Dynamical Systems on Nam Le

$C^r$ Stability Conjecture

Thu, 28 May 2026 00:00:00 +0000

Structural stability is a global topological property: a dynamical system is structurally stable if all nearby systems have the same orbit structure, up to continuous reparametrisation. Hyperbolicity is a local differential property: the tangent bundle over the recurrent set splits into uniformly contracting and expanding directions. That these two conditions should be equivalent is one of the deepest principles in smooth dynamics.

Conjecture ($C^r$ Stability Conjecture, Palis–Smale, ~1970)

Let $M$ be a closed smooth manifold and $r \geq 1$. If $f \in \mathrm{Diff}^r(M)$ is $C^r$-structurally stable, then $f$ is hyperbolic, i.e., it satisfies Axiom A and the Strong Transversality Condition.

The problem is rated L3 on UnsolvedMath and sits at the heart of the global theory of smooth dynamical systems. The case $r = 1$ is resolved. The case $r \geq 2$ is open, and even basic consequences of structural stability that are elementary for $r = 1$ remain unknown for $r = 2$.

Key Definitions #

Structural stability. A diffeomorphism $f \in \mathrm{Diff}^r(M)$ is $C^r$-structurally stable if there exists a $C^r$-neighborhood $\mathcal{U}$ of $f$ such that every $g \in \mathcal{U}$ is topologically conjugate to $f$: there is a homeomorphism $h : M \to M$ with $h \circ f = g \circ h$. The system is therefore robust under $C^r$-small perturbations in the strongest possible sense: topology, not just orbit counts, is preserved.

Axiom A. The diffeomorphism $f$ satisfies Axiom A if:

the non-wandering set $\Omega(f)$ is hyperbolic: there is a $Df$-invariant splitting $T_x M = E^s_x \oplus E^u_x$ over $\Omega(f)$ with uniform exponential contraction on $E^s$ and expansion on $E^u$;
the periodic points of $f$ are dense in $\Omega(f)$.

Strong Transversality Condition (STC). For every $x, y \in \Omega(f)$, the stable manifold $W^s(x)$ and the unstable manifold $W^u(y)$ intersect transversally. Tangential intersections, namely homoclinic or heteroclinic tangencies, are forbidden.

Together, Axiom A and the STC constitute what is usually meant by saying $f$ is hyperbolic in the sense of the stability conjecture.

The Two Directions #

The conjecture, as an equivalence, has an easy direction and a hard direction.

Structural stability follows from hyperbolicity (the easy direction). Robbin (1971) proved this for $C^2$ diffeomorphisms; Robinson (1976) extended it to $C^1$. Both proofs use the implicit function theorem on an appropriate space of conjugacies, and work for all $r \geq 1$ since Axiom A + STC is the hypothesis.

Theorem (Robbin 1971, Robinson 1976)

For every $r \geq 1$, if $f \in \mathrm{Diff}^r(M)$ satisfies Axiom A and the Strong Transversality Condition, then $f$ is $C^r$-structurally stable.

Hyperbolicity follows from structural stability (the hard direction) is the conjecture itself. It requires understanding what structural stability forces on the dynamics, ruling out every non-hyperbolic mechanism compatible with stability. This is where the difficulty lies, and where the gap between $r = 1$ and $r \geq 2$ opens.

The $C^1$ Case: Mañé’s Theorem #

The $C^1$ stability conjecture was fully proved by Mañé in 1987.

Theorem (Mañé, 1987)

Every $C^1$-structurally stable diffeomorphism of a closed manifold satisfies Axiom A and the Strong Transversality Condition.

The proof, published in Publ. Math. IHÉS 66 (1987), 161–210, is a tour de force of $C^1$ perturbation theory. It rests on several tools that are available only in the $C^1$ topology:

Pugh’s $C^1$ closing lemma (1967): Given a non-wandering point $x$ of $f$, one can make an arbitrarily small $C^1$ perturbation of $f$ to create a periodic orbit passing near $x$. This is the essential mechanism for showing that periodic points are dense in $\Omega(f)$.
Mañé’s ergodic closing lemma (1982): A more refined version that controls the Lyapunov exponents of the created periodic orbit, allowing the construction of hyperbolic periodic points that shadow the orbit of an ergodic measure.
Franks’ lemma (1971): Linear maps along periodic orbits can be prescribed independently (up to $C^1$ conjugacy), allowing one to test whether a given splitting is genuinely hyperbolic or can be destroyed by a small $C^1$ perturbation.

The strategy is to assume structural stability and use these tools to show, step by step, that the non-wandering set must be hyperbolic and that tangencies cannot persist. Mañé had proved the surface case ($\dim M = 2$, $r = 1$) earlier, with the full higher-dimensional result completed in the 1987 paper. Aoki (1992) and Hayashi (1992) subsequently settled the closely related Mañé conjecture on the $C^1$ interior of the set of diffeomorphisms with all hyperbolic periodic points.

The Wall at $r \geq 2$ #

The $C^r$ case for $r \geq 2$ is not merely an incremental extension. The tools that power Mañé’s proof are fundamentally $C^1$ phenomena.

The $C^r$ closing lemma is open for $r \geq 2$. Pugh’s closing lemma fails for $r \geq 2$ in general: Gutierrez showed that the local perturbation argument used for $C^1$ does not work in the $C^2$ topology. A $C^r$ closing lemma is available only for specific classes of diffeomorphisms:

Conservative (volume-preserving) diffeomorphisms on surfaces: Asaoka–Irie ($C^\infty$, 2015), Cristofaro-Gardiner–Prasad–Zhang (2023).
Partially hyperbolic diffeomorphisms with one-dimensional center bundle (all $r \geq 2$ including $r = \infty$): Gan–Shi (2022) and the follow-up $C^r$-chain closing lemma of Shi–Wang (Ergodic Theory Dynam. Syst. 44, 2024).

In the absence of a general $C^r$ closing lemma, the first step of Mañé’s proof, showing that periodic points are dense in $\Omega(f)$ under $C^r$ structural stability, is not known for $r \geq 2$.

Mañé himself underscored this gap. In the 1987 paper, immediately after the proof of Theorem A, he writes that for $r > 1$ “not even [being] known whether a $C^2$ structurally stable diffeomorphism has at least one periodic point, it seems, to say the least, difficult to prove that they are dense.”

Franks’ lemma also fails for $r \geq 2$. Controlling linear maps along periodic orbits requires $C^1$ perturbations; in higher regularity the ambient perturbation must be smooth and the constraints on higher derivatives can prevent the desired linear behaviour from being achieved.

Research Directions #

1. The $C^r$ Closing Lemma for General Diffeomorphisms #

The most direct path to the $C^r$ stability conjecture passes through a general $C^r$ closing lemma. For $r \geq 2$ this asks: given any non-wandering point of a $C^r$ diffeomorphism, can one make an arbitrarily small $C^r$ perturbation to close the orbit? Answering this in the affirmative for all closed manifolds and all $r \geq 2$ would be a landmark result, and would immediately advance the stability conjecture. The recent progress in conservative surface dynamics (Cristofaro-Gardiner et al., 2023) and partially hyperbolic settings shows the question is not hopeless, but the general dissipative case remains untouched.

2. The Surface Case $\dim M = 2$, $r \geq 2$ #

On surfaces the dynamics is simpler: the non-wandering set has lower-dimensional structure, and the absence of a center bundle means “partially hyperbolic” reduces to “hyperbolic.” Mañé settled the surface case for $r = 1$. The $C^r$ stability conjecture for surfaces and $r \geq 2$ is already an important open target and may be the most accessible subcase. Recent $C^\infty$ closing lemmas for conservative surface diffeomorphisms (Asaoka–Irie) suggest that the conservative surface case may be reachable.

3. Partially Hyperbolic Diffeomorphisms #

A diffeomorphism is partially hyperbolic if the tangent bundle splits as $TM = E^{ss} \oplus E^c \oplus E^{uu}$ with uniform contraction on $E^{ss}$, uniform expansion on $E^{uu}$, and an intermediate “center” bundle $E^c$. For these systems, Gan–Shi (2022) and Shi–Wang (2024) have established $C^r$ closing and chain-closing lemmas when $\dim E^c = 1$. The question is whether $C^r$-structural stability of a partially hyperbolic diffeomorphism forces the center bundle to also become hyperbolic, that is, whether partial hyperbolicity implies full hyperbolicity under stability.

4. The Palis Global Conjecture #

Palis proposed that the complement of the hyperbolic diffeomorphisms is exactly the closure of systems exhibiting homoclinic tangencies or heteroclinic cycles. This is a positive description of non-hyperbolic dynamics, and is a strengthening of the $C^r$ stability conjecture (it would also characterise what structural stability forbids). In $C^1$ topology this programme is largely complete through Bonatti– Crovisier’s connecting lemma (2004) and related results. For $r \geq 2$ it is wide open, and progress on the Palis conjecture in $C^r$ would likely resolve the stability conjecture as a corollary.

5. Flows and the Vector Field Analogue #

The stability conjecture has a natural analogue for $C^r$ vector fields: a $C^r$-structurally stable flow should satisfy Axiom A and the strong transversality condition. For $r = 1$ this is also proved. For $r \geq 2$ it is open. The vector field setting introduces additional complications from singular points (zeros of the vector field), as Labarca–Pacifico showed that on manifolds with boundary stable flows can fail Axiom A, so the correct formulation may need adaptation. Progress on the diffeomorphism case would likely shed light on the flow case as well.

References #

Palis, J. & Smale, S. (1970). Structural stability theorems. Proc. Sympos. Pure Math., 14, 223–231.
Robbin, J. W. (1971). A structural stability theorem. Annals of Mathematics, 94(2), 447–493.
Robinson, C. (1976). Structural stability of $C^1$ diffeomorphisms. Journal of Differential Equations, 22(1), 28–73.
Mañé, R. (1987). A proof of the $C^1$ stability conjecture. Publications Mathématiques de l’IHÉS, 66, 161–210.
Aoki, N. (1992). The set of Axiom A diffeomorphisms with no cycles. Bol. Soc. Brasil. Mat., 23(1–2), 21–65.
Hayashi, S. (1992). Diffeomorphisms in $\mathcal{F}^1(M)$ satisfy Axiom A. Ergodic Theory Dynam. Systems, 12(2), 233–253.
Gan, S. & Shi, Y. (2022). $C^r$-closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle. Journal of Differential Equations, 334, 337–363.
Shi, Y. & Wang, X. (2024). $C^r$-chain closing lemma for certain partially hyperbolic diffeomorphisms. Ergodic Theory Dynam. Systems, 44(7), 1923–1944.
Bonatti, C. & Crovisier, S. (2004). Récurrence et généricité. Inventiones Mathematicae, 158(1), 33–104.
Berger, P. (2017). Lectures on structural stability in dynamics. arXiv:1703.00092.

Free Books on Dynamical Systems

Thu, 27 Jun 2024 23:14:15 +0800

Arxiv/ Free Books #

1. Lectures on Neural Dynamics - Francesco Bullo #

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Chapter 2: Stability in dynamic neural networks using Lyapunov methods, multistability, and energy functions
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Chapter 4: Unsupervised learning via neural dynamics, linking Hebbian rules to tasks like PCA, clustering, and similarity-based representation learning.

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6. Symmetries of Living Systems: Symmetry Fibrations and Synchronization in Biological Networks - Hernan A. Makse, Paolo Boldi, Francesco Sorrentino, Ian Stewart #

Abstract: A symmetry is a `change without change’. As simple as it sounds, this concept is the fundamental cornerstone that unifies all branches of theoretical physics. Virtually all physical laws – ranging from classical mechanics and electrodynamics to relativity, quantum mechanics, and the standard model – can be expressed in terms of symmetry invariances. In this book, we explore whether the same principle can also explain the emergent laws of biological systems. We introduce a new geometry for biological networks and AI architectures, drawing inspiration from the mystic genius of Grothendieck’s fibrations in category theory. We attempt to bridge the gap between physics and biology using symmetries but with a twist. The traditional symmetry groups of physics are global and too rigid to describe biology. Instead, the novel notion of symmetry fibration is local, flexible, and adaptable to evolutionary pressures, providing the right framework for understanding biological complexity. In other words, this more general symmetry invariance is necessary and sufficient to ensure that a given biological network configuration can support a synchronized function. In this book, we review the theoretical progress over the last decades from mathematics, physics, computer science, dynamical systems, and graph theory that has led to the discovery of symmetry fibrations in biological networks. These symmetries act as organizing principles for biological networks. They serve as effective tools for describing the structure of these networks, blending geometry and topology. Fibrations explain how structure dictates function across various biological domains, including the transcriptome, proteome, metabolome, and connectome. Additionally, they facilitate a reduction in the dimensionality of the network, simplifying it into its fundamental building blocks for biological computation.

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8. A gentle invitation to the fractional world - Nicola Abatangelo, Serena Dipierro, Enrico Valdinoci #

Abstract: This book is intended as a self-contained introduction to selected topics in the fractional world, focusing particularly on aspects that arise in the study of equations driven by the fractional Laplacian. The scope of this work is not intended to be exhaustive or all-encompassing. We have chosen topics that we believe will appeal to readers embarking on their journey into fractional analysis. It requires only fundamental calculus and a basic understanding of measure theory. In Chapter 1, we introduce the primary object of study, the fractional Laplacian. This operator appears in diverse contexts, prompting multiple definitions and viewpoints, many of which we explore, along with some key identities. A notable distinction between local and nonlocal analysis is that in the latter, explicit calculations are often impractical or impossible. There are anyway some fortunate exceptions which are gathered in Chapter 2, providing useful and instructive examples. Chapter 3 presents an introduction to the important aspect of Liouville-type results. A large portion of this book is devoted to the regularity theory of solutions in Lebesgue spaces. Chapter 4 examines global solutions using Riesz and Bessel potential analysis, capturing the impact of both low and high frequencies on smoothness, decay, and oscillations. These spaces are also flexible enough to provide, as a byproduct, a solid regularity theory in the more commonly used fractional Sobolev spaces. In Chapter 5 we derive the corresponding interior regularity theory for solutions within a bounded domain using appropriate cutoffs and localization techniques. Additionally, technical appendices include auxiliary results used in key proofs.

9. Kinetically constrained models - Ivailo Hartarsky, Cristina Toninelli #

Abstract: The goal of this book is to provide an introduction to the mathematical theory of Kinetically constrained models developed in the last twenty years, intended for both mathematicians and physicists.

10. What is Entropy? - John C. Baez #

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11. Alice’s Adventures in a Differentiable Wonderland – Volume I, A Tour of the Land - Simone Scardapane #

Abstract: Neural networks surround us, in the form of large language models, speech transcription systems, molecular discovery algorithms, robotics, and much more. Stripped of anything else, neural networks are compositions of differentiable primitives, and studying them means learning how to program and how to interact with these models, a particular example of what is called differentiable programming. This primer is an introduction to this fascinating field imagined for someone, like Alice, who has just ventured into this strange differentiable wonderland. I overview the basics of optimizing a function via automatic differentiation, and a selection of the most common designs for handling sequences, graphs, texts, and audios. The focus is on a intuitive, self-contained introduction to the most important design techniques, including convolutional, attentional, and recurrent blocks, hoping to bridge the gap between theory and code (PyTorch and JAX) and leaving the reader capable of understanding some of the most advanced models out there, such as large language models (LLMs) and multimodal architectures.

12. Inverse Problems and Data Assimilation: A Machine Learning Approach - Eviatar Bach, Ricardo Baptista, Daniel Sanz-Alonso, Andrew Stuart #

Abstract: The aim of these notes is to demonstrate the potential for ideas in machine learning to impact on the fields of inverse problems and data assimilation. The perspective is one that is primarily aimed at researchers from inverse problems and/or data assimilation who wish to see a mathematical presentation of machine learning as it pertains to their fields. As a by-product, we include a succinct mathematical treatment of various topics in machine learning.

13. The Lanczos algorithm for matrix functions: a handbook for scientists - Tyler Chen #

Abstract: Lanczos-based methods have become standard tools for tasks involving matrix functions. Progress on these algorithms has been driven by several largely disjoint communities, resulting many innovative and important advancements which would not have been possible otherwise. However, this also has resulted in a somewhat fragmented state of knowledge and the propagation of a number of incorrect beliefs about the behavior of Lanczos-based methods in finite precision arithmetic. This monograph aims to provide an accessible introduction to Lanczos-based methods for matrix functions. The intended audience is scientists outside of numerical analysis, graduate students, and researchers wishing to begin work in this area. Our emphasis is on conceptual understanding, with the goal of providing a starting point to learn more about the remarkable behavior of the Lanczos algorithm. Hopefully readers will come away from this text with a better understanding of how to think about Lanczos for modern problems involving matrix functions, particularly in the context of finite precision arithmetic.

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15. Calculus and applications - Teo Banica #

Abstract: This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f’(x)$ and of the integral $\int_a^bf(x)dx$. Then we investigate the case of the complex functions $f:\mathbb C\to\mathbb C$, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, $f:\mathbb R^N\to\mathbb R^M$ or $f:\mathbb R^N\to\mathbb C^M$ or $f:\mathbb C^N\to\mathbb C^M$, with general theory, integration results, maximization questions, and basic applications to physics.

16. Stochastic Partial Differential Equations, Space-time White Noise and Random Fields - Robert C. Dalang, Marta Sanz-Solé #

Abstract: This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). The volume consists of two blocks: the core matter (Chapters 1 to 5) and the appendices (A, B and C). Chapter 1 introduces the subject, with a discussion of isonormal Gaussian processes, space-time white noise, and motivating examples of SPDEs. Chapter 2 presents a theory of stochastic integration with respect to space-time white noise. Chapter 3 deals with SPDEs with additive noise. In Chapter 4, we study a general class of SPDEs, in which additive and multiplicative nonlinearities appear. In Chapter 5, we present a selection of important topics in the theory of SPDEs, that have been the subject of much research over the last twenty years. Appendix A summarises the main results from the theory of stochastic processes and stochastic analysis that are used throughout the book. Appendix B is devoted to a systematic presentation of properties of fundamental solutions and Green’s functions associated to the classical linear differential operators (heat, fractional heat and wave operators). Appendix C is a toolbox section. Each chapter is followed by a “Notes” section, which gives historically important references, original sources and points towards other related important contributions.

17. Dynamic Programming: Finite States - Thomas J. Sargent, John Stachurski #

Abstract: This book is about dynamic programming and its applications in economics, finance, and adjacent fields. It brings together recent innovations in the theory of dynamic programming and provides applications and code that can help readers approach the research frontier. The book is aimed at graduate students and researchers, although most chapters are accessible to undergraduate students with solid quantitative backgrounds.

18. Resources of the Quantum World - Gilad Gour #

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19. Funktionalanalysis Teil I - Christoph Bock #

Abstract: Roughly spoken, Functionalanalysis means the study of the category of infinite-dimensional vectorspaces over the field of real or complex numbers, together with their linear maps. In most cases, one further needs a topological structure on such a vectorspace, because then, you can consider the continuous linear maps between such spaces. The name Functionalanalysis is due to the fact, that in the beginning of the theory, the authors wanted to expand Calculus onto functionals of spaces of functions. Functionalanalytical results give the possibility to solve problems in the Theory of (Partial) Differential Equations, in Complex Analysis or in Quantum Mechanics. But the aim of this lines is not to explain the applications. We will discuss the mathematical theory of almost metric spaces, normed vector spaces and algebras, spaces of continuous resp. $p$-integrable functions as well as reflexive and uniformly convex spaces.

We added that, in the case $p \in {]}0,1{[}$, $L^p$ is the completion of the compactly supported continuous functions (with the obvious metric), too. Actually, the proof is the same as in the case $p \in {[}1, \infty{[}$.

20. Algebraic Topology for Data Scientists - Michael S. Postol #

Abstract: This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science. Algebraic topology is traditionally a very specialized field of math, and most mathematicians have never been exposed to it, let alone data scientists, computer scientists, and analysts. I have three goals in writing this book. The first is to bring people up to speed who are missing a lot of the necessary background. I will describe the topics in point-set topology, abstract algebra, and homology theory needed for a good understanding of TDA. The second is to explain TDA and some current applications and techniques. Finally, I would like to answer some questions about more advanced topics such as cohomology, homotopy, obstruction theory, and Steenrod squares, and what they can tell us about data. It is hoped that readers will acquire the tools to start to think about these topics and where they might fit in.

21. Discrete and Continuous Weak KAM Theory: an introduction through examples and its applications to twist maps - Maxime Zavidovique #

Abstract: The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear, but where central ideas and results persist. It can therefore serve as a good introduction to (continuous) weak KAM theory. After a general exposition of the general abstract theory, several examples are studied. The last section is devoted to the historical problem of conservative twist maps of the annulus. At the end of the first three Chapters, the relations between the results proved in the discrete setting and the analogous theorems of classical weak KAM theory are discussed. Some key differences are also highlighted between the discrete and classical theory. Those results are new. The text also contains other results never published before, such as the convergence of solutions of discounted equations for degenerate perturbations.

Mathematics - Dynamical Systems

Thu, 27 Jun 2024 23:14:15 +0800