Cherry Maps and Toral Dynamics

• Cherry maps are circle endomorphisms with a collapsed interval and power-law singularities, serving as one-dimensional models for toral recurrence and non-wandering dynamics.
• They are characterized through both smooth flat-interval and discontinuous gap formulations, which capture transitions from continuous Poincaré return maps to singular, border-collision behaviors.
• Their analysis reveals critical geometric scaling laws and thresholds for Hausdorff dimension, providing actionable insights into bifurcation transitions and conjugacy in dynamical systems.

Cherry maps are circle endomorphisms associated with Cherry flows on the torus. In the flat-interval formulation, they are order-preserving degree-one maps of $S^1$ that collapse an open arc to a point and have controlled power-law singularities at the endpoints of that flat interval; in related work on structural transitions of Cherry flows, the relevant return maps are monotone but discontinuous circle maps with a gap. Across these formulations, Cherry maps serve as one-dimensional models for toral recurrence, rotation theory, non-wandering sets, and bifurcation geometry near singular return mechanisms (Palmisano, 2012, Derks et al., 2019).

1. Definitions and model classes

Several papers study a class $\mathcal L$ or $\mathscr L$ of circle maps with a flat interval $U=(a,b)\subset S^1$ such that $f(U)$ is a single point, $f$ is a diffeomorphism on the complement of $U$, and the boundary of $U$ carries prescribed power-law behavior. In the symmetric case the critical exponent is written $(\ell,\ell)$; in the asymmetric case it is written $(\ell_1,\ell_2)$. The regularity hypotheses vary across the literature: $\mathcal L$0 weakly order preserving maps with a flat interval, $\mathcal L$1 order-preserving circle maps with a flat piece, and $\mathcal L$2 non-decreasing degree-one maps with a flat interval all appear as principal objects of study (Palmisano, 2012, Ndawa, 2021, Palmisano, 2015, Ndawa, 17 Aug 2025).

The lift formalism is standard. A degree-one circle map is represented by a lift $\mathcal L$3 satisfying

$\mathcal L$4

This framework accommodates continuous and discontinuous maps, as well as injective and non-injective cases. In the flat-interval setting, non-injectivity is concentrated on the collapsed interval $\mathcal L$5; in the discontinuous setting, non-surjectivity produces a genuine gap in the image (Derks et al., 2019).

In Palmisano’s $\mathcal L$6 construction, “flat” is literal: for the lift $\mathcal L$7, one has

$\mathcal L$8

That construction shows that a flat interval is compatible with irrational rotation number and Denjoy-type behavior, so the class is much broader than smooth monotone circle diffeomorphisms (Palmisano, 2015).

2. Return maps of Cherry flows

A Cherry flow is described in the cited literature as a flow on $\mathcal L$9 with no closed trajectories and exactly two singularities, one sink and one saddle, both hyperbolic. Every Cherry flow admits a Poincaré global section $\mathscr L$0, and the first return map on this section is order-preserving, constant on an interval $\mathscr L$1, $\mathscr L$2 away from $\mathscr L$3, and near the endpoints of $\mathscr L$4 behaves like a power law whose exponent is determined by the saddle eigenvalues. If the saddle has eigenvalues

$\mathscr L$5

then the singularity exponent is

$\mathscr L$6

The quasi-minimal set of the flow is locally homeomorphic to $\mathscr L$7, where $\mathscr L$8 is an interval and $\mathscr L$9 is the non-wandering set of the return map (Palmisano, 2012).

The return-map viewpoint also appears in a transition problem from Poincaré flows to Cherry flows. In that setting, a continuous monotone return map for a Poincaré flow becomes discontinuous when a parameter crosses a saddle-node mechanism. The local model is

$U=(a,b)\subset S^1$0

followed by the quadratic coordinate change

$U=(a,b)\subset S^1$1

This produces a local normal form that can be embedded into a global flow on the torus by patching with simple vector fields in three regions, allowing the transition from continuous Poincaré behavior to discontinuous Cherry behavior to be tracked explicitly (Derks et al., 2019).

These two return-map descriptions are compatible at the level of dynamical purpose rather than literal map class. The flat-interval theory emphasizes continuous non-invertible circle maps, whereas the discontinuity-creation theory emphasizes injective but discontinuous maps with gaps. Both are organized around how singular flow geometry is encoded by a degree-one circle map (Palmisano, 2012, Derks et al., 2019).

3. Rotation number, quasi-minimal sets, and Denjoy phenomena

For a lift $U=(a,b)\subset S^1$2 of a Cherry map, the rotation number is defined by

$U=(a,b)\subset S^1$3

independently of $U=(a,b)\subset S^1$4, and $U=(a,b)\subset S^1$5. An equivalent formulation used elsewhere is

$U=(a,b)\subset S^1$6

The irrational case is central throughout the literature. One paper states that a Cherry map has a periodic point if and only if its rotation number is rational; another studies exclusively the irrational case in order to analyze geometric scaling and Hausdorff dimension (Palmisano, 2012, Ndawa, 2022).

Cherry maps with a flat interval are not generally conjugate to rigid rotations even when the rotation number is irrational. Palmisano proved that for any irrational number $U=(a,b)\subset S^1$7 there exists a $U=(a,b)\subset S^1$8, non-decreasing circle map $U=(a,b)\subset S^1$9 of degree one and an arc $f(U)$0 such that $f(U)$1 is a point, $f(U)$2 has rotation number $f(U)$3, and $f(U)$4 is a Denjoy counterexample. In the same paper, for continuous non-decreasing degree-one circle maps with irrational rotation number, the following are equivalent: being a Denjoy counterexample, not having dense orbits, having a wandering interval, and the existence of an interval $f(U)$5 with $f(U)$6 as $f(U)$7 (Palmisano, 2015).

The discontinuous Cherry-flow return maps studied in the gap setting retain a well-defined rotation number but are not surjective. Because some orbit segments never return through a certain interval, the map has a gap, and irrational rotation number can coexist with non-dense orbits. The cited work describes this as a natural route to Denjoy-like behavior for injective discontinuous circle maps (Derks et al., 2019).

At the flow level, Palmisano used the flat-interval construction to obtain Cherry flows whose quasi-minimal set is an attractor in the sense of Milnor: its basin has strictly positive Lebesgue measure, and no proper closed subset has the same basin up to a null set. The resulting Cherry flow has the prescribed irrational rotation number, and the basin of attraction of the quasi-minimal set has non-empty interior (Palmisano, 2015).

4. Geometric scaling and Hausdorff dimension

A major theme in the theory of Cherry maps is the dichotomy between degenerate geometry and bounded geometry. In the symmetric $f(U)$8 flat-interval setting, the relevant scale ratios are

$f(U)$9

where $f$0 are the continued-fraction denominators of the irrational rotation number. Degenerate geometry means $f$1; bounded geometry means that $f$2 stays bounded away from $f$3. The central theorem gives a sharp transition at $f$4: if $f$5, then $f$6 at least exponentially fast, whereas if the rotation number is of bounded type and $f$7, then $f$8 is bounded away from zero (Palmisano, 2012).

This geometric transition controls the Hausdorff dimension of the non-wandering set. If $f$9 has critical exponent $U$0, then

$U$1

If $U$2 and the irrational rotation number is of bounded type, then

$U$3

For Cherry flows with saddle eigenvalues $U$4, the same threshold becomes $U$5, and under bounded type one obtains

$U$6

for the quasi-minimal set $U$7 (Palmisano, 2012).

The asymmetric theory generalizes this picture to critical exponents $U$8. For order-preserving $U$9 circle maps with a flat piece, irrational rotation number, and critical exponents $U$0, the non-wandering set is written

$U$1

Under the negative Schwarzian assumption (A1), the geometry is degenerate for

$U$2

and for bounded-type rotation numbers it is bounded for

$U$3

When the rotation number is bi-periodic,

$U$4

the geometry is bounded above a curve $U$5 defined on $U$6. The corresponding Hausdorff-dimension statement is parallel: $U$7 in the degenerate regime and $U$8 in the bounded regime (Ndawa, 2021).

The cited proofs rely on dynamical partitions, cross-ratio inequalities, and recursive estimates for scale ratios. A notable conclusion in the small-exponent regime is that the relevant subsequences of scaling ratios go to zero at least exponentially fast, and in the strictly intermediate case $U$9 at least double exponentially fast. This identifies the critical exponent $(\ell,\ell)$0 as the threshold between degenerate Cantor geometry and bounded Cantor geometry (Ndawa, 2021).

5. Discontinuity creation and the amended Arnold-tongue picture

In the discontinuous return-map formulation, the central question is how a Cherry-flow return map acquires a gap. The cited analysis shows that the natural mechanism is not an arbitrary jump but a singular transition controlled by the saddle-node structure of the flow. As $(\ell,\ell)$1, the map first develops a very steep region, then a discontinuity of finite size. More precisely: for $(\ell,\ell)$2, the map is continuous but becomes extremely steep near the would-be gap; at $(\ell,\ell)$3, the gap opens with finite size; for $(\ell,\ell)$4, the return map is discontinuous, with the slope blowing up at both ends of the jump (Derks et al., 2019).

The asymptotics quantify this transition. Writing

$(\ell,\ell)$5

the local return time satisfies

$(\ell,\ell)$6

so $(\ell,\ell)$7 as $(\ell,\ell)$8. In a very small neighborhood of the critical initial condition, the map stretches by a factor $(\ell,\ell)$9 with

$(\ell_1,\ell_2)$0

Thus the slope becomes enormous as the bifurcation is approached from the Poincaré side, but only on an exponentially narrow set of initial conditions. For $(\ell_1,\ell_2)$1, the map has a finite jump, and near the discontinuity the leading non-constant term behaves like

$(\ell_1,\ell_2)$2

with $(\ell_1,\ell_2)$3, so the derivative diverges at the edge of the gap. The paper describes this as a square-root singularity mechanism; in Cherry flows the singularity appears on both sides of the gap, unlike threshold systems where it is typically one-sided (Derks et al., 2019).

This local singularity changes the organization of phase locking. In smooth monotone circle maps without gaps, the classical Arnold tongue picture has tongue boundaries formed by saddle-node bifurcations. Once a gap is present, the boundary of a tongue may instead be formed partly by border collisions with the endpoints of the gap. The cited work distinguishes type I border collisions, where the derivative is infinite at the gap endpoint, from type II border collisions, where the derivative is finite. For monotone maps with a gap and a square-root singularity, periodic orbits in a phase-locked region can be organized by sequences such as

$(\ell_1,\ell_2)$4

or

$(\ell_1,\ell_2)$5

This is the precise sense in which the Arnold tongue picture is amended once gaps are present (Derks et al., 2019).

6. Conjugacy, bi-Lagrangian structures, and associated connections

A separate line of work places Cherry maps inside a symplectic and bi-Lagrangian framework. On $(\ell_1,\ell_2)$6, a pair of transversal vector fields endowed with a symplectic form defines a bi-Lagrangian structure $(\ell_1,\ell_2)$7. This viewpoint motivates constructions in which dynamics on $(\ell_1,\ell_2)$8 are transported to tangent and cotangent bundles and related back to Cherry maps. In particular, a Cherry vector field $(\ell_1,\ell_2)$9 on $\mathcal L$00, with flow $\mathcal L$01, determines a return-type map

$\mathcal L$02

where $\mathcal L$03 is the minimal positive time carrying $\mathcal L$04 to $\mathcal L$05. The cited proposition states that $\mathcal L$06 belongs to the class $\mathcal L$07 (Ndawa, 2022).

The same papers show that diffeomorphism actions on vector fields induce conjugacy actions on Cherry maps. The push-forward

$\mathcal L$08

induces

$\mathcal L$09

More specifically, if $\mathcal L$10 generates $\mathcal L$11, then $\mathcal L$12 generates $\mathcal L$13. For irrational rotation number, one cited paper states that the conjugacy orbit of a map in $\mathcal L$14 is determined by the rotation number (Ndawa, 2022).

Pairs of Cherry vector fields can also be combined to generate new Cherry maps with asymmetric critical exponents. If $\mathcal L$15 and $\mathcal L$16 generate $\mathcal L$17 with flat pieces $\mathcal L$18, $\mathcal L$19, with $\mathcal L$20 and

$\mathcal L$21

then the piecewise-defined map belongs to $\mathcal L$22, has flat piece $\mathcal L$23, and has critical exponents $\mathcal L$24. This provides an explicit vector-field mechanism for producing left-right asymmetry in a Cherry map (Ndawa, 2022).

The more recent bi-Lagrangian extension develops this correspondence further. For a bi-Lagrangian manifold $\mathcal L$25, the tangent lift

$\mathcal L$26

is again bi-Lagrangian, with Hess connection satisfying

$\mathcal L$27

On the cotangent bundle, the conormal foliations define

$\mathcal L$28

In the Cherry-map setting, for a pair $\mathcal L$29 of transversal Cherry vector fields with the same singular set, the complement of the singularities carries a symplectic form and a bi-Lagrangian structure, hence a Hess connection $\mathcal L$30. The paper then defines the linear connection associated to the Cherry map as the common restriction

$\mathcal L$31

provided these restrictions agree for all generating pairs. This suggests a program in which a Cherry map is studied not only through its one-dimensional dynamics but also through a connection induced by the geometry of the generating flows (Ndawa, 17 Aug 2025).