November  2021, 41(11): 5455-5471. 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  November 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (11) : 5455-5471. 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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

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

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

[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. 

[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. 

[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.

[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.

[14]

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

[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/

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[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. 

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

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

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

[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. 

[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. 

[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.

[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.

[14]

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

[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/

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[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. 

[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.

[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.

[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.

[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.

[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.

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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