Accumulation sets and zero entropy dynamics in the Lozi map

Kristijan Kilassa Kvaternik

arxiv: 2604.22632 · v1 · submitted 2026-04-24 · 🧮 math.DS

Accumulation sets and zero entropy dynamics in the Lozi map

Kristijan Kilassa Kvaternik This is my paper

Pith reviewed 2026-05-08 09:35 UTC · model grok-4.3

classification 🧮 math.DS

keywords Lozi maptopological entropyaccumulation setunstable manifoldnon-wandering settrapping regionpiecewise linear map

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The pith

The Lozi map restricted to the complement of an accumulation set on the unstable manifold has zero topological entropy for an open range of parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the Lozi family of piecewise-linear maps for parameters where a fixed point has no homoclinic points and a period-two orbit is attracting. It builds a polygon whose forward images create nested trapping regions whose intersection exactly equals the accumulation set ℓ of the unstable manifold of the fixed point. This geometric description shows that every non-wandering point of the second iterate of the map lies either in ℓ or at one of the fixed points. As a direct result, the map has zero topological entropy when restricted to the plane minus ℓ. The argument extends an earlier entropy result to a larger open set of parameters.

Core claim

For parameter pairs (a, b) where the fixed point X has no homoclinic points and the period-two orbit {P, P'} is attracting, the non-wandering set of L_{a,b}^2 is contained in the union of ℓ and the fixed points of L_{a,b}, where ℓ is the set of accumulation points of the unstable manifold W_X^u that do not lie on W_X^u. Consequently the map has zero topological entropy when restricted to the complement of ℓ.

What carries the argument

The polygon D whose forward images under L_{a,b} form nested trapping sequences whose intersection is exactly the accumulation set ℓ of the unstable manifold.

If this is right

The non-wandering set of L_{a,b}^2 lies inside ℓ union the fixed points.
Topological entropy vanishes on the complement of ℓ in the plane.
The trapping iterates of the polygon characterize ℓ completely.
The zero-entropy conclusion holds throughout the open parameter region defined by the no-homoclinic and attracting-orbit conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same trapping-region method could locate accumulation sets and zero-entropy regions in other piecewise-linear maps that satisfy analogous homoclinic conditions.
Numerical orbit sampling outside ℓ should show all trajectories eventually escape or approach fixed points or the period-two orbit.
The result suggests that positive entropy in the Lozi family may be confined to thin invariant sets for wider parameter intervals than previously known.

Load-bearing premise

The parameters must be restricted to those where the fixed point has no homoclinic points and the period-two orbit attracts.

What would settle it

Exhibiting a non-wandering point of L_{a,b}^2 that lies outside both ℓ and the fixed points would contradict the claimed containment.

Figures

Figures reproduced from arXiv: 2604.22632 by Kristijan Kilassa Kvaternik.

**Figure 1.** Figure 1: The stable (red) and unstable (blue) manifold of X for parameter values a = 1.06, b = 0.96, together with some iterates of Z. Proof. In general, if a point lies in the first quadrant below ZZ−1 , then its image under La,b lies in the first or second quadrant below ZZ−1 and above the line x = 1 − a b y. By contradiction, assume that all forward iterates of A lie in the first quadrant below ZZ−1 , that is, … view at source ↗

**Figure 2.** Figure 2: Construction of the point T and the polygon D. In both cases, we define D to be the polygon with boundary ∂D = [Z, T] u ∪ ZT . (3.1) Note that in both cases, T lies on Wu+ X , satisfies Wu X ∩ ZT = {Z, T}, and has the property that Wu+ X \ [X, T] u ⊂ Int D. Geometrically, D is a region bounded by a portion of Wu X and the segment ZT, which contains the forward branch Wu+ X \ [X, T] u . Corollary 3.3. Let D… view at source ↗

**Figure 3.** Figure 3: Polygon K and its image under L 2 a,b (Lemma 3.7). Lemma 3.7. Let the point T and polygon D be as in (3.1). Then ℓR = \∞ k=0 L 2k a,b(D). Proof. Since ℓR ⊆ D, D is L 2 a,b-invariant (Corollary 3.3) and ℓR is both L 2 a,b- and L −2 a,binvariant, it directly follows that ℓR ⊆ T∞ k=0 L 2k a,b(D). It remains to show that T∞ k=0 L 2k a,b(D) ⊆ ℓR. To prove that, let K be the polygon with the boundary ∂K = T Z ∪… view at source ↗

**Figure 4.** Figure 4: Construction of a path from Q to X′ (Lemma 3.10). The path from Q′ to X′ is contained in an open neighborhood of (X, Q1) u which contains only one arc component of Wu X (Corollary 3.6). Lemma 3.10. The set U ′ is path-connected. Proof. Let the polygon D and the point T be as in (3.1). Notice that the complement of the set N ∪ D ∪ La,b(D) in the extended plane is path-connected since it is open and connecte… view at source ↗

read the original abstract

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 \v{S}timac to a broader set of parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends zero-entropy results for the Lozi map with a new polygon trapping construction for the accumulation set, but the link from non-wandering containment to zero entropy on the non-compact complement needs explicit justification.

read the letter

The paper's main contribution is a polygon D whose forward iterates under the Lozi map nest and trap, with their intersection giving a precise description of the accumulation set ℓ of the unstable manifold. This lets them prove that for the square of the map, the non-wandering set lives inside ℓ together with the fixed points, under the stated parameter conditions. From there they conclude zero topological entropy for the dynamics off ℓ. What is new is the explicit polygon construction and the resulting characterization of ℓ. It goes beyond the earlier Misiurewicz-Štimac result by covering a larger open set of parameters where the period-two orbit attracts and the fixed point has no homoclinics. The argument relies on standard unstable manifold properties plus this new trapping region, so it is not just a rehash. The construction itself is the strong part. Building nested trapping sets from a polygon and showing they intersect exactly at the accumulation points gives a tangible geometric picture that was only partial before. The potential issue is whether the zero-entropy statement follows cleanly. In the plane minus ℓ, which is non-compact, having the non-wandering set confined to ℓ plus fixed points does not rule out positive entropy from orbits that wander for a long time before approaching ℓ. The abstract calls it a consequence, but a full argument would need to control the number of (n,ε)-separated points during those transients, maybe by tracking the branches of the map or the shape of the iterates of D. If the paper supplies that control, fine; if it leans only on the containment, the step is not automatic and could be a gap. The parameter window is narrow but clearly described, so no overclaim there. This work is aimed at specialists in one-dimensional and piecewise-linear dynamics who care about entropy boundaries and manifold accumulation. A reader interested in concrete examples for chaotic maps would get value from the explicit sets. It is worth sending to peer review because the geometric novelty is real and the extension is substantive, even if the entropy implication needs a referee to verify the transient handling.

Referee Report

2 major / 2 minor

Summary. The manuscript considers the Lozi family L_{a,b} restricted to parameter pairs where the fixed point X has no homoclinic points and the period-two orbit {P, P'} is attracting. It defines ℓ as the set of accumulation points of the unstable manifold W_X^u that lie off the manifold itself. A polygon D is constructed whose forward images under L_{a,b} form nested trapping regions; the authors show that ℓ coincides with the intersection of these iterates. They then prove that the non-wandering set of the second iterate L_{a,b}^2 is contained in ℓ union the fixed points of L_{a,b}. As a consequence, the restriction of L_{a,b} to the complement of ℓ in the plane has zero topological entropy. The result extends a theorem of Misiurewicz and Štimac to a larger open set of parameters.

Significance. If the containment and entropy claims are fully rigorous, the work supplies an explicit geometric characterization of the accumulation set ℓ together with a concrete trapping construction that controls the dynamics outside it. This extends the range of parameters for which zero-entropy behavior on the complement is known and illustrates how piecewise-linear geometry can be used to analyze non-wandering sets in non-compact phase space. The explicit polygon D and the nesting argument constitute a verifiable, constructive contribution to the study of Lozi maps.

major comments (2)

[the paragraph containing the entropy conclusion (immediately following the non-wandering-set containment statement)] The deduction that zero topological entropy on ℝ² ∖ ℓ follows from the non-wandering set of L_{a,b}^2 being contained in ℓ ∪ {fixed points} (stated as a consequence after the containment theorem) is not automatic in a non-compact space. Topological entropy is the exponential growth rate of (n,ε)-separated sets; finite non-wandering sets do not preclude positive entropy during transients (e.g., expanding maps on ℝ). The manuscript must supply an explicit bound on the number of (n,ε)-separated orbits in the complement, using either the piecewise-linear branches of L_{a,b} or the geometry of the nested trapping regions generated by D, rather than treating the implication as immediate.
[the section presenting the polygon D and the intersection characterization of ℓ] In the geometric construction of the polygon D and the proof that its forward images intersect precisely at ℓ (the characterization theorem), it is necessary to confirm that every orbit starting in the complement eventually enters the trapping regions and that no additional accumulation points arise outside the stated intersection. The nesting argument should be checked against possible escapes along the unstable manifold branches before the trapping takes effect.

minor comments (2)

[abstract] The abstract contains LaTeX artifacts (e.g., “fmbox{}ixed”) that should be removed in the final version.
[figures] Ensure that every figure is referenced in the text and that the caption explicitly indicates which iterates of D are shown and how they relate to the trapping property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying points where additional rigor is needed to handle the non-compact setting and to fully verify the trapping construction. We address each major comment below and will revise the paper accordingly to strengthen the arguments.

read point-by-point responses

Referee: The deduction that zero topological entropy on ℝ² ∖ ℓ follows from the non-wandering set of L_{a,b}^2 being contained in ℓ ∪ {fixed points} (stated as a consequence after the containment theorem) is not automatic in a non-compact space. Topological entropy is the exponential growth rate of (n,ε)-separated sets; finite non-wandering sets do not preclude positive entropy during transients (e.g., expanding maps on ℝ). The manuscript must supply an explicit bound on the number of (n,ε)-separated orbits in the complement, using either the piecewise-linear branches of L_{a,b} or the geometry of the nested trapping regions generated by D, rather than treating the implication as immediate.

Authors: We agree that the zero-entropy conclusion does not follow automatically from the non-wandering-set containment in a non-compact space, and that an explicit bound on separated orbits is required. The nested trapping regions produced by the iterates of D ensure that every orbit outside ℓ enters a fixed bounded domain after finitely many steps. Within this domain the second iterate has only the fixed points as non-wandering points, and the piecewise-linear branches have controlled expansion. We will add a new lemma that uses the geometry of the trapping regions to give an explicit upper bound on the cardinality of any (n,ε)-separated set in the complement, showing that the exponential growth rate is zero. This revision will make the entropy claim fully rigorous. revision: yes
Referee: In the geometric construction of the polygon D and the proof that its forward images intersect precisely at ℓ (the characterization theorem), it is necessary to confirm that every orbit starting in the complement eventually enters the trapping regions and that no additional accumulation points arise outside the stated intersection. The nesting argument should be checked against possible escapes along the unstable manifold branches before the trapping takes effect.

Authors: The polygon D is constructed to intersect the unstable manifold branches so that the forward images trap all orbits whose accumulation behavior is not that of ℓ. To address the referee’s concern explicitly, we will insert an additional proposition that verifies two facts: (i) no orbit in the complement can escape indefinitely along the unstable manifold branches without entering the nested sequence, and (ii) the intersection of the forward images coincides exactly with ℓ, with no extraneous accumulation points outside this intersection. The proof will rely on the absence of homoclinic points at X together with the attracting character of the period-two orbit. This addition will confirm that the characterization theorem is complete. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit geometric construction of trapping polygon independent of entropy conclusion

full rationale

The derivation begins with the explicit definition of ℓ as accumulation points of W_X^u excluding points on W_X^u itself. It then introduces a concrete polygon D and proves that its forward iterates under L_{a,b} are nested and trapping, with their intersection exactly equal to ℓ. From this geometric characterization the paper derives the containment of the non-wandering set of L_{a,b}^2 inside ℓ union the fixed points, and states the zero-entropy consequence on the complement. None of these steps reduces to a self-definition, a fitted parameter re-used as a prediction, or a load-bearing self-citation; the cited Misiurewicz–Štimac result is treated as an external benchmark being extended rather than an input that forces the present conclusion. The argument is therefore self-contained via direct construction and standard manifold properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard facts about hyperbolic fixed points, unstable manifolds, and topological entropy in piecewise-smooth maps; no new entities are postulated and no parameters are fitted to data.

axioms (2)

standard math Unstable manifolds of hyperbolic fixed points are well-defined immersed curves whose forward iterates accumulate on invariant sets.
Invoked when defining ℓ as the accumulation points of W_X^u not on the manifold itself.
standard math Topological entropy is zero on a set if the non-wandering set of the iterate is finite or consists of isolated points.
Used to conclude zero entropy once the non-wandering set is shown to lie in ℓ union fixed points.

pith-pipeline@v0.9.0 · 5492 in / 1562 out tokens · 35410 ms · 2026-05-08T09:35:30.066792+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Ishii, D

Y. Ishii, D. Sands,Monotonicity of the Lozi family near the tent maps, Commun. Math. Phys.198 (1998), 397–406

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K. Kilassa Kvaternik,Tangential Homoclinic Points for Lozi Maps, J. Dyn. Diff. Equat. (2006), https://doi.org/10.1007/s10884-026-10505-2

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K. Kilassa Kvaternik,Tangential homoclinic points locus of the Lozi maps and applications, Doctoral Thesis, University of Zagreb, Croatia (2022), available online:https://repozitorij.pmf.unizg. hr/islandora/object/pmf:11546

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Misiurewicz,Strange attractor for the Lozi mappings, Ann

M. Misiurewicz,Strange attractor for the Lozi mappings, Ann. New York Acad. Sci.357(1980) (Nonlinear Dynamics), 348–358

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Misiurewicz, S

M. Misiurewicz, S. ˇStimac,The zero entropy locus for the Lozi maps, Monatsh. Math.208(2025), 471–484

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I. B. Yildiz,Discontinuity of Topological Entropy for the Lozi Maps, Ergod. Theory Dyn. Syst.32 (2012), 1783–1800 18 KRISTIJAN KILASSA KV ATERNIK (K. Kilassa Kvaternik)University of Zagreb, Faculty of Electrical Engineering and Com- puting, Department of Applied Mathematics, Unska 3, 10 000 Zagreb, Croatia – and – Jagiellonian University, Institute of Mat...

work page 2012

[1] [1]

M. Brin, G. Stuck,Introduction to dynamical systems, Cambridge University Press, 2002

work page 2002

[2] [2]

Burns, H

K. Burns, H. Weiss,A geometric criterion for positive topological entropy, Communications in math- ematical physics172(1995), 95–118

work page 1995

[3] [3]

Franks,A New Proof of the Brouwer Plane Translation Theorem, Ergod

J. Franks,A New Proof of the Brouwer Plane Translation Theorem, Ergod. Theory Dyn. Syst.12 (1992), 217–226

work page 1992

[4] [4]

Ishii,Towards a kneading theory for Lozi mappings I: A solution of the prunning front conjecture and the first tangency problem, Nonlinearity10(1997), 731–747

Y. Ishii,Towards a kneading theory for Lozi mappings I: A solution of the prunning front conjecture and the first tangency problem, Nonlinearity10(1997), 731–747

work page 1997

[5] [5]

Ishii,Towards a kneading theory for Lozi mappings II: Monotonicity of the topological entropy and Hausdorff dimension of attractors, Commun

Y. Ishii,Towards a kneading theory for Lozi mappings II: Monotonicity of the topological entropy and Hausdorff dimension of attractors, Commun. Math. Phys.190(1997), 375–394

work page 1997

[6] [6]

Ishii, D

Y. Ishii, D. Sands,Monotonicity of the Lozi family near the tent maps, Commun. Math. Phys.198 (1998), 397–406

work page 1998

[7] [7]

Katok, B

A. Katok, B. Hasselblatt,Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995

work page 1995

[8] [8]

Kilassa Kvaternik,Tangential Homoclinic Points for Lozi Maps, J

K. Kilassa Kvaternik,Tangential Homoclinic Points for Lozi Maps, J. Dyn. Diff. Equat. (2006), https://doi.org/10.1007/s10884-026-10505-2

work page doi:10.1007/s10884-026-10505-2 2006

[9] [9]

K. Kilassa Kvaternik,Tangential homoclinic points locus of the Lozi maps and applications, Doctoral Thesis, University of Zagreb, Croatia (2022), available online:https://repozitorij.pmf.unizg. hr/islandora/object/pmf:11546

work page 2022

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Lozi,Un attracteur ´ etrange(?) du type attracteur de H´ enon, J

R. Lozi,Un attracteur ´ etrange(?) du type attracteur de H´ enon, J. Physique (Paris)39(Coll. C5) (1978), 9–10

work page 1978

[11] [11]

Misiurewicz,Strange attractor for the Lozi mappings, Ann

M. Misiurewicz,Strange attractor for the Lozi mappings, Ann. New York Acad. Sci.357(1980) (Nonlinear Dynamics), 348–358

work page 1980

[12] [12]

Misiurewicz, S

M. Misiurewicz, S. ˇStimac,The zero entropy locus for the Lozi maps, Monatsh. Math.208(2025), 471–484

work page 2025

[13] [13]

I. B. Yildiz,Monotonicity of the Lozi family and the zero entropy locus, Nonlinearity24(2011), 1613–1628

work page 2011

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I. B. Yildiz,Discontinuity of Topological Entropy for the Lozi Maps, Ergod. Theory Dyn. Syst.32 (2012), 1783–1800 18 KRISTIJAN KILASSA KV ATERNIK (K. Kilassa Kvaternik)University of Zagreb, Faculty of Electrical Engineering and Com- puting, Department of Applied Mathematics, Unska 3, 10 000 Zagreb, Croatia – and – Jagiellonian University, Institute of Mat...

work page 2012