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November  2021, 41(11): 5037-5055. doi: 10.3934/dcds.2021067

A phase transition for circle maps with a flat spot and different critical exponents

Liviana Palmisano 1,, and  Bertuel Tangue Ndawa 2,

1. 

Stockton Road, Other lines, Durham, DH1 3LE, United Kingdom's
2. 

Route Melen, Yaounde, MO 8390, Cameroon

* Corresponding author: Liviana Palmisano

Received  July 2019 Revised  January 2021 Published  November 2021 Early access  April 2021

Fund Project: The authors would like to thank the referees whose valuable comments helped to improve the exposition of the paper. The first author is supported by the Trygger Foundation. The second author is supported by the Centre d'Excellence Africain en Science Mathématiques et Applications (CEA-SMA). Part of the research for this paper took place at ICTP. The authors would like to thank the ICTP and in particular Prof. Stefano Luzzatto for their hospitality and support

We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the critical exponents. At this boundary there is a phase transition in the geometry of the system. Differently from the previous approaches, this is achieved by studying the asymptotical behavior of the renormalization operator.

Keywords: Cicle map, flat piece, critical exponent, geometry, renormalization.
Mathematics Subject Classification: Primary: 37E05, 37E10, 37E20; Secondary: 37C05.
Citation: Liviana Palmisano, Bertuel Tangue Ndawa. A phase transition for circle maps with a flat spot and different critical exponents. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5037-5055. doi: 10.3934/dcds.2021067
References:
[1]

S. Guarino, P. Crovisier and L. Palmisano, Ergodic properties of bimodal circle endomorphisms satisfying a Diophantine condition, Ergodic Theory Dynam. Systems, 39 (2019), 1462-1500.  doi: 10.1017/etds.2017.80.  Google Scholar

[2]

W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete, 25. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[3]

J. Graczyk, Dynamics of circle maps with flat spots, Fund. Math., 209 (2010), 267-290.  doi: 10.4064/fm209-3-4.  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, Commun. Math. Phys., 173 (1995), 599-622.  doi: 10.1007/BF02101658.  Google Scholar

[5]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233.   Google Scholar

[6]

O. Kozlovski and S. van Strien, Asymmetric unimodal maps with non-universal period-doubling scaling laws, Comm. Math. Phys., 379 (2020), 103–143, arXiv: math/1907.05812. doi: 10.1007/s00220-020-03835-9.  Google Scholar

[7]

M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc., 6 (1993), 425-457.  doi: 10.1090/S0894-0347-1993-1182670-0.  Google Scholar

[8]

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

[9]

M. Martens and B. Winckler, Instability of renormalization, preprint, arXiv: 1609.04473. Google Scholar

[10]

M. Martens and L. Palmisano, Invariant manifolds for non-differentiable operators, preprint, arXiv: 1704.06328. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

L. Palmisano, Cherry Flows with non-trivial attractors, Fund. Math., 244 (2019), 243-253.  doi: 10.4064/fm531-3-2018.  Google Scholar

[15]

L. Palmisano, On physical measures for Cherry flows, Fund. Math., 232 (2016), 167-179.  doi: 10.4064/fm232-2-5.  Google Scholar

[16]

L. Palmisano, Quasi-symmetric conjugacy for circle maps with a flat interval, Ergodic Theory Dynam. Systems, 39 (2019), 425-445.  doi: 10.1017/etds.2017.36.  Google Scholar

[17]

L. Palmisano, Unbounded regime for circle maps with a flat interval, Discrete Contin. Dyn. Syst., 35 (2015), 2099-2122.  doi: 10.3934/dcds.2015.35.2099.  Google Scholar

[18]

J. J. P. Veerman and F. M. Tangerman, Scalings in circle maps. Ⅰ, Comm. Math. Phys., 134 (1990), 89-107.  doi: 10.1007/BF02102091.  Google Scholar

[19]

F. M. Tangerman and J. J. P. Veerman, Scalings in circle maps. Ⅱ, Comm. Math. Phys., 141 (1991), 279-291.  doi: 10.1007/BF02101506.  Google Scholar

[20]

B. N. Tangue, Cherry maps with different critical exponents: Bifurcation of geometry, Rus. J. Nonlin. Dyn., 16 (2020), 651-672.  doi: 10.20537/nd200409.  Google Scholar

[21]

B. N. Tangue, Rigidity of fibonacci circle maps with a flat piece and different critical exponents, preprint, arXiv: math/2103.02347. Google Scholar

[22]

J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419-428.  doi: 10.1088/0951-7715/2/3/003.  Google Scholar

show all references

References:
[1]

S. Guarino, P. Crovisier and L. Palmisano, Ergodic properties of bimodal circle endomorphisms satisfying a Diophantine condition, Ergodic Theory Dynam. Systems, 39 (2019), 1462-1500.  doi: 10.1017/etds.2017.80.  Google Scholar

[2]

W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete, 25. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[3]

J. Graczyk, Dynamics of circle maps with flat spots, Fund. Math., 209 (2010), 267-290.  doi: 10.4064/fm209-3-4.  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, Commun. Math. Phys., 173 (1995), 599-622.  doi: 10.1007/BF02101658.  Google Scholar

[5]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233.   Google Scholar

[6]

O. Kozlovski and S. van Strien, Asymmetric unimodal maps with non-universal period-doubling scaling laws, Comm. Math. Phys., 379 (2020), 103–143, arXiv: math/1907.05812. doi: 10.1007/s00220-020-03835-9.  Google Scholar

[7]

M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc., 6 (1993), 425-457.  doi: 10.1090/S0894-0347-1993-1182670-0.  Google Scholar

[8]

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

[9]

M. Martens and B. Winckler, Instability of renormalization, preprint, arXiv: 1609.04473. Google Scholar

[10]

M. Martens and L. Palmisano, Invariant manifolds for non-differentiable operators, preprint, arXiv: 1704.06328. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

L. Palmisano, Cherry Flows with non-trivial attractors, Fund. Math., 244 (2019), 243-253.  doi: 10.4064/fm531-3-2018.  Google Scholar

[15]

L. Palmisano, On physical measures for Cherry flows, Fund. Math., 232 (2016), 167-179.  doi: 10.4064/fm232-2-5.  Google Scholar

[16]

L. Palmisano, Quasi-symmetric conjugacy for circle maps with a flat interval, Ergodic Theory Dynam. Systems, 39 (2019), 425-445.  doi: 10.1017/etds.2017.36.  Google Scholar

[17]

L. Palmisano, Unbounded regime for circle maps with a flat interval, Discrete Contin. Dyn. Syst., 35 (2015), 2099-2122.  doi: 10.3934/dcds.2015.35.2099.  Google Scholar

[18]

J. J. P. Veerman and F. M. Tangerman, Scalings in circle maps. Ⅰ, Comm. Math. Phys., 134 (1990), 89-107.  doi: 10.1007/BF02102091.  Google Scholar

[19]

F. M. Tangerman and J. J. P. Veerman, Scalings in circle maps. Ⅱ, Comm. Math. Phys., 141 (1991), 279-291.  doi: 10.1007/BF02101506.  Google Scholar

[20]

B. N. Tangue, Cherry maps with different critical exponents: Bifurcation of geometry, Rus. J. Nonlin. Dyn., 16 (2020), 651-672.  doi: 10.20537/nd200409.  Google Scholar

[21]

B. N. Tangue, Rigidity of fibonacci circle maps with a flat piece and different critical exponents, preprint, arXiv: math/2103.02347. Google Scholar

[22]

J. J. P. Veerman, Irrational rotation numbers, Nonlinearity, 2 (1989), 419-428.  doi: 10.1088/0951-7715/2/3/003.  Google Scholar

Figure 1.  A function in $ \mathscr{ L}^{(X)} $
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Figure 2.  The curve $ \Gamma $. The quadrant $ Q_- $ is below $ \Gamma $ and the quadrant $ Q_+ $ is above $ \Gamma $
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