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Projection methods and discrete gradient methods for preserving first integrals of ODEs
Unbounded regime for circle maps with a flat interval
1. | Institute of Mathematics of PAN, ul. Śniadeckich 8, 00-956 Warszawa, Poland |
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. |
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. |
[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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