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S. Štimac

Publications and source records attributed to S. Štimac.

5 recordsLinked to original sources
arXiv2025

The zero entropy locus for the Lozi maps

We study the zero entropy locus for the Lozi maps. We first define a region $R$ in the parameter space and prove that for the parameters in $R$, the Lozi maps have the topological entropy zero. $R$ is contained in a larger region where every Lozi map has a unique period-two orbit, and that orbit is attracting. It is easy to see that the zero entropy locus cannot coincide with that larger region since it contains parameters for which the fixed point of the corresponding Lozi map has homoclinic points.

math.DS↗
arXiv2023

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

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 the Hénon maps this applies to a positive Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters, the Wang-Young 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 the Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967. We also show that no simpler 1-dimensional models exist.

math.DS↗
arXiv2021

On conjugacy between natural extensions of 1-dimensional maps

We prove that for any nondegenerate dendrite $D$ there exist topologically mixing maps $F : D \to D$ and $f : [0, 1] \to [0, 1]$, such that the natural extensions (aka shift homeomorphisms) $σ_F$ and $σ_f$ are conjugate, and consequently the corresponding inverse limits are homeomorphic. Moreover, the map $f$ does not depend on the dendrite $D$, and can be selected so that the inverse limit $\underleftarrow{\lim} (D, F)$ is homeomorphic to the pseudo-arc. The result extends to any finite number of dendrites. Our work is motivated by, but independent of, the recent result of the first and third author on conjugation of Lozi and Hénon maps to natural extensions of dendrite maps.

math.DS↗
arXiv2009

The Ingram Conjecture

We prove the Ingram Conjecture, i.e., we show that the inverse limit spaces of every two tent maps with different slopes in the interval [1, 2] are non-homeomorphic. Based on the structure obtained from the proof, we also show that every self-homeomorphism of the inverse limit space of the tent map is pseudo-isotopic, on the core, to some power of the shift homeomorphism.

math.DS↗