We extend the result of Michał Misiurewicz assuring the existence of strange attractors for the parametrized family { f ( a , b ) } of orientation reversing Lozi maps to the orientation preserving case. That is, we rigorously determine an open subset of the parameter space for which an attractor A ( a , b ) of f ( a , b ) always exists and exhibits chaotic properties. Moreover, we prove that the attractor is maximal in some open parameter region and arises as the closure of the unstable manifold of a fixed point on which f ( a , b ) | A ( a , b ) is mixing. We also show that A ( a , b ) vary continuously with parameter ( a , b ) in the Hausdorff metric.

1.
M.
Hénon
, “
Numerical study of quadratic area-preserving mappings
,”
Quart. Appl. Math.
27
,
291
312
(
1969
).
2.
M.
Hénon
, “
A two-dimensional mapping with a strange attractor
,”
Comm. Math. Phys.
50
(
1
),
69
77
(
1976
).
3.
R.
Lozi
, Un attracteur étrange (?) du type attracteur de Hénon. http://dx.doi.org/10.1051/jphyscol:1978505, 39, 1978.
4.
O. E.
Rössler
, “
An equation for continuous chaos
,”
Phys. Lett. A
57
(
5
),
397
398
(
1976
).
5.
L. O.
Chua
,
M.
Komuro
, and
T.
Matsumoto
, “
The double scroll family. I. Rigorous proof of chaos
,”
IEEE Trans. Circuits Syst.
33
(
11
),
1072
1097
(
1986
).
6.
M.
Misiurewicz
, “
Strange attractors for the Lozi mappings
,”
Ann. N. Y. Acad. Sci.
357
(
1
),
348
358
(
1980
).
7.
M.
Misiurewicz
and
S.
Štimac
, “
Lozi-like maps
,”
Discrete Contin. Dyn. Syst.
38
(
6
),
2965
2985
(
2018
).
8.
Y.
Ishii
, “
Towards a kneading theory for Lozi mappings I: A solution of the pruning front conjecture and the first tangency problem
,”
Nonlinearity
10
(
3
),
731
747
(
1997
).
9.
Y.
Ishii
, “
Towards a kneading theory for Lozi mappings. II: Monotonicity of the topological entropy and Hausdorff dimension of attractors
,”
Commun. Math. Phys.
190
(
2
),
375
394
(
1997
).
10.
M.
Misiurewicz
and
S.
Štimac
, “
Symbolic dynamics for Lozi maps
,”
Nonlinearity
29
(
10
),
3031
3046
(
2016
).
11.
I.
Burak Yildiz
, “
Monotonicity of the Lozi family and the zero entropy locus
,”
Nonlinearity
24
(
5
),
1613
1628
(
2011
).
12.
I.
Burak Yildiz
, “
Discontinuity of topological entropy for Lozi maps
,”
Ergod. Theory Dyn. Syst.
32
(
5
),
1783
1800
(
2012
).
13.
P. A.
Glendinning
and
D. J. W.
Simpson
, “
Robust chaos and the continuity of attractors
,”
Trans. Math. Appl.
4
(
1
),
tnaa002
(
2020
).
14.
J.
Boroński
and
S.
Štimac
, “Densely branching trees as models for Hénon-like and Lozi-like attractors”
Adv. Math.
429
,
109191
(
2023
).
15.
I.
Ghosh
and
D. J. W.
Simpson
, “
Robust Devaney chaos in the two-dimensional border-collision normal form
,”
Chaos
32
(
4
),
043120
(
2022
).
16.
P. A.
Glendinning
and
D. J. W.
Simpson
, “
A constructive approach to robust chaos using invariant manifolds and expanding cones
,”
Discrete Contin. Dyn. Syst.
41
(
7
),
3367
3387
(
2021
).
17.
M. R.
Rychlik
, “Invariant measures and the variational principle for Lozi mappings,” Ph.D. thesis (University of California, Berkeley, CA, 1983).
18.
Y.
Cao
and
Z.
Liu
, “
Strange attractors in the orientation preserving Lozi map
,”
Chaos, Solitons Fractals
9
(
11
),
1857
1863
(
1998
).
19.
P.
Glendinning
, “
Robust chaos revisited
,”
Eur. Phys. J. Spec. Top.
226
(
9
),
1721
1738
(
2017
).
20.
S.
Banerjee
and
C.
Grebogi
, “
Border collision bifurcations in two-dimensional piecewise smooth maps
,”
Phys. Rev. E
59
,
4052
4061
(
1999
).
21.
H. E.
Nusse
and
J. A.
Yorke
, “
Border-collision bifurcations including “Period two to period three” for piecewise smooth systems
,”
Phys. D: Nonlinear Phenom.
57
(
1
),
39
57
(
1992
).
22.
M.
Benedicks
and
M.
Viana
, “
Solution of the basin problem for Hénon-like attractors
,”
Invent. Math.
143
,
375
434
(
2001
).
23.
Y.
Cao
and
J.
Min Mao
, “
The non-wandering set of some Hénon maps
,”
Chaos, Solitons Fractals
11
(
13
),
2045
2053
(
2000
).
24.
Y.
Cao
and
Z.
Liu
, “
The geometric structure of strange attractors in the Lozi map
,”
Commun. Nonlinear Sci. Numer. Simul.
3
(
2
),
119
123
(
1998
).
25.
J.
Buzzi
, “
Maximal entropy measures for piecewise affine surface homeomorphisms
,”
Ergod. Theory Dyn. Syst.
29
(
6
),
1723
1763
(
2009
).
26.
L.
Mora
and
M.
Viana
, “
Abundance of strange attractors
,”
Acta Math.
171
(
1
),
1
71
(
1993
).
27.
J.-P.
Eckmann
and
D.
Ruelle
,
Ergodic Theory of Chaos and Strange Attractors
(
Springer
,
New York
,
2004
), pp. 273–312.
28.
D.
Ruelle
and
F.
Takens
, “
On the nature of turbulence
,”
Commun. Math. Phys.
20
(
3
),
167
192
(
1971
).
29.
J.
Banks
,
J.
Brooks
,
G.
Cairns
,
G.
Davis
, and
P.
Stacey
, “
On Devaney’s definition of chaos
,”
Am. Math. Mon.
99
(
4
),
332
334
(
1992
).
30.
M.
Viana
, “
Strange attractors in higher dimensions
,”
Bol. Soc. Brasil. Mat. (N.S.)
24
(
1
),
13
62
(
1993
).
31.
D.-S.
Ou
, “Critical points in higher dimensions, I: Reverse order of periodic orbit creations in the Lozi family,” arXiv:2203.02326 (2022).
32.
F.
Hausdorff
, “Grundzüge der Mengenlehre.” in Göschens Lehrbücherei/Gruppe I: Reine und Angewandte Mathematik Series (Von Veit, 1914).
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