arXiv subjects

Kristijan Kilassa Kvaternik

Publications and source records attributed to Kristijan Kilassa Kvaternik.

3 recordsLinked to original sources

arXiv2026

Basins of attraction and escape in the Lozi map

For the Lozi map $L_{a,b}$, we consider parameter pairs for which the fixed point $X$ in the first quadrant has no homoclinic points and the period-two orbit $\{P,P'\}$ is attracting. For such parameters, let $\ell$ denote the set of accumulation points of the unstable manifold $W_X^u$ that do not belong to $W_X^u$. We completely classify the forward asymptotic behavior of points in the phase space. The forward orbit of every point in the plane either converges to $X$, to the other fixed point $Y$ in the third quadrant, or to $\ell$, or it escapes to infinity. The global phase space is organized by the stable manifolds of the fixed points: $W_Y^s$ separates the basin of $\ell$ from the region of escaping orbits, while $W_X^s$ is the exceptional set of points whose orbits converge to $X$. In particular, if $\mathcal{A}_1$ denotes the component of $\mathbb{R}^2 \setminus W_Y^s$ containing $X$, then $\mathcal{A}_1 \setminus W_X^s$ is precisely the basin of attraction of $\ell$.

math.DS↗

arXiv2026

Accumulation sets and zero entropy dynamics in the Lozi map

For the Lozi map $L_{a,b}$, we consider parameter pairs for which the fixed 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 fixed 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.

math.DS↗

arXiv2026

Tangential homoclinic points for Lozi maps

For the family of Lozi maps, we study homoclinic points for the saddle fixed point $X$ in the first quadrant. Specifically, in the parameter space, we examine the boundary of the region in which homoclinic points for $X$ exist. For all parameters on that boundary, all intersections of the stable and unstable manifold of $X$, apart from $X$, are tangential, or these manifolds intersect along a segment. We ultimately prove that for such parameters, all possible homoclinic points for $X$ are iterates of two special points $Z$ and $V$, or iterates of points on a segment joining $V$ with an iterate of $Z$. Additionally, we describe the parameter curves that form the boundary and provide explicit equations for several of them.

math.DS↗