Maximal attractors for perturbations of unimodal maps near a homoclinic tangency

Abstract: We show that small planar perturbations of families of unimodal maps have maximal (equal to the intersection of the iterates of their trapping region) attractors at parameters near homoclinic tangency of one of the fixed points. The results hold for general unimodal maps exhibiting certain hyperbolic properties. Hence, admissible families considered include Hénon-like families, Lozi-like families, or perturbations of border collision normal forms. Therefore, our results generalize results of Viana in the case of Hénon-like families and Glendinning and Simpson in the case of border collision normal forms.

• The paper develops admissible families and cone-field estimates to prove that the closure of the unstable manifold is a maximal invariant attractor near a homoclinic tangency.
• Under piecewise uniform hyperbolicity and expansion exceeding √2, the attractor is mixing, has dense periodic orbits, exhibits positive Lyapunov exponents, and qualifies as a topological strange attractor.
• The framework unifies Hénon-like, Lozi-like, and border-collision maps while proving Hausdorff continuity of attractors and identifying unresolved non-wandering-set questions in the orientation-preserving case.

Overview

This paper, by Przemysław Kucharski (2608.18761), studies two-parameter families of planar homeomorphisms obtained as small perturbations of one-dimensional unimodal families, parameterized so as to model the unfolding of a homoclinic tangency of the fixed point $z_{+}$. The main contribution is a proof that, at parameters near the tangency, the closure of the unstable manifold $\overline{W^{u}(z_{+})}$ is not merely an attractor but a maximal invariant set in some trapping region — equivalently, it is a topological attractor equal to $\bigcap_{n\geq 0} f^{n}(T)$ for a trapping region $T$. Under additional piecewise uniform hyperbolicity and high expansion, this maximal set is moreover a topological strange attractor (mixing with positive Lyapunov exponent along a dense orbit) that varies continuously in the Hausdorff metric. The framework, called admissible families, is deliberately broad: it encompasses Wang–Young (Hénon-like) families, Lozi-like families, and strongly dissipative border-collision normal forms (BCNFs), thereby unifying and generalizing earlier results of Viana on Hénon-like maps, Cao–Liu and Kucharski on Lozi maps, and Glendinning–Simpson on BCNFs.

Admissible families and hyperbolicity

An admissible family $\{f_{a,b}\}_{(a,b)\in A\times B}$ consists of a unimodal one-dimensional family $f_{a}$ satisfying exponential expansion outside a neighborhood of the critical point $c$ (with a Misiurewicz-type condition $f_{a_{*}}(c)=q_{+}$), embedded as $(x,y)\mapsto(f_{a}(x),0)$ at $b=0$, plus a $\overline{W^{u}(z_{+})}$0 perturbation whose $\overline{W^{u}(z_{+})}$1 norm is uniformly $\overline{W^{u}(z_{+})}$2. The definition generalizes the Wang–Young setup to include piecewise hyperbolic systems such as BCNFs and Lozi-like maps.

The core technical estimate (Lemma 3.1) establishes invariance of unstable cone fields with coefficient $\overline{W^{u}(z_{+})}$3 under $\overline{W^{u}(z_{+})}$4 and of stable cones under $\overline{W^{u}(z_{+})}$5 away from a neighborhood of the vertical line over the critical point, together with exponential expansion rates inherited (up to constants vanishing with $\overline{W^{u}(z_{+})}$6) from the one-dimensional dynamics. As a consequence, any orbit remaining outside the critical strip lies in a genuinely hyperbolic set with almost-vertical local stable manifolds. The paper notes candidly that while a sketch of this argument exists in Wang–Young's work, full details were absent there; the complete proof is supplied here.

A structural lemma then fixes the geometry for $\overline{W^{u}(z_{+})}$7 small: there exist intervals $\overline{W^{u}(z_{+})}$8 on which the forward-invariant rectangle $\overline{W^{u}(z_{+})}$9 contains both fixed points, $\bigcap_{n\geq 0} f^{n}(T)$0 is an $\bigcap_{n\geq 0} f^{n}(T)$1-curve crossing $\bigcap_{n\geq 0} f^{n}(T)$2, and the local unstable manifolds of $\bigcap_{n\geq 0} f^{n}(T)$3 cross the critical line $\bigcap_{n\geq 0} f^{n}(T)$4. Orientation preserving ($\bigcap_{n\geq 0} f^{n}(T)$5) and reversing ($\bigcap_{n\geq 0} f^{n}(T)$6) cases are treated separately throughout, since the relative positions of the invariant manifolds of the two fixed points differ.

Chaotic attractors in the piecewise hyperbolic case

Assuming piecewise uniform hyperbolicity (uniform exponential growth whenever orbits avoid $\bigcap_{n\geq 0} f^{n}(T)$7) together with the high-expansion condition $\bigcap_{n\geq 0} f^{n}(T)$8, the paper proves that $\bigcap_{n\geq 0} f^{n}(T)$9 is a chaotic attractor carrying Devaney chaos, and indeed that $T$0 is mixing. The key step is showing that every smooth unstable curve in $T$1 eventually crosses both $T$2 and its image simultaneously, forcing intersection with $T$3; otherwise, a length-growth comparison between the bound $T$4 (from folding into at most $T$5 pieces per two iterates) and the lower bound $T$6 contradicts high expansion. This density of stable-manifold intersections implies dense homoclinic intersections, hence horseshoes and dense periodic points; transitivity follows from mixing, and sensitivity from the Banks–Brooks–Cairns–Davis–Stacey criterion.

The paper emphasizes that these results are novel in the stated generality and generalize prior work restricted to specific families. Notably, applying the results to BCNFs yields a smaller chaotic-attractor region than Glendinning and Simpson's explicit region — the authors state this limitation plainly — but additionally produces regions where a topological strange attractor exists, a conclusion absent from Glendinning–Simpson's work. The reduction of strongly dissipative BCNFs (with Jacobian $T$7 decaying faster than $T$8) to admissible families via a linear conjugacy $T$9 is given explicitly, and the Lozi family is recovered as a special case.

Renormalization model

The technical heart of the paper is a first-return construction generalizing Dyi-Shing Ou's renormalization model for orientation-preserving Lozi maps. Pulling back the local stable manifold of $\{f_{a,b}\}_{(a,b)\in A\times B}$0 by the two branches $\{f_{a,b}\}_{(a,b)\in A\times B}$1 yields sequences of $\{f_{a,b}\}_{(a,b)\in A\times B}$2-curves $\{f_{a,b}\}_{(a,b)\in A\times B}$3 (subsets of $\{f_{a,b}\}_{(a,b)\in A\times B}$4) and $\{f_{a,b}\}_{(a,b)\in A\times B}$5 (subsets of $\{f_{a,b}\}_{(a,b)\in A\times B}$6); the strips between consecutive $\{f_{a,b}\}_{(a,b)\in A\times B}$7-curves partition a rectangle $\{f_{a,b}\}_{(a,b)\in A\times B}$8 on which the first return map $\{f_{a,b}\}_{(a,b)\in A\times B}$9 is defined, each return domain $f_{a}$0 being separated by a preimage of the critical line.

The central quantitative result (Proposition 5.2) compares the ordering of the return domains and their images: the position of $f_{a}$1 is order-isomorphic to the geometric sequence $f_{a}$2, where $f_{a}$3 is the stable contraction constant. In particular, in the orientation reversing case the images $f_{a}$4 accumulate on $f_{a}$5 in alternating fashion, whereas in the orientation preserving case they accumulate monotonically — a distinction with quantitative content beyond Ou's original qualitative model. This ordering is precisely what drives the maximality arguments.

Maximality and Hausdorff continuity

Maximality proceeds by a case analysis on the renormalization dynamics. A general principle states that if a disk has boundary contained in $f_{a}$6 and absorbs a point's orbit segment, its $f_{a}$7-limit lies in $f_{a}$8. In the orientation reversing case, maximality holds for open parameter sets once $f_{a}$9 creates a horseshoe, since all returns either remain in the horseshoe's non-wandering set or land in regions bounded by invariant manifolds. In the orientation preserving case, the paper identifies parameters $c$0 where $c$1 lands exactly on the $c$2-th pullback $c$3 of the stable manifold; near such parameters, the critical strip is mapped deep into the return structure, and every orbit is channeled either through a region bounded by arcs of $c$4 and $c$5, or onto almost-vertical stable manifolds intersecting $c$6. Consequently $c$7 for all $c$8, which forces equality of $c$9 and $f_{a_{*}}(c)=q_{+}$0 for any trapping region containing the latter. Combined with mixing, this upgrades the attractor to a topological strange attractor: a point outside the countable union of critical-line images has a dense orbit with positive lower Lyapunov exponent.

A separate section establishes Hausdorff continuity of the attractors as functions of the parameter, provided the attractor equals $f_{a_{*}}(c)=q_{+}$1 on the parameter set. The argument sandwiches $f_{a_{*}}(c)=q_{+}$2 between ascending images of the local unstable manifold and descending images of a continuously varying trapping region, using a lemma on Hausdorff continuity of complements of tubular neighborhoods of continuously varying curves. The authors note that Barge's earlier semi-continuity approach was difficult to extract and that the author's own previous argument was insufficient; the present proof is self-contained and also completes the case of the orientation-preserving Lozi family.

Limitations and open questions

Several restrictions qualify the main results. All conclusions require sufficiently small $f_{a_{*}}(c)=q_{+}$3 (the perturbative regime), so for BCNFs the resulting chaotic/strange-attractor regions do not cover the full parameter regions established by Glendinning and Simpson. The strange-attractor statement requires the strong assumptions of piecewise uniform hyperbolicity and high expansion ($f_{a_{*}}(c)=q_{+}$4). Most significantly, the paper resolves only part of the non-wandering-set problem in the orientation preserving case: it exhibits open parameter sets accumulating on the tangency where $f_{a_{*}}(c)=q_{+}$5 is maximal, but leaves open whether there exist parameters where $f_{a_{*}}(c)=q_{+}$6 is a chaotic attractor yet the non-wandering set contains points outside it. An affirmative answer in the Hénon-like setting would bear directly on coexistence phenomena within the Palis program, and remains unresolved here.

Conclusion

The paper delivers a unified perturbative theory of maximal, topologically strange attractors for planar perturbations of unimodal families near homoclinic tangencies, resting on a quantitative generalization of Ou's renormalization model and on careful cone-field and manifold-position analysis valid across both smooth non-uniformly hyperbolic and piecewise affine settings. Its principal achievement — maximality, with attendant Hausdorff continuity — extends Viana's global-attractor results for Hénon-like maps, Cao–Liu and Kucharski's results for Lozi maps, and Glendinning–Simpson's robust chaos for BCNFs into a single framework, while leaving the fine structure of the non-wandering set in the orientation preserving case as the natural outstanding question.