# American Institute of Mathematical Sciences

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

## Lozi-like maps

 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.

Citation: Michał Misiurewicz, Sonja Štimac. Lozi-like maps. Discrete & Continuous Dynamical Systems - A, 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
Positions of some distinguished points
The set of parameters
The triangle $\Theta$ and positions of some distinguished points
Attractor for the Lozi map with parameters described by (F1') and (F2'). The $y$-coordinate is stretched by factor $7/4$
Graphs of (F1') and (F2')
Equations (F1') and (F2') as inequalities
 [1] David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873 [2] Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652 [3] Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313 [4] Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006 [5] Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403 [6] Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255 [7] Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1 [8] John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723 [9] Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927 [10] Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939 [11] Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555 [12] Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725 [13] Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940 [14] Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399 [15] Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75 [16] Jim Wiseman. Symbolic dynamics from signed matrices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621 [17] George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43 [18] Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301 [19] Miaohua Jiang, Qiang Zhang. A coupled map lattice model of tree dispersion. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 83-101. doi: 10.3934/dcdsb.2008.9.83 [20] Kokum R. De Silva, Shigetoshi Eda, Suzanne Lenhart. Modeling environmental transmission of MAP infection in dairy cows. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1001-1017. doi: 10.3934/mbe.2017052

2018 Impact Factor: 1.143