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Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation
On the derivative of the $\alpha$-Farey-Minkowski function
1. | Department of Mathematics, University Walk, Clifton, Bristol BS8 1TW, United Kingdom |
References:
[1] |
L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.
doi: 10.1007/BF02773211. |
[2] |
J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arith., 74 (1996), 311-327. |
[3] |
K. Falconer, "Techniques in Fractal Geometry,'' John Wiley & Sons, Ltd., Chichester, 1997. |
[4] |
J. Jaerisch and M. Kesseböhmer, Regularity of multifractal spectra of conformal iterated function systems, Trans. Amer. Math. Soc., 363 (2011), 313-330.
doi: 10.1090/S0002-9947-2010-05326-7. |
[5] |
M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems, Ergodic Theory Dynam. Systems, 32 (2012), 989-1017.
doi: 10.1017/S0143385711000186. |
[6] |
M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., 605 (2007), 133-163.
doi: 10.1515/CRELLE.2007.029. |
[7] |
M. Kesseböhmer and B. O. Stratmann, Fractal analysis for sets of non-differentiability of Minkowski's question mark function, J. Number Theory, 128 (2008), 2663-2686.
doi: 10.1016/j.jnt.2007.12.010. |
[8] |
H. Minkowski, "Geometrie der Zahlen,'' Gesammelte Abhandlungen, Vol. 2, 1911; reprinted by Chelsea, New York, (1967), 43-52. |
[9] |
S. Munday, "Finite and Infinite Ergodic Theory for Linear and Conformal Dynamical Systems,'' Ph.D Thesis, University of St Andrews, 2011. |
[10] |
J. Paradís and P. Viader, The derivative of Minkowski's $?(x)$ function, J. Math. Anal. Appl., 253 (2001), 107-125.
doi: 10.1006/jmaa.2000.7064. |
[11] |
J. Paradís, P. Viader and L. Bibiloni, Riesz-Nágy singular functions revisited, J. Math. Anal. Appl., 329 (2007), 592-602.
doi: 10.1016/j.jmaa.2006.06.082. |
[12] |
H. L. Royden, "Measure Theory,'' Third edition, Prentice-Hall, New Jersey, 1988. |
[13] |
R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.
doi: 10.1090/S0002-9947-1943-0007929-6. |
[14] |
J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543.
doi: 10.1088/0951-7715/22/3/001. |
show all references
References:
[1] |
L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.
doi: 10.1007/BF02773211. |
[2] |
J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Ergodic properties of generalised Lüroth series, Acta Arith., 74 (1996), 311-327. |
[3] |
K. Falconer, "Techniques in Fractal Geometry,'' John Wiley & Sons, Ltd., Chichester, 1997. |
[4] |
J. Jaerisch and M. Kesseböhmer, Regularity of multifractal spectra of conformal iterated function systems, Trans. Amer. Math. Soc., 363 (2011), 313-330.
doi: 10.1090/S0002-9947-2010-05326-7. |
[5] |
M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems, Ergodic Theory Dynam. Systems, 32 (2012), 989-1017.
doi: 10.1017/S0143385711000186. |
[6] |
M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., 605 (2007), 133-163.
doi: 10.1515/CRELLE.2007.029. |
[7] |
M. Kesseböhmer and B. O. Stratmann, Fractal analysis for sets of non-differentiability of Minkowski's question mark function, J. Number Theory, 128 (2008), 2663-2686.
doi: 10.1016/j.jnt.2007.12.010. |
[8] |
H. Minkowski, "Geometrie der Zahlen,'' Gesammelte Abhandlungen, Vol. 2, 1911; reprinted by Chelsea, New York, (1967), 43-52. |
[9] |
S. Munday, "Finite and Infinite Ergodic Theory for Linear and Conformal Dynamical Systems,'' Ph.D Thesis, University of St Andrews, 2011. |
[10] |
J. Paradís and P. Viader, The derivative of Minkowski's $?(x)$ function, J. Math. Anal. Appl., 253 (2001), 107-125.
doi: 10.1006/jmaa.2000.7064. |
[11] |
J. Paradís, P. Viader and L. Bibiloni, Riesz-Nágy singular functions revisited, J. Math. Anal. Appl., 329 (2007), 592-602.
doi: 10.1016/j.jmaa.2006.06.082. |
[12] |
H. L. Royden, "Measure Theory,'' Third edition, Prentice-Hall, New Jersey, 1988. |
[13] |
R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.
doi: 10.1090/S0002-9947-1943-0007929-6. |
[14] |
J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543.
doi: 10.1088/0951-7715/22/3/001. |
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