Smoothing Schemes for Topological Conjugacies

• The paper demonstrates that minimal local smoothness, under conditions like uniform expansivity and bounded distortion, can explode into global Cʳ regularity.
• It employs cohomological and geometric techniques to systematically propagate local differentiability across various dynamical systems including interval maps and tilings.
• The findings establish that strict criteria—such as the (D)-property for circle maps and polyhedral windows for model sets—are essential for smoothing rigidity and effective classification.

A smoothing scheme for topological conjugacies refers to the set of methods, criteria, and structural results that guarantee the regularity enhancement (“smoothing”) of a conjugating homeomorphism between dynamical systems, often elevating it from mere continuity (or low regularity) to full differentiability or analytic smoothness, conditional on certain dynamical and geometric properties. Such schemes are foundational in rigidity theory, classification of dynamical systems, and in understanding when qualitative equivalence implies quantitative or geometric equivalence.

1. Rigidity Phenomena in One-Dimensional Dynamics

In the context of one-dimensional multimodal and unimodal interval maps, the explosion of smoothness theorem (Alves et al., 2011) encapsulates a canonical smoothing scheme. Given two $C^r$ multimodal maps $f, g$ (without periodic attractors or neutral points), a topological conjugacy $h$ that is merely $C^1$ at a single point in the expanding set of $f$ undergoes an "explosion" in regularity: $h$ becomes $C^r$ on a dynamically large domain (the basin of a renormalization interval, or globally in the unimodal case). The mechanism is rooted in the property that local differentiability at a point, combined with uniform expansivity and bounded distortion, propagates global smoothness along dynamically defined sets using the "zooming pairs" and "uniformly asymptotically affine" (uaa) control. This rigidity extends to classification, as topological conjugacy suffices to specify differentiable structure under strict dynamical constraints.

Map type	Smoothing criterion	Outcome
Unimodal, no attractors	$h$ is $C^1$ at any boundary	$h$ is $C^r$ globally
Multimodal, no attractors	$h$ is $C^1$ at $p\in$ expanding	$h$ is $C^r$ on basin
With attractors/neutrals	--	No explosion

The general theme is that minimal local regularity, plus sufficient dynamical mixing, induces global smoothness in the conjugacy.

2. Smoothing via Cohomological and Geometric Structures

A distinct smoothing mechanism emerges in the paper of conjugacies for model sets and tilings (Kellendonk et al., 2014). Here, smoothing is interpreted through the lens of mutual local derivability (MLD) and pattern-equivariant cohomology. Under regularity constraints on the internal space ($H\cong \mathbb{R}^n$ or product with a finite group) and polyhedral window, any topological conjugacy between finite local complexity (FLC) model sets is MLD-equivalent to a reprojection—i.e., a smooth change given by a linear homomorphism $L:H\to\mathbb{R}^d$, plus strongly pattern-equivariant corrections.

A conjugacy deformation $F$ is "nonslip" (the smoothing criterion), and its cohomology class is represented by $F(x) = L(o^M(x)) + \phi(x)$, with $L$ linear and $\phi$ pattern-equivariant (locally determined). The group of asymptotically negligible deformations, $H^1_{an}(M,\mathbb{R})$, exactly measures the "degree of smoothing freedom," and its dimension matches that of $H$. When the window fails to be polyhedral, smoothing can collapse: conjugacies need not be reprojections and may fall far outside Meyer sets, emphasizing the geometric necessity for smoothing rigidity.

3. Smoothing Schemes in Circle and Surface Dynamics

Rigidity and smoothing in topological conjugacies also manifest in higher-dimensional or circle scenarios:

• For circle homeomorphisms with breaks (Adouani et al., 2015), the presence of the (D)-property (trivial product of jumps across break points) enables a reduction to piecewise analytic or piecewise linear conjugacies. When the (D)-property fails, smoothing by conjugacy is impossible; singularities persist, and the conjugating homeomorphism is not absolutely continuous.
• In Morse–Smale or Smale-type systems on surfaces or higher manifolds (Medvedev et al., 2018, Grines et al., 2016, Grines et al., 2017), smoothing schemes relate to the algebraic-geometric-moduli scheme describing the dynamics, with conjugacy classes determined by combinatorial invariants such as intersection patterns and numerical moduli at tangencies. Any smoothing of topological conjugacies across these systems must preserve schemes (the entire algebraic-geometric-moduli data).

4. Methodological Core: Propagation, Approximation, and Cohomological Control

Several methodological motifs recur in smoothing schemes:

• Propagation via dynamical structures: Differentiability, when present locally, is spread outwards by dynamical pullbacks and the action of expanding branches.
• Approximation by piecewise linear or monotone functions: As in tent maps and carcass maps (Plakhotnyk, 2017, Plakhotnyk, 2016), piecewise linear approximations ($h_n$) of the conjugacy (constructed from preimages of critical values) converge to $h$, and limits of slopes encode differentiability. Smoothing can then be achieved by modifying these finite-scale approximants appropriately.
• Cohomological obstruction theory: Smoothing is controlled entirely by cohomological classes (pattern-equivariant, asymptotically negligible), with explicit formulas for deformations and rigorous measurement of the available degrees of freedom (Kellendonk et al., 2014).
• Geometric tameness and isotopy: In higher-dimensional topology, particularly for 4-manifolds, smoothing schemes via controlled isotopy (Quinn's finger moves and Gabai's light bulb trick (Cha et al., 2023)) can convert a topologically wild embedding into a smooth one, conditional on the existence of a geometric dual. Smoothing here is constructive and local, emphasizing isotopy obstructions codified by algebraic invariants like the Dax invariant.

5. Smoothing Criteria and Counterexamples

A crucial aspect is the identification of sharp criteria (often expressed as structural obstructions or required numerical equivalences):

• In one-dimensional maps, the preservation of critical orders and absence of neutral/attractor points are necessary for explosion of smoothness.
• In model sets, polyhedral structure of the window (H2) is essential; fractal or irregular windows defeat smoothing rigidity.
• For circle maps, the (D)-property (trivial products over break orbits) gates smoothing of conjugacies; its failure entails non-smooth or singular behavior.
• For group actions on manifolds, bilipschitz bounds ($\epsilon$ sufficiently small, e.g., $1/4000$ (Grillet, 2022)) guarantee tameness and thus the ability to smooth a topological conjugacy to a smooth one.

Counterexamples abound when these criteria are violated: model sets lacking polyhedral windows yield conjugacies that are not reprojections, and circle homeomorphisms with nontrivial jump products cannot be smoothed.

6. Algebraic and Analytical Implications

Smoothing schemes for topological conjugacies not only resolve questions about regularity of conjugacies but also underpin classification theorems and provide explicit avenues for practical computations, rigorous numerical schemes, and rigidity results (Mostow–Sullivan-type rigidity, spectral equivalence, normal forms for bifurcations (Glendinning et al., 2023)). The connection with analytic methods (KAM iterations, cohomological equations (Kalinin et al., 2021, O'Hare, 9 Sep 2024)) links local regularity improvements with deep global structure via Fourier analysis and foliation regularity.

7. Applications and Future Directions

Applications of smoothing schemes span:

• Classification of dynamical systems up to smooth conjugacy and orbit equivalence.
• The determination of when topological equivalence implies measure-theoretic or differentiable equivalence (SRB measures, Lyapunov spectra).
• Construction and understanding of "collapsed" flows and diffeomorphisms in partial hyperbolicity, using smoothing to realize self orbit equivalences (Bowden et al., 17 Oct 2025).
• Topological invariants (e.g., Liouville structures for foliations; Dax invariants for disks/spheres in 4-manifolds) with direct implications for the structure of mapping class groups and isotopy classes.
• Quantitative analysis of rigidity and near-rigidity for systems with matching finite periodic data, yielding effective C^{1+\alpha} conjugacies (O'Hare, 9 Sep 2024), with exponential control of errors.

Further directions include exploration of smoothing schemes for lower regularity systems, extension to actions by non-cyclic groups and systems with complex singularities, and investigations into finer invariants for dynamically intricate systems (e.g., partially hyperbolic diffeomorphisms and higher rank lattice actions).

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In summary, smoothing schemes for topological conjugacies constitute a central paradigm by which local regularity, cohomological structure, geometric constraints, and analytic techniques combine to rigidly upgrade qualitative equivalence to quantitative and geometric equivalence, subject to sharp dynamical and geometric criteria. This interplay not only classifies systems but also exposes fundamental obstructions and provides a constructive framework for resolving regularity questions throughout dynamics, topology, and geometry.