Unbounded regime for circle maps with a flat interval

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May 2015, 35(5): 2099-2122. doi: 10.3934/dcds.2015.35.2099

Unbounded regime for circle maps with a flat interval

Liviana Palmisano ^1,

1.	Institute of Mathematics of PAN, ul. Śniadeckich 8, 00-956 Warszawa, Poland

Received May 2014 Revised October 2014 Published December 2014

Abstract
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We study $\mathcal{C}^2$ weakly order preserving circle maps with a flat interval. In particular we are interested in the geometry of the mapping near to the singularities at the boundary of the flat interval. Without any assumption on the rotation number we show that the geometry is degenerate when the degree of the singularities is less than or equal to two and becomes bounded when the degree goes to three. As an example of application, the result is applied to study Cherry flows.

Keywords: topological transitivity, Cherry flows., circle mappings, Low-dimensional dynamics, non-wandering set.

Mathematics Subject Classification: Primary: 37E1.

Citation: Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099

References:

[1]	S. K. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, vol. 153 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996.
[2]	T. M. Cherry, Analytic Quasi-Periodic Curves of Discontinuous Type on a Torus, Proc. London Math. Soc., S2-44 (1938), 175. doi: 10.1112/plms/s2-44.3.175.
[3]	W. de Melo and S. van Strien, One-dimensional Dynamics, vol. 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.
[4]	J. Graczyk, L. B. Jonker, G. Świątek, F. M. Tangerman and J. J. P. Veerman, Differentiable circle maps with a flat interval, Comm. Math. Phys., 173 (1995), 599-622, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104274914. doi: 10.1007/BF02101658.
[5]	J. Graczyk, Dynamics of circle maps with flat spots, Fund. Math., 209 (2010), 267-290. doi: 10.4064/fm209-3-4.
[6]	J. Graczyk, D. Sands and G. Świątek, Metric attractors for smooth unimodal maps, Ann. of Math. (2), 159 (2004), 725-740. doi: 10.4007/annals.2004.159.725.
[7]	M. Martens, S. van Strien, W. de Melo and P. Mendes, On Cherry flows, Ergodic Theory Dynam. Systems, 10 (1990), 531-554. doi: 10.1017/S0143385700005733.
[8]	P. Mendes, A metric property of Cherry vector fields on the torus, J. Differential Equations, 89 (1991), 305-316. doi: 10.1016/0022-0396(91)90123-Q.
[9]	P. C. Moreira and A. A. G. Ruas, Metric properties of Cherry flows, J. Differential Equations, 97 (1992), 16-26. doi: 10.1016/0022-0396(92)90081-W.
[10]	L. Palmisano, On physical measures for cherry flows, Preprint.
[11]	L. Palmisano, Sur les Applications du Cercle Avec un Intervalle Plat et Flots de Cherry, PhD thesis, Université Paris-Sud XI, 2013.
[12]	L. Palmisano, A phase transition for circle maps and cherry flows, Comm. Math. Phys., 321 (2013), 135-155. doi: 10.1007/s00220-013-1685-2.
[13]	R. Saghin and E. Vargas, Invariant measures for Cherry flows, Comm. Math. Phys., 317 (2013), 55-67. doi: 10.1007/s00220-012-1611-z.
[14]	G. Świątek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104162273. doi: 10.1007/BF01218263.
[15]	F. M. Tangerman and J. J. P. Veerman, Scalings in circle maps. II, Comm. Math. Phys., 141 (1991), 279-291, URL http://projecteuclid.org/euclid.cmp/1104248301. doi: 10.1007/BF02101506.
[16]	S. van Strien, Hyperbolicity and invariant measures for general $C^2$ interval maps satisfying the Misiurewicz condition, Comm. Math. Phys., 128 (1990), 437-495, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104180533. doi: 10.1007/BF02096868.
[17]	J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419-428, URL http://stacks.iop.org/0951-7715/2/419. doi: 10.1088/0951-7715/2/3/003.
[18]	J. J. P. Veerman and F. M. Tangerman, Scalings in circle maps. I, Comm. Math. Phys., 134 (1990), 89-107, URL http://projecteuclid.org/euclid.cmp/1104201615. doi: 10.1007/BF02102091.

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References:

[1]	S. K. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, vol. 153 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996.
[2]	T. M. Cherry, Analytic Quasi-Periodic Curves of Discontinuous Type on a Torus, Proc. London Math. Soc., S2-44 (1938), 175. doi: 10.1112/plms/s2-44.3.175.
[3]	W. de Melo and S. van Strien, One-dimensional Dynamics, vol. 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.
[4]	J. Graczyk, L. B. Jonker, G. Świątek, F. M. Tangerman and J. J. P. Veerman, Differentiable circle maps with a flat interval, Comm. Math. Phys., 173 (1995), 599-622, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104274914. doi: 10.1007/BF02101658.
[5]	J. Graczyk, Dynamics of circle maps with flat spots, Fund. Math., 209 (2010), 267-290. doi: 10.4064/fm209-3-4.
[6]	J. Graczyk, D. Sands and G. Świątek, Metric attractors for smooth unimodal maps, Ann. of Math. (2), 159 (2004), 725-740. doi: 10.4007/annals.2004.159.725.
[7]	M. Martens, S. van Strien, W. de Melo and P. Mendes, On Cherry flows, Ergodic Theory Dynam. Systems, 10 (1990), 531-554. doi: 10.1017/S0143385700005733.
[8]	P. Mendes, A metric property of Cherry vector fields on the torus, J. Differential Equations, 89 (1991), 305-316. doi: 10.1016/0022-0396(91)90123-Q.
[9]	P. C. Moreira and A. A. G. Ruas, Metric properties of Cherry flows, J. Differential Equations, 97 (1992), 16-26. doi: 10.1016/0022-0396(92)90081-W.
[10]	L. Palmisano, On physical measures for cherry flows, Preprint.
[11]	L. Palmisano, Sur les Applications du Cercle Avec un Intervalle Plat et Flots de Cherry, PhD thesis, Université Paris-Sud XI, 2013.
[12]	L. Palmisano, A phase transition for circle maps and cherry flows, Comm. Math. Phys., 321 (2013), 135-155. doi: 10.1007/s00220-013-1685-2.
[13]	R. Saghin and E. Vargas, Invariant measures for Cherry flows, Comm. Math. Phys., 317 (2013), 55-67. doi: 10.1007/s00220-012-1611-z.
[14]	G. Świątek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104162273. doi: 10.1007/BF01218263.
[15]	F. M. Tangerman and J. J. P. Veerman, Scalings in circle maps. II, Comm. Math. Phys., 141 (1991), 279-291, URL http://projecteuclid.org/euclid.cmp/1104248301. doi: 10.1007/BF02101506.
[16]	S. van Strien, Hyperbolicity and invariant measures for general $C^2$ interval maps satisfying the Misiurewicz condition, Comm. Math. Phys., 128 (1990), 437-495, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104180533. doi: 10.1007/BF02096868.
[17]	J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419-428, URL http://stacks.iop.org/0951-7715/2/419. doi: 10.1088/0951-7715/2/3/003.
[18]	J. J. P. Veerman and F. M. Tangerman, Scalings in circle maps. I, Comm. Math. Phys., 134 (1990), 89-107, URL http://projecteuclid.org/euclid.cmp/1104201615. doi: 10.1007/BF02102091.

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