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June  2018, 38(6): 2965-2985. doi: 10.3934/dcds.2018127

Lozi-like maps

Michał Misiurewicz 1, and  Sonja Štimac 2,**,

1. 

Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, USA
2. 

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10 000 Zagreb, Croatia

**Supported in part by the NEWFELPRO Grant No. 24 HeLoMa, and in part by the Croatian Science Foundation grant IP-2014-09-2285

Received  September 2017 Published  April 2018

Fund Project: This work was partially supported by a grant number 426602 from the Simons Foundation to Michał Misiurewicz.

We define a broad class of piecewise smooth plane homeomorphisms which have properties similar to the properties of Lozi maps, including the existence of a hyperbolic attractor. We call those maps Lozi-like. For those maps one can apply our previous results on kneading theory for Lozi maps. We show a strong numerical evidence that there exist Lozi-like maps that have kneading sequences different than those of Lozi maps.

Keywords: Lozi map, Lozi-like map, attractor, symbolic dynamics, kneading theory.
Mathematics Subject Classification: Primary: 37D45, 37E30; Secondary: 37B10.
Citation: Michał Misiurewicz, Sonja Štimac. Lozi-like maps. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2965-2985. doi: 10.3934/dcds.2018127
References:
[1]

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.  Google Scholar

[2]

Z. Elhadj, Lozi Mappings: Theory and Applications, CRC Press, Boca Raton, FL, 2014.  Google Scholar

[3]

Y. Ishii, Towards a kneading theory for Lozi mappings Ⅰ. A solution of the pruning front conjecture and the first tangency problem, Nonlinearity, 10 (1997), 731-747.   Google Scholar

[4]

R. Lozi, Un attracteur etrange(?) du type attracteur de Hénon, J. Phys. Colloques(Coll. C5), 39 (1978), 9-10.  doi: 10.1051/jphyscol:1978505.  Google Scholar

[5]

M. Misiurewicz, Strange attractor for the Lozi mappings, Ann. New York Acad. Sci., 357 (1980), 348-358.   Google Scholar

[6]

M. Misiurewicz and S. Štimac, Symbolic dynamics for Lozi maps, Nonlinearity, 29 (2016), 3031-3046.  doi: 10.1088/0951-7715/29/10/3031.  Google Scholar

show all references

References:
[1]

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.  Google Scholar

[2]

Z. Elhadj, Lozi Mappings: Theory and Applications, CRC Press, Boca Raton, FL, 2014.  Google Scholar

[3]

Y. Ishii, Towards a kneading theory for Lozi mappings Ⅰ. A solution of the pruning front conjecture and the first tangency problem, Nonlinearity, 10 (1997), 731-747.   Google Scholar

[4]

R. Lozi, Un attracteur etrange(?) du type attracteur de Hénon, J. Phys. Colloques(Coll. C5), 39 (1978), 9-10.  doi: 10.1051/jphyscol:1978505.  Google Scholar

[5]

M. Misiurewicz, Strange attractor for the Lozi mappings, Ann. New York Acad. Sci., 357 (1980), 348-358.   Google Scholar

[6]

M. Misiurewicz and S. Štimac, Symbolic dynamics for Lozi maps, Nonlinearity, 29 (2016), 3031-3046.  doi: 10.1088/0951-7715/29/10/3031.  Google Scholar

Figure 1.  Positions of some distinguished points
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Figure 2.  The set of parameters
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Figure 3.  The triangle $\Theta$ and positions of some distinguished points
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Figure 4.  Attractor for the Lozi map with parameters described by (F1') and (F2'). The $y$-coordinate is stretched by factor $7/4$
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Figure 5.  Graphs of (F1') and (F2')
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Figure 6.  Equations (F1') and (F2') as inequalities
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