Topological growth without orbit entropy at one derivative

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Family 151 · Dynamical systems and ergodic theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

If a map repeatedly wraps a space around itself, must its trajectories become exponentially complicated? Shub’s entropy conjecture makes this intuition precise by comparing two kinds of growth: the number of distinguishable orbit histories and the expansion of topological cycles under iteration.

For infinitely differentiable maps on compact manifolds, the proposed inequality is a theorem. Family 151 claims that it fails for maps with only one continuous derivative. The example has zero topological entropy, yet its action on second homology has eigenvalue two.

Two qualifications are central. The constructed map is noninvertible, and the expanding topological information occurs in an intermediate homology degree. The manuscript therefore does not claim a counterexample among diffeomorphisms, nor does it contradict known inequalities for first homology or topological degree.

2. Background and essential definitions

Counting distinguishable histories

Let $f:M\to M$ be continuous on a compact metric space with distance $d$. To compare two starting points for $n$ steps, use

$$d_n(x,y)=\max_{0\le j<n}d(f^j(x),f^j(y)).$$

Two histories are distinguishable at resolution $\varepsilon$ when this distance exceeds $\varepsilon$. Let $s_f(n,\varepsilon)$ be the largest size of a set of pairwise distinguishable histories. Topological entropy is

$$h_{\mathrm{top}}(f)= \lim_{\varepsilon\downarrow0}\limsup_{n\to\infty} \frac{1}{n}\log s_f(n,\varepsilon).$$

Thus positive entropy means exponential growth in possible histories at sufficiently fine observation scales. Zero entropy permits nontrivial dynamics, but their distinguishable histories have no positive exponential growth rate. This is a property of the map on the whole space, without selecting a probability measure.

Tracking cycles instead of individual points

Real homology $H_k(M;\mathbb R)$ is a finite-dimensional vector space recording $k$-dimensional cycles modulo those bounding higher-dimensional pieces. First homology records loop information; second homology can record sphere-like surfaces. A continuous map induces a linear map $f_*$ on these spaces.

The spectral radius $\rho(f_*)$ is the largest absolute value of an eigenvalue on their direct sum. An eigenvalue two means that repeated application multiplies some homological information at rate $2^n$. The conjectured inequality is

$$h_{\mathrm{top}}(f)\ge\log\rho(f_*).$$

The model example is the sphere map $z\mapsto z^2$, viewing the sphere as the complex plane together with infinity. A generic target has two preimages, each preserving orientation, so the map has degree two and multiplies its second homology by two.

A $C^1$ map has a continuous first derivative. A $C^\infty$ map has continuous derivatives of every order. A diffeomorphism must additionally be invertible with a differentiable inverse. These regularity and invertibility distinctions affect the entropy question.

3. Development history and previous partial results

Shub formulated the connection between entropy and homological growth in the 1970s. His later account explicitly discusses both endomorphisms, which need not be invertible, and diffeomorphisms. It also distinguishes earlier Lipschitz counterexamples from the question at $C^1$ regularity. A Lipschitz map controls distances but need not have a continuous derivative.

Manning established an entropy lower bound from the induced action on first real homology for continuous maps. Thus a counterexample cannot simply hide exponentially growing loop information in a continuous zero-entropy map.

Misiurewicz and Przytycki established a $C^1$ lower bound involving topological degree. On an oriented closed manifold, degree controls the action on the highest-dimensional homology. This leaves intermediate homology as a different issue.

Yomdin’s 1987 work proved the full inequality for $C^\infty$ maps by relating entropy to volume growth. Gromov’s exposition explains the geometric estimates behind this result. High regularity bounds the complexity that can be hidden in successively finer geometric features.

Positive results also cover important restricted classes at lower regularity. Saghin and Xia proved the conjectured bound for partially hyperbolic diffeomorphisms with one-dimensional center. Liao, Viana, and Yang proved it for diffeomorphisms away from homoclinic tangencies. These hypotheses impose dynamical structure absent from an arbitrary $C^1$ self-map.

The claimed counterexample occupies the gap between those results: finite regularity, noninvertibility, and growth in an intermediate-dimensional homology group. It changes the general self-map statement while leaving the cited positive theorems intact.

4. What the new manuscripts claim

The single manuscript, dated 25 September 2026, claims that for some positive integer $q$, there is a $C^1$ map on the compact boundaryless smooth manifold

$$M=(\mathbb R/100\mathbb Z)\times(S^2)^{q+1}$$

with

$$h_{\mathrm{top}}(f)=0, \qquad 2\text{ an eigenvalue of }f_*:H_2(M;\mathbb R)\to H_2(M;\mathbb R).$$

The circle factor is a clock with circumference 100; its particular scale is a construction convenience. The remaining factors are spheres. The manifold has dimension $2q+3$, at least five, so second homology is neither first nor top-dimensional homology.

Consequently the claimed example satisfies

$$0=h_{\mathrm{top}}(f)<\log2\le\log\rho(f_*).$$

The map is explicitly noninvertible. The claim is an existence statement for a finite-dimensional example, not an assertion that every manifold supports one or that the displayed dimension is minimal.

The construction supplies both parts of the conclusion: a classification of forward-orbit behavior that gives zero entropy, and a homotopy calculation retaining the degree-two action of squaring on one sphere coordinate.

5. Why the advance matters

Provisional importance: 5/5. If correct, this gives a negative answer to a central general formulation of the entropy conjecture. It shows that one continuous derivative does not suffice to convert all homological expansion into orbit complexity.

The example also sharpens the role of regularity. The difference between $C^1$ and $C^\infty$ can look technical in elementary calculus, but here it separates a claimed counterexample from a known universal theorem. Infinitely many increasingly fine features can be compatible with a continuous first derivative while evading the stronger geometric control used in the smooth theorem.

Its significance is not that topology becomes irrelevant to dynamics. First homology and degree still impose their established constraints. The result instead identifies a particular failure of a proposed universal bridge between intermediate topology and entropy.

6. The main idea

A squaring coordinate with delayed interventions

Write a point as $(t,z,v_1,\ldots,v_q)$. The coordinate $t$ is the circular clock, $z$ is the main sphere coordinate, and the other sphere coordinates store small complex signals. Most of the time, $z$ evolves approximately by squaring.

A clock passage has preparation, possible delay, readout, and reset stages. Near one clock position, its advance can become arbitrarily slow depending on $z$. During preparation, the auxiliary coordinates receive signals designed for possible later readouts. A signal meant for a delay of $m$ steps has an angular oscillation of order $2^m$, together with an amplitude of order $L^{-m}$ for a sufficiently large $L>2$.

The oscillation predicts the direction of a future square. The very small amplitude controls its first derivative. Intervening multiplication by $L$ restores the signal’s size when it is read. The manuscript chooses $L$ larger than both the angular-growth rate and a derivative bound for the clock-prediction map, making the infinite prediction series converge with its first derivatives.

The auxiliary coordinates live on spheres, so they must use a bounded extension of multiplication away from their small linear region. A separate estimate keeps each relevant signal in that linear region until readout. Each channel is read at most once per passage, and all channels are reset before the next preparation stage.

This is a mathematical construction with infinitely many possible delay terms, not a finite algorithm for predicting arbitrary dynamics. A finite truncation would miss sufficiently long delays. The controlled infinite series is the feature that supplies a response at every required delay while retaining $C^1$ regularity.

Force simple asymptotic behavior

At readout, contributions align with the current square rather than cancel it. The manuscript claims that a completed passage forces the main coordinate outside the unit circle. Thereafter repeated squaring drives its modulus toward infinity in the sphere coordinate.

Every orbit approaches one of two compact invariant sets. In one, the clock is fixed at its slow position and $z=0$; the remaining coordinates have radial dynamics with fixed angles and a nondecreasing radius map. In the other, $z=\infty$, the auxiliary coordinates vanish, and only a periodic clock rotation remains.

Both restrictions have zero entropy. The first has only polynomially many distinguishable radial itineraries at fixed resolution, and the second is periodic.

An additional entropy-localization lemma handles the fact that entry is not uniform in the initial point. The manuscript derives a bound on the proportion of orbit blocks spent away from neighborhoods of the two invariant sets. Encoding the rare exceptional blocks contributes arbitrarily little exponential complexity. Thus arbitrarily long transients do not create positive entropy in the claimed example.

Retain homological doubling

Let $\pi$ project to the main sphere and let $g(z)=z^2$. The main-coordinate update is squaring plus a bounded correction from the auxiliary coordinates. Continuously fading that correction to zero gives a homotopy between $\pi\circ f$ and $g\circ\pi$.

Homotopic maps induce the same homology map, so

$$\pi_*f_*=g_*\pi_*=2\pi_* \quad\text{on second homology}.$$

Since the projection has a section obtained by fixing the other coordinates, $\pi_*$ is onto. The displayed relation therefore forces eigenvalue two for $f_*$.

This relation holds on homology. It is not a dynamical factor relation saying that every actual projected orbit follows squaring. The time-dependent corrections can change trajectories drastically while remaining invisible to this homotopy calculation.

7. Scope, limitations, and open questions

The counterexample is a $C^1$ self-map, not a diffeomorphism. Its loss of invertibility is explicit: some auxiliary-coordinate regions are collapsed to identical outputs. The manuscript does not settle the unrestricted diffeomorphism version through this construction.

It also does not establish the same phenomenon at an arbitrarily prescribed higher finite regularity. Convergence of the first-derivative series is the stated regularity result. Questions about stronger derivative classes require their own estimates, and the $C^\infty$ case is already governed by Yomdin’s positive theorem.

The homology coefficients are real, and the claimed expansion occurs in degree two. No violation of Manning’s first-homology bound or the degree bound is intended. Minimal dimension and simpler explicit realizations are separate questions.

Finally, zero topological entropy does not mean that all points become fixed after a bounded time. The construction uses long, nonuniform delays. Its claim is the more precise one that the global exponential rate of distinguishable histories vanishes despite those delays.

8. Sources and further reading

Shub’s Dynamical systems, filtrations and entropy (1974) is the original conceptual source; All, most, some differentiable dynamical systems (2007) discusses the formulations and regularity distinctions.

Manning’s Topological entropy and the first homology group (1975), Misiurewicz and Przytycki’s Topological entropy and degree of smooth mappings (1977), and Yomdin’s Volume growth and entropy (1987) are primary references for the positive results. Gromov’s Entropy, homology and semialgebraic geometry (1987) explains the smooth geometric framework.

For restricted diffeomorphism classes, see Saghin and Xia’s The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center (2010) and Liao, Viana, and Yang’s The entropy conjecture for diffeomorphisms away from tangencies (2013). Hatcher’s Algebraic Topology provides background on homology, degree, and homotopy invariance.

Source manuscripts

• M1. OpenAI, A C^1 Counterexample to the Entropy Conjecture (September 25, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.