Accumulation sets and zero entropy dynamics in the Lozi map
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917433600638976 |
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| author | Kvaternik, Kristijan Kilassa |
| author_facet | Kvaternik, Kristijan Kilassa |
| contents | For the family of Lozi maps $L_{a,b}$, we consider parameter pairs for which the f\mbox{}ixed point $X$ has no homoclinic points and the period-two orbit $\{P,P'\}$ is attracting. For such parameters, let $\ell$ be the set of accumulation points of the unstable manifold $W_X^u$ that do not lie on $W_X^u$. We construct a polygon $\mathcal{D}$ whose forward images under $L_{a,b}$ form nested sequences of sets that eventually become trapping. We show that this geometric construction gives a characterization of $\ell$ as the intersection of these iterates. Using this structure, we prove that the non-wandering set for $L_{a,b}^2$ is contained in the union of $\ell$ and the set of f\mbox{}ixed points of $L_{a,b}$. As a consequence, the Lozi map, restricted to the complement of $\ell$ in the plane, has zero topological entropy. This result extends a recent one of Misiurewicz and Štimac to a broader set of parameters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22632 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accumulation sets and zero entropy dynamics in the Lozi map Kvaternik, Kristijan Kilassa Dynamical Systems 37B40, 37B20, 37E30 For the family of Lozi maps $L_{a,b}$, we consider parameter pairs for which the f\mbox{}ixed point $X$ has no homoclinic points and the period-two orbit $\{P,P'\}$ is attracting. For such parameters, let $\ell$ be the set of accumulation points of the unstable manifold $W_X^u$ that do not lie on $W_X^u$. We construct a polygon $\mathcal{D}$ whose forward images under $L_{a,b}$ form nested sequences of sets that eventually become trapping. We show that this geometric construction gives a characterization of $\ell$ as the intersection of these iterates. Using this structure, we prove that the non-wandering set for $L_{a,b}^2$ is contained in the union of $\ell$ and the set of f\mbox{}ixed points of $L_{a,b}$. As a consequence, the Lozi map, restricted to the complement of $\ell$ in the plane, has zero topological entropy. This result extends a recent one of Misiurewicz and Štimac to a broader set of parameters. |
| title | Accumulation sets and zero entropy dynamics in the Lozi map |
| topic | Dynamical Systems 37B40, 37B20, 37E30 |
| url | https://arxiv.org/abs/2604.22632 |