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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Graczyk, Dynamics of circle maps with flat spots, Fund. Math., 209 (2010), 267-290. doi: 10.4064/fm209-3-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Palmisano, On physical measures for cherry flows,, Preprint., ().   Google Scholar

[11]

L. Palmisano, Sur les Applications du Cercle Avec un Intervalle Plat et Flots de Cherry, PhD thesis, Université Paris-Sud XI, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Graczyk, Dynamics of circle maps with flat spots, Fund. Math., 209 (2010), 267-290. doi: 10.4064/fm209-3-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Palmisano, On physical measures for cherry flows,, Preprint., ().   Google Scholar

[11]

L. Palmisano, Sur les Applications du Cercle Avec un Intervalle Plat et Flots de Cherry, PhD thesis, Université Paris-Sud XI, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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