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July  2019, 39(7): 3767-3787. doi: 10.3934/dcds.2019153

## 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

Received  February 2018 Revised  November 2018 Published  April 2019

Fund Project: Work supported by the Maria de Maeztu Excellence Grant MDM-2014-0445 and the grant 2017 SGR 1374. N. Fagella has been partially supported by the grants MTM2014-52209-C2-2-P and MTM2017-86795-C3-3-P, A. Jorba, M. Jorba-Cuscó and J.C. Tatjer have been supported by the grant MTM2015-67724-P.

Linear skew products of the complex plane,
 $\left. \begin{array}{rcl} \theta & \mapsto & \theta+\omega,\\ z & \mapsto & a(\theta)z, \end{array} \right\}$
where
 $\theta\in {\mathbb T}$
,
 $z\in {\mathbb C}$
,
 $\frac{\omega}{2\pi}$
is irrational, and
 $\theta\mapsto a(\theta) \in {\mathbb C}\setminus \{0\}$
is a smooth map, appear naturally when linearizing dynamics around an invariant curve of a quasi-periodically forced complex map. In this paper we study linear and topological equivalence classes of such maps through conjugacies which preserve the skewed structure, relating them to the Lyapunov exponent and the winding number of
 $\theta\mapsto a(\theta)$
. We analyze the transition between these classes by considering one parameter families of linear skew products. Finally, we show that, under suitable conditions, an affine variation of the maps above has a non-reducible invariant curve that undergoes a fractalization process when the parameter goes to a critical value. This phenomenon of fractalization of invariant curves is known to happen in nonlinear skew products, but it is remarkable that it also occurs in simple systems as the ones we present.
Citation: Núria Fagella, Àngel Jorba, Marc Jorba-Cuscó, Joan Carles Tatjer. Classification of linear skew-products of the complex plane and an affine route to fractalization. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3767-3787. doi: 10.3934/dcds.2019153
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] P. Glendinning, Global attractors of pinched skew products, Dyn. Syst., 17 (2002), 287-294.  doi: 10.1080/14689360210160878.  Google Scholar [9] G. H. Hardy and J. E. Littlewood, Some problems of diophantine approximation, Acta Math., 37 (1914), 193-239.  doi: 10.1007/BF02401834.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] S. Lang, Introduction to Diophantine Approximations, Springer-Verlag, New York, second edition, 1995. doi: 10.1007/978-1-4612-4220-8.  Google Scholar [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.  Google Scholar [21] M. Ponce, Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955.  doi: 10.1088/0951-7715/20/12/011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179.  doi: 10.1016/S0167-2789(97)00167-X.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] P. Glendinning, Global attractors of pinched skew products, Dyn. Syst., 17 (2002), 287-294.  doi: 10.1080/14689360210160878.  Google Scholar [9] G. H. Hardy and J. E. Littlewood, Some problems of diophantine approximation, Acta Math., 37 (1914), 193-239.  doi: 10.1007/BF02401834.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] S. Lang, Introduction to Diophantine Approximations, Springer-Verlag, New York, second edition, 1995. doi: 10.1007/978-1-4612-4220-8.  Google Scholar [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.  Google Scholar [21] M. Ponce, Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955.  doi: 10.1088/0951-7715/20/12/011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179.  doi: 10.1016/S0167-2789(97)00167-X.  Google Scholar
Invariant curve of (3) for c = 1. Plots for µ = 0:5, µ = 0:9, µ = 0:99 and µ = 0:999
Asymptotic growth of the invariant curve of (3) w.r.t. $\mu$ when $\mu\nearrow 1$, for $c = 1$. The horizontal axis shows $1-\mu$ and the symbols "+'' denote the computed values. The dotted line is the fitting function. Top: On the left, fitting $\|z_{\mu}\|_{\infty}$ by $1.54(1-\mu)^{-1/2}$. On the right, fitting of $\|z_{\mu}'\|_{\infty}$ by $0.41(1-\mu)^{-3/2}$. Bottom: On the left, fitting of the length of $z_{\mu}$ by $3.1(1-\mu)^{-3/2}$. On the right, fitting of $(z_{\mu}, 0)$ by $0.5(1-\mu)^{-1}$
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