpha$, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1 (1958), 127-134.   Google Scholar

[22]

C. Jr. Sudler, An estimate for a restricted partition function, Quart. J. Math. Oxford Ser., 15 (1964), 1-10.  doi: 10.1093/qmath/15.1.1.  Google Scholar

[23]

J. Surányi, Über die Anordnung der Vielfachen einer reelen Zahl $ \rm mod \; 1 $, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1 (1958), 107–111. Google Scholar

[24]

S. Świerczkowski, On successive settings of an arc on the circumference of a circle, Fundamenta Mathematicae, 46 (1959), 187-189.  doi: 10.4064/fm-46-2-187-189.  Google Scholar

[25]

P. Verschueren and B. Mestel, Growth of the Sudler product of sines at the golden rotation number, J. Math. Anal. Appl., 433 (2016), 200-226.  doi: 10.1016/j.jmaa.2015.06.014.  Google Scholar

[26]

P. B. Wiegmann and A. V. Zabrodin, Quantum group and magnetic translations. Bethe ansatz solution for the Harper's equation, Modern Phys. Lett. B, 8 (1994), 311-318.  doi: 10.1142/S0217984994000315.  Google Scholar

show all references

References:
[1]

C. AistleitnerG. LarcherF. PillichshammerS. S. Eddin and R. F. Tichy, On Weyl products and uniform distribution modulo one, Monatshefte für Math., 185 (2018), 365-395.  doi: 10.1007/s00605-017-1100-8.  Google Scholar

[2]

P. ArnouxS. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith., 87 (1999), 209-217.  doi: 10.4064/aa-87-3-209-217.  Google Scholar

[3]

A. Avila and S. Jitomirskaya, The ten martini problem, Ann. Math., 170 (2009), 303-342.  doi: 10.4007/annals.2009.170.303.  Google Scholar

[4]

A. Avila and R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. Math., 164 (2006), 911-940.  doi: 10.4007/annals.2006.164.911.  Google Scholar

[5]

J. Beck, The modulus of polynomials with zeros on the unit circle: a problem of Erdos, Ann. Math., 134 (1991), 609-651.  doi: 10.2307/2944358.  Google Scholar

[6]

V. Berthé, S. Ferenczi and L. Q. Zamboni, Interactions between dynamics, arithmetic and combinatorics: The good, the bad, and the ugly, Algebraic and Topological Dynamics, Contemp. Math., 385 (2005), 333–364; Amer. Math. Soc., Providence.  Google Scholar

[7]

S. Grepstad and M. Neumüller, Asymptotic behaviour of the Sudler product of sines for quadratic irrationals, J. Math. Anal. Appl., 465 (2018), 928-960.  doi: 10.1016/j.jmaa.2018.05.045.  Google Scholar

[8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford Univ. Press, New York, 2008.   Google Scholar
[9]

P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. Lond. A, 68 (1955), 874-892.   Google Scholar

[10]

Y. HatsugaiM. Kohmoto and Y.-S. Wu, Quantum group, Bethe ansatz equations, and Bloch wave functions in magnetic fields, Phys. Rev. B, 53 (1996), 9697-9712.   Google Scholar

[11]

D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B, 14 (1976), 2239-2249.   Google Scholar

[12]

O. Knill and F. Tangerman, Self-similarity and growth in Birkhoff sums for the golden rotation, Nonlinearity, 24 (2011), 3115-3127.  doi: 10.1088/0951-7715/24/11/006.  Google Scholar

[13]

H. Koch, Golden mean renormalization for the almost Mathieu operator and related skew products, J. Math. Phys., 62 (2021), 042702, 12 pp. doi: 10.1063/5.0005429.  Google Scholar

[14]

H. Koch and S. Kocić, Renormalization and universality of the Hofstadter spectrum, Nonlinearity, 33 (2020), 4381-4389.  doi: 10.1088/1361-6544/ab8693.  Google Scholar

[15]

H. Koch, Asymptotic scaling and universality for skew products with factors in ${{{\rm SL}}}(2, {\mathbb{R}})$, Preprint 2021, Available e.g. at http://web.ma.utexas.edu/users/koch/papers/skewunivers/ Google Scholar

[16]

Y. Last, Zero measure spectrum for the almost Mathieu operator, Comm. Math. Phys., 164 (1994), 421-432.   Google Scholar

[17]

M. Lerch, Question 1547, L'Intermédiaire des Mathématiciens, 11 (1904), 144-145.   Google Scholar

[18]

D. S. Lubinsky, The size of $(q; q)_n$ for $q$ on the unit circle, J. Number Theory, 76 (1999), 217-247.  doi: 10.1006/jnth.1998.2365.  Google Scholar

[19]

A. M. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Semin. Univ. Hamburg, 1 (1922), 77-98.  doi: 10.1007/BF02940581.  Google Scholar

[20]

J. M. Thuswaldner, $S$-adic sequences. A bridge between dynamics, arithmetic, and geometry, Preprint 2019. Google Scholar

[21]

V. T. Sós, On the distribution $ \rm mod \; 1 $ of the sequence $n\alpha$, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1 (1958), 127-134.   Google Scholar

[22]

C. Jr. Sudler, An estimate for a restricted partition function, Quart. J. Math. Oxford Ser., 15 (1964), 1-10.  doi: 10.1093/qmath/15.1.1.  Google Scholar

[23]

J. Surányi, Über die Anordnung der Vielfachen einer reelen Zahl $ \rm mod \; 1 $, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1 (1958), 107–111. Google Scholar

[24]

S. Świerczkowski, On successive settings of an arc on the circumference of a circle, Fundamenta Mathematicae, 46 (1959), 187-189.  doi: 10.4064/fm-46-2-187-189.  Google Scholar

[25]

P. Verschueren and B. Mestel, Growth of the Sudler product of sines at the golden rotation number, J. Math. Anal. Appl., 433 (2016), 200-226.  doi: 10.1016/j.jmaa.2015.06.014.  Google Scholar

[26]

P. B. Wiegmann and A. V. Zabrodin, Quantum group and magnetic translations. Bethe ansatz solution for the Harper's equation, Modern Phys. Lett. B, 8 (1994), 311-318.  doi: 10.1142/S0217984994000315.  Google Scholar

Figure 1.  The orbit $ j\mapsto y^s_j $ for the inverse golden mean (left) and its absolute value (log scale, right)
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