Crooked Horseshoes in Dynamical Systems

• Crooked horseshoes are invariant sets that extend the classical Smale horseshoe concept by exhibiting non-planar, bent, and twisted geometries.
• They arise in settings ranging from non-uniform hyperbolicity near tangencies to rotational dynamics in annular systems and higher-dimensional Hénon-type maps.
• Key analyses reveal their fractal structure, zero Lebesgue measure, and symbolic dynamics that encode folding, stacking, and complex return maps.

“Crooked horseshoes” is an informal term in dynamical systems for invariant sets that retain the combinatorics or symbolic dynamics of a Smale horseshoe while losing its nearly rectangular geometric realization. In the sources considered here, the phrase covers several related phenomena: non-uniformly hyperbolic horseshoes produced near homoclinic or heteroclinic tangencies on surfaces; rotational horseshoes in annular dynamics, where strips wind around the annulus; and higher-dimensional Hénon-type or return-map constructions in which folding occurs along multiple independent crease directions or stacking directions. The common feature is not a single formal definition, but a combination of horseshoe-type symbolic dynamics with strongly bent, twisted, stacked, or near-tangent geometry (Matheus, 2013, Labouriau et al., 2021, Li et al., 10 Sep 2025).

1. Classical horseshoes and the departure from planarity

The reference model is the classical Smale horseshoe. One starts with a rectangle $R\subset \mathbb{R}^2$, stretches it in one unstable direction, contracts it in one stable direction, folds the elongated strip, and places it back so that two strips of $f(R)$ lie inside $R$. The maximal invariant set

$\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$

is a totally disconnected basic set of saddle type; on $\Lambda$, the dynamics is conjugate, or at least semi-conjugate, to a full shift or a subshift of finite type. In the uniformly hyperbolic setting, one has a continuous splitting $T_xM=E^s(x)\oplus E^u(x)$ with uniform contraction on $E^s$ and uniform expansion on $E^u$ (Matheus, 2013).

The adjective “crooked” becomes relevant when that standard picture ceases to be geometrically planar. In higher-dimensional formulations, one still writes phase space as $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$ with a product domain $R=I^u\times I^s$, but the image of a quasi-$f(R)$0-dimensional sheet need not be folded along a single crease direction. In the language of recent Hénon-type constructions, crookedness refers to sheets that are bent, twisted, stretched in multiple directions, or stacked in complicated ways, with folds that may occur along independent creases and, in four dimensions, even along independent stacking directions. This distinguishes genuinely higher-dimensional horseshoes from trivial products of a planar horseshoe with linear directions (Li et al., 10 Sep 2025, Li et al., 2023).

2. Non-uniformly hyperbolic crooked horseshoes near tangencies

A major surface-dynamical usage of the term refers to the non-uniformly hyperbolic horseshoes constructed by Palis–Yoccoz and studied fractally by Matheus–Palis–Yoccoz. The setting begins with a uniformly hyperbolic horseshoe $f(R)$1 for a surface diffeomorphism and a heteroclinic quadratic tangency between $f(R)$2 and $f(R)$3 for periodic points in $f(R)$4. After localizing the dynamics in $f(R)$5 and unfolding the tangency, one defines

$f(R)$6

For most parameters $f(R)$7 in the side of the unfolding with two transverse intersections, and under the Palis–Yoccoz “slightly fat” condition

$f(R)$8

$f(R)$9 is transitive, totally disconnected, of saddle type, and topologically horseshoe-like, but it is not uniformly hyperbolic because stable and unstable directions become nearly tangent in a small critical region (Matheus, 2013).

Geometrically, the defining objects are the parabolic tongues created by the unfolding. Outside those tongues the dynamics remains uniformly hyperbolic, while passages through the tongues implement a folding map and generate near-critical returns. The resulting invariant set is “crooked” because stable and unstable laminations are threaded through these tongues and acquire complicated near-tangent geometry. Hyperbolicity survives only in the non-uniform sense: the set supports geometric SRB measures with nonzero Lyapunov exponents, yet uniform bounds fail because some orbits make recurrent visits to the critical region (Matheus, 2013).

The fractal geometry of these crooked horseshoes is unusually rigid. Matheus–Palis–Yoccoz prove that

$R$0

strengthening the earlier statement that both stable and unstable sets have Lebesgue measure zero. In a substantial subregion $R$1 of the Palis–Yoccoz domain they further obtain

$R$2

Thus, although the geometry is bent by tangency and parabolic return, the stable and unstable sets still obey dimension formulas close to the uniformly hyperbolic model (Matheus, 2013).

3. Rotational horseshoes and annular crookedness

In annular dynamics, the closely related formal notion is a rotational horseshoe. For an orientation-preserving homeomorphism of the infinite annulus $R$3, a compact invariant set $R$4 is a rotational horseshoe if there exists a finite partition of $R$5, an itinerary map $R$6 that is a semiconjugacy to the full shift, and a lift $R$7 such that the displacement of $R$8 is approximated, up to uniformly bounded error, by a sum of symbol-dependent horizontal vectors $R$9. The symbolic dynamics is therefore coupled to a controlled winding in the angular direction (Labouriau et al., 2021).

The geometry motivating the “crooked” label is explicit in the return map derived for periodically forced flows on $\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$0. Starting from an attracting heteroclinic network, one first breaks the network to obtain attracting invariant tori, then turns on periodic forcing. Near the former torus, the Poincaré map on an annulus has the form

$\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$1

where $\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$2 contains a logarithmic twist term

$\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$3

and $\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$4 is strongly contracting in the radial variable. For large forcing frequency $\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$5, horizontal segments are sent to spiral-like curves that wrap around the annulus before reentering the domain. Conley–Moser conditions then produce a compact invariant set conjugate to the full shift on two symbols, with topological entropy at least $\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$6 (Labouriau et al., 2021).

In this usage, crookedness is not produced by tangency or by multidimensional folding of a cube, but by twist. The image of a strip is no longer a straight horseshoe strip inside a rectangle; it winds around the annulus and intersects the domain after one or more turns. The route described in the source is

$\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$7

so the crooked horseshoe appears as an intermediate hyperbolic stage between quasiperiodic dynamics and dissipative strange attractors (Labouriau et al., 2021).

4. Higher-dimensional folded horseshoes in Hénon-type maps

A different line of work constructs explicit polynomial models of crooked horseshoes in three and four dimensions. Li–Fujioka–Shudo introduce Hénon-type maps whose intersections $\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$8 realize new topological horseshoe types impossible in two dimensions. Their 3D map $\Lambda=\bigcap_{n\in\mathbb{Z}} f^n(R)$9 gives a singly folded horseshoe; $\Lambda$0 gives a genuinely 3D doubly folded horseshoe with independent creases but a common stacking direction; the coupled 4D map $\Lambda$1 has a doubly folded regime, Topology III-A, and a singly folded regime, Topology III-B; and their 4D Topology IV thickens a 3D doubly folded horseshoe by a contracting direction (Li et al., 2023).

The coupled Hénon map of Part I supplies the explicit anti-integrable organization of two of these regimes. In the weak-coupling limit, one obtains a four-symbol horseshoe, with four local states $\Lambda$2; in the strong-coupling limit $\Lambda$3, only two real local states remain, yielding a two-symbol horseshoe. The same 4D symplectic normal form therefore supports two topologically different horseshoes, and in the parameter wedges identified in the source both are uniformly hyperbolic (Fujioka et al., 2023).

The later paper on paperfolding templates reframes these constructions as explicit geometric templates. Foldings such as $\Lambda$4 and $\Lambda$5 encode which direction is creased and along which axis the sheet is stacked. In three dimensions, the composition

$\Lambda$6

models a doubly folded sheet with two orthogonal crease directions and a common stacking direction. In four dimensions, the composition

$\Lambda$7

produces independent folding and stacking in the $\Lambda$8 and $\Lambda$9 directions; and the triple composition

$T_xM=E^s(x)\oplus E^u(x)$0

gives an 8-layer stack (Li et al., 10 Sep 2025).

Construction	Geometric feature	Symbolic dynamics
3D $T_xM=E^s(x)\oplus E^u(x)$1	Single fold	Full shift on 2 symbols
3D $T_xM=E^s(x)\oplus E^u(x)$2	Double fold, common stacking direction $T_xM=E^s(x)\oplus E^u(x)$3	Full shift on 4 symbols
4D coupled Hénon, Type A / Topology III-A	Double fold, independent stacking directions	Full shift on 4 symbols
4D coupled Hénon, Type B / Topology III-B	Single effective fold after coordinate change	Full shift on 2 symbols
4D triple-fold map	Three orthogonal folds, 8-layer stack	Full shift on 8 symbols

In these models, “crooked horseshoe” names a specifically geometric generalization of the Smale picture. The invariant set is still built from horizontal disks, horizontal slices, and slabs, but the count of branches is determined by the number of independent foldings: one fold gives $T_xM=E^s(x)\oplus E^u(x)$4 branches, two folds give $T_xM=E^s(x)\oplus E^u(x)$5, and three folds give $T_xM=E^s(x)\oplus E^u(x)$6. A plausible implication is that paperfolding operations provide a combinatorial language for classifying families of higher-dimensional horseshoes that would otherwise be visible only through complicated slices or projections (Li et al., 10 Sep 2025).

5. Three-dimensional horseshoes near a Hopf–Hopf network

A further higher-dimensional realization occurs near an unfolding of a Hopf–Hopf singularity in $T_xM=E^s(x)\oplus E^u(x)$7. The setting is a one-parameter family of $T_xM=E^s(x)\oplus E^u(x)$8 vector fields with a heteroclinic cycle involving a bifocus equilibrium $T_xM=E^s(x)\oplus E^u(x)$9 and two hyperbolic periodic solutions $E^s$0. At the organizing center, the network is

$E^s$1

For $E^s$2 small, the two-dimensional connections persist, while the three-dimensional manifolds $E^s$3 and $E^s$4 intersect transversely along two distinct tubular sets $E^s$5 (Ibáñez et al., 30 Apr 2025).

The local maps near $E^s$6, $E^s$7, and $E^s$8 all contain logarithmic angular terms. When composed with the global transitions, they produce a return map whose action on cross-sections sends annuli to spiralling sheets and bounded regions to scrolls. Proposition 2.1 in the source shows that, for $E^s$9 small, there exist infinitely many two-dimensional heteroclinic connections from $E^u$0 to $E^u$1. These are encoded in the repeated intersections of spiralling sheets with transverse tori (Ibáñez et al., 30 Apr 2025).

The horseshoe construction then proceeds by decomposing the return geometry into horizontal and vertical slabs in the sense of Wiggins. For $E^u$2 large, one obtains pairwise disjoint compact invariant sets $E^u$3, each supported on two branches and each conjugate to the full shift on two symbols. The union

$E^u$4

has zero Lebesgue measure and accumulates on the circles $E^u$5 associated with the transverse tubes $E^u$6. The source further identifies $E^u$7 with the local heteroclinic class of $E^u$8 and $E^u$9, and proves infinite forward and backward switching along the network (Ibáñez et al., 30 Apr 2025).

These objects are crooked in a distinct sense. The horseshoe branches are not merely multiple strips inside a fixed box; they are narrow, highly distorted pieces inside scrolls winding around invariant tori. The source explicitly attributes that distortion to rotation: without the complex eigenvalues and Floquet multipliers, the logarithmic spiralling would be absent, and the return would degenerate toward a simpler one-dimensional mechanism (Ibáñez et al., 30 Apr 2025).

6. Boundaries of the horseshoe locus and thermodynamic aspects

The Hénon family

$\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$0

provides a parameter-space description of where a uniformly hyperbolic horseshoe ceases to be uniformly hyperbolic. Let $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$1 denote the hyperbolic horseshoe locus and $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$2 the maximal entropy locus. The source proves that there is a real-analytic tangency curve $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$3 such that $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$4 is a hyperbolic horseshoe iff $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$5, and $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$6 iff $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$7. At $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$8, the map has exactly one orbit of tangencies between invariant manifolds of suitable fixed points, homoclinic for $\mathbb{R}^{n_u}\oplus \mathbb{R}^{n_s}$9 and heteroclinic for $R=I^u\times I^s$0 (Arai et al., 2018).

The geometry of this boundary is unusually precise. The function $R=I^u\times I^s$1 is strictly monotone decreasing on $R=I^u\times I^s$2 and strictly monotone increasing on $R=I^u\times I^s$3; the Chebyshev parameter $R=I^u\times I^s$4 is the unique corner of the extended boundary; and

$R=I^u\times I^s$5

This gives a concrete parameter-space model of the first bifurcation from a clean Smale horseshoe to a tangency-ridden, non-uniformly hyperbolic geometry with the same maximal entropy $R=I^u\times I^s$6 (Arai et al., 2018).

The same source connects this first-bifurcation geometry to zero-temperature thermodynamic formalism. Writing $R=I^u\times I^s$7 for the unstable Lyapunov exponent of an invariant measure $R=I^u\times I^s$8, and considering equilibrium measures for the potential $R=I^u\times I^s$9, it proves that near $f(R)$00, every $f(R)$01-ground state is Lyapunov minimizing and entropy maximizing among Lyapunov minimizing measures. This does not define crooked horseshoes by itself, but it identifies a distinguished variational structure precisely at parameters where a uniformly hyperbolic horseshoe first becomes tangential and geometrically distorted (Arai et al., 2018).