Symbolic dynamics for Lozi maps

Misiurewicz, Michał; Štimac, Sonja

doi:10.1088/0951-7715/29/10/3031

Cited by 21 publications

(21 citation statements)

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“…Note that for the Lozi-like maps the analogue of the y-axis is a smooth curve called the divider, see [23,Definition 2.5], and the analog of E is the intersection of the divider and ∆.…”

Section: Geometry Of the Trees Tmentioning

confidence: 99%

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Densely branching trees as models for Hénon-like and Lozi-like attractors

Boroński¹,

Štimac²

2021

Preprint

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Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For Hénon maps this applies to Benedicks-Carleson positive Lebesgue measure parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. In general, under mild dissipation, conjugacy holds if the attractor does not have an arc in common with a stable manifold, and the density of branch points holds when the attractor is a homoclinic class. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967.

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“…Note that for the Lozi-like maps the analogue of the y-axis is a smooth curve called the divider, see [23,Definition 2.5], and the analog of E is the intersection of the divider and ∆.…”

Section: Geometry Of the Trees Tmentioning

confidence: 99%

“…Parametric families of maps in the plane, and especially the Hénon and Lozi maps, have been widely studied in many different contexts and a lot of important results have been proved; see for example [5], [6], [10], [17], [18], [19], [23], [26], [29], [32].…”

Section: Introductionmentioning

confidence: 99%

Densely branching trees as models for Hénon-like and Lozi-like attractors

Boroński¹,

Štimac²

2021

Preprint

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“…This paper can be considered the second part of our paper [5]. In [5] we developed three equivalent approaches to compressing information about the symbolic dynamics of Lozi maps. Let us recall that Lozi maps are maps of the Euclidean plane to itself, given by the formula L a,b (x, y) = (1 + y − a|x|, b).…”

Section: Introductionmentioning

confidence: 99%

“…This paper can be considered the second part of our paper [5]. In [5] we developed three equivalent approaches to compressing information about the symbolic dynamics of Lozi maps.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore for the Lozi-like maps (under suitable assumptions) all results of that paper hold, and the proofs are the same, subject only to obvious modification of terminology. There are only two exceptions, where in [5] we used the results of [2]. For those exceptions we provide new, general proofs in the section about symbolic dynamics.The paper is organized as follows.…”

mentioning

confidence: 99%

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Lozi-like maps

Misiurewicz¹,

Štimac²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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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. dimension for unimodal maps, there is a big difference. The one-dimensional analogue of the Lozi family is the family of tent maps. There we have one parameter and one kneading sequence. For the Lozi family we have two parameters, but infinitely many kneading sequences. Thus, by using a concrete formula (1), we immensely restrict the possible sets of kneading sequences. It makes sense to conjecture that in a generic nparameter subfamily of Lozi-like maps, n kneading sequences determine all other kneading sequences, at least locally. In fact, in our example at the end of this paper, we see that for the Lozi family two kneading sequences may determine the parameter values, and thus all kneading sequences.As we mentioned, we wrote [5] thinking about possible generalizations. Therefore for the Lozi-like maps (under suitable assumptions) all results of that paper hold, and the proofs are the same, subject only to obvious modification of terminology. There are only two exceptions, where in [5] we used the results of [2]. For those exceptions we provide new, general proofs in the section about symbolic dynamics.The paper is organized as follows. In Section 2 we provide the definitions and prove the basic properties of Lozi-like maps. In Section 3 we prove the existence of an attractor. In Section 4 we give two proofs about symbolic dynamics, that are different than in [5]. Finally, in Section 5 we present an example of a three-parameter family of Lozi-like maps. We also show that it is essentially larger than the Lozi family. This last result uses a computer in a not completely rigorous way, so strictly speaking it is not a proof, but a strong numerical evidence. DefinitionsA cone K in R 2 is a set given by a unit vector v and a number ∈ (0, 1) bywhere || · || denotes the usual Euclidean norm. The straight line {tv : t ∈ R} is the axis of the cone. Two cones will be called disjoint if their intersection consists only of the vector 0. Clearly, the image under an invertible linear transformation of R 2 of a cone is a cone, although the image of the axis is not necessarily the axis of the image.Lemma 2.1. Let K u and K s be disjoint cones. Then there is an invertible linear transformation T : R 2 → R 2 such that the x-axis is the axis of T (K u ) and the y-axis is the axis of T (K s ).Proof. We will define T as the composition of three linear transformations. Choose two of the four components of R 2 (K u ∪ K s ), such that they are not images of each other under the central symmetry with respect to the origin. Then choose one vector in each of those components. There is an invertible linear transformation T...

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