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Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces
Classification of linear skew-products of the complex plane and an affine route to fractalization
Departament de Matemàtiques i Informàtica, Barcelona Graduate School of Mathematics (BGSMath), Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain |
$ \left. \begin{array}{rcl} \theta & \mapsto & \theta+\omega,\\ z & \mapsto & a(\theta)z, \end{array} \right\} $ |
$ \theta\in {\mathbb T} $ |
$ z\in {\mathbb C} $ |
$ \frac{\omega}{2\pi} $ |
$ \theta\mapsto a(\theta) \in {\mathbb C}\setminus \{0\} $ |
$ \theta\mapsto a(\theta) $ |
References:
[1] |
L. V. Ahlfors, Complex Analysis: An Introduction of the Theory of Analytic Functions of one Complex Variable, Second edition. McGraw-Hill Book Co., New York-Toronto-London, 1966. |
[2] |
K. Bjerklöv,
SNA's in the quasi-periodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161.
doi: 10.1007/s00220-008-0626-y. |
[3] |
J.-L. Figueras and À. Haro, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16pp.
doi: 10.1063/1.4938185. |
[4] |
J.-L. Figueras and À. Haro,
A note on the fractalization of saddle invariant curves in quasiperiodic systems, Discrete and Continuous Dynamical Systems - S, 9 (2016), 1095-1107.
doi: 10.3934/dcdss.2016043. |
[5] |
J.-L. Figueras and T. O. Timoudas, Sharp $\frac{1}{2}$-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles, Preprint, 2018. Google Scholar |
[6] |
G. Fuhrmann, M. Gröger and T. Jäger,
Non-smooth saddle-node bifurcations Ⅱ: Dimensions of strange attractors, Ergodic Theory Dynam. Systems, 38 (2018), 2989-3011.
doi: 10.1017/etds.2017.4. |
[7] |
G. Fuhrmann and J. Wang, Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent, Discrete Contin. Dyn. Syst. Ser. A, 37 92017), 5747–5761.
doi: 10.3934/dcds.2017249. |
[8] |
P. Glendinning,
Global attractors of pinched skew products, Dyn. Syst., 17 (2002), 287-294.
doi: 10.1080/14689360210160878. |
[9] |
G. H. Hardy and J. E. Littlewood,
Some problems of diophantine approximation, Acta Math., 37 (1914), 193-239.
doi: 10.1007/BF02401834. |
[10] |
À. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8pp.
doi: 10.1063/1.2150947. |
[11] |
À. Haro and C. Simó, To be or not to be a SNA: That is the question, Preprint, 2006. Google Scholar |
[12] |
T. H. Jäger,
Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255.
doi: 10.1088/0951-7715/16/4/303. |
[13] |
T. H. Jäger,
On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510.
doi: 10.1017/S0143385706000745. |
[14] |
À. Jorba, C. Núñez, R. Obaya and J. C. Tatjer,
Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3895-3928.
doi: 10.1142/S0218127407019780. |
[15] |
À. Jorba,
Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps, Nonlinearity, 14 (2001), 943-976.
doi: 10.1088/0951-7715/14/5/303. |
[16] |
À. Jorba and J. C. Tatjer,
A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567.
doi: 10.3934/dcdsb.2008.10.537. |
[17] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, volume 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187.![]() ![]() |
[18] |
A. Ya. Khinchin, Continued Fractions, Dover Publications, Inc., Mineola, NY, russian edition, 1997. With a preface by B. V. Gnedenko, Reprint of the 1964 translation. |
[19] |
S. Lang, Introduction to Diophantine Approximations, Springer-Verlag, New York, second edition, 1995.
doi: 10.1007/978-1-4612-4220-8. |
[20] |
L. Nirenberg, A proof of the Malgrange preparation theorem, In Proceedings of Liverpool Singularities–-Symposium, I (1969/70), pages 97–105. Lecture Notes in Mathematics, Vol. 192. Springer, Berlin, 1971. |
[21] |
M. Ponce,
Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955.
doi: 10.1088/0951-7715/20/12/011. |
[22] |
A. Prasad, V. Mehra and R. Ramaskrishna,
Strange nonchaotic attractors in the quasiperiodically forced logistic map, Phys. Rev. E, 57 (1998), 1576-1584.
doi: 10.1103/PhysRevE.57.1576. |
[23] |
A. Prasad, S. S. Negi and R. Ramaswamy,
Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291-309.
doi: 10.1142/S0218127401002195. |
[24] |
H. Rüssmann,
On optimal estimates for the solutions of linear difference equations on the circle, Celestial Mech., 14 (1976), 33-37.
doi: 10.1007/BF01247129. |
[25] |
J. Stark,
Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179.
doi: 10.1016/S0167-2789(97)00167-X. |
show all references
References:
[1] |
L. V. Ahlfors, Complex Analysis: An Introduction of the Theory of Analytic Functions of one Complex Variable, Second edition. McGraw-Hill Book Co., New York-Toronto-London, 1966. |
[2] |
K. Bjerklöv,
SNA's in the quasi-periodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161.
doi: 10.1007/s00220-008-0626-y. |
[3] |
J.-L. Figueras and À. Haro, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16pp.
doi: 10.1063/1.4938185. |
[4] |
J.-L. Figueras and À. Haro,
A note on the fractalization of saddle invariant curves in quasiperiodic systems, Discrete and Continuous Dynamical Systems - S, 9 (2016), 1095-1107.
doi: 10.3934/dcdss.2016043. |
[5] |
J.-L. Figueras and T. O. Timoudas, Sharp $\frac{1}{2}$-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles, Preprint, 2018. Google Scholar |
[6] |
G. Fuhrmann, M. Gröger and T. Jäger,
Non-smooth saddle-node bifurcations Ⅱ: Dimensions of strange attractors, Ergodic Theory Dynam. Systems, 38 (2018), 2989-3011.
doi: 10.1017/etds.2017.4. |
[7] |
G. Fuhrmann and J. Wang, Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent, Discrete Contin. Dyn. Syst. Ser. A, 37 92017), 5747–5761.
doi: 10.3934/dcds.2017249. |
[8] |
P. Glendinning,
Global attractors of pinched skew products, Dyn. Syst., 17 (2002), 287-294.
doi: 10.1080/14689360210160878. |
[9] |
G. H. Hardy and J. E. Littlewood,
Some problems of diophantine approximation, Acta Math., 37 (1914), 193-239.
doi: 10.1007/BF02401834. |
[10] |
À. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8pp.
doi: 10.1063/1.2150947. |
[11] |
À. Haro and C. Simó, To be or not to be a SNA: That is the question, Preprint, 2006. Google Scholar |
[12] |
T. H. Jäger,
Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255.
doi: 10.1088/0951-7715/16/4/303. |
[13] |
T. H. Jäger,
On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510.
doi: 10.1017/S0143385706000745. |
[14] |
À. Jorba, C. Núñez, R. Obaya and J. C. Tatjer,
Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3895-3928.
doi: 10.1142/S0218127407019780. |
[15] |
À. Jorba,
Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps, Nonlinearity, 14 (2001), 943-976.
doi: 10.1088/0951-7715/14/5/303. |
[16] |
À. Jorba and J. C. Tatjer,
A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567.
doi: 10.3934/dcdsb.2008.10.537. |
[17] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, volume 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187.![]() ![]() |
[18] |
A. Ya. Khinchin, Continued Fractions, Dover Publications, Inc., Mineola, NY, russian edition, 1997. With a preface by B. V. Gnedenko, Reprint of the 1964 translation. |
[19] |
S. Lang, Introduction to Diophantine Approximations, Springer-Verlag, New York, second edition, 1995.
doi: 10.1007/978-1-4612-4220-8. |
[20] |
L. Nirenberg, A proof of the Malgrange preparation theorem, In Proceedings of Liverpool Singularities–-Symposium, I (1969/70), pages 97–105. Lecture Notes in Mathematics, Vol. 192. Springer, Berlin, 1971. |
[21] |
M. Ponce,
Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955.
doi: 10.1088/0951-7715/20/12/011. |
[22] |
A. Prasad, V. Mehra and R. Ramaskrishna,
Strange nonchaotic attractors in the quasiperiodically forced logistic map, Phys. Rev. E, 57 (1998), 1576-1584.
doi: 10.1103/PhysRevE.57.1576. |
[23] |
A. Prasad, S. S. Negi and R. Ramaswamy,
Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291-309.
doi: 10.1142/S0218127401002195. |
[24] |
H. Rüssmann,
On optimal estimates for the solutions of linear difference equations on the circle, Celestial Mech., 14 (1976), 33-37.
doi: 10.1007/BF01247129. |
[25] |
J. Stark,
Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179.
doi: 10.1016/S0167-2789(97)00167-X. |


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