A C1 counterexample to the entropy conjecture

1 · The essentials

Can the way a map transforms a space force its repeated action to be complicated? The manuscript answers no for the general class of continuously differentiable maps considered here. It constructs a map from a compact, smooth space without boundary back to itself. The map cannot be reversed, and a measure of the complexity of its repeated action is zero. Yet its effect on a particular kind of two-dimensional structure in the space includes a multiplication by 2. This contradicts the proposed lower bound linking those two kinds of behavior. The main result has a Lean formalization, although the supplied source does not specify its detailed coverage.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a machine whose bookkeeping records a doubling, while watching it run reveals no growth in complexity according to the chosen measure. The example separates two mathematical measurements in a similar way: growth recorded by the shape of the space does not necessarily force complexity in repeated motion.

Why it matters

The result rules out the general $C^{1}$ self-map formulation of Shub’s entropy conjecture. It shows that growth in the map’s action on homology need not force positive topological entropy. Any lower bound of this kind must therefore have a scope that excludes this example.

2 · Going further

The precise question concerns a self-map $f : M \to M$ of a compact smooth manifold without boundary. The homological entropy lower bound would require the topological entropy to be at least the logarithm of the homological spectral radius. The manuscript identifies the general $C^{1}$ self-map version of this statement as a formulation of Shub’s entropy conjecture.

The construction is a noninvertible $C^{1}$ map with $h_{top} (f) = 0$ , while the induced map on second real homology, $f_{*} : H_{2} (M; R) \to H_{2} (M; R)$ , has eigenvalue $2$ . This eigenvalue makes the homological spectral radius at least $2$ , so its logarithm is positive. The zero topological entropy is therefore strictly below the proposed bound.

The central mechanism reported in the abstract is this separation between homological growth and orbit complexity. The supplied abstract does not describe how the map is built or how its zero entropy is established, so a more detailed construction argument cannot be given from this source. Only the main manuscript is supplied; no companion manuscripts or relationships among them are described.

Useful notions

Repeated application of a map
Continuous differentiability
Compact smooth manifolds
Topological entropy
Homology and eigenvalues

What was known before / what is new

Before

The source presents a proposed lower bound comparing topological entropy with the logarithm of the homological spectral radius. It calls the general $C^{1}$ self-map formulation Shub’s entropy conjecture. It gives no further history or earlier partial results.

New

The manuscript supplies a noninvertible $C^{1}$ self-map of a compact smooth manifold without boundary that has zero topological entropy and eigenvalue $2$ on second real homology. Together, these properties provide a counterexample to the general homological entropy lower bound in this class.

Limits and assumptions

The example is specifically noninvertible and $C^{1}$ , on a compact smooth manifold without boundary, with the stated homological behavior occurring in second real homology. It does not by itself settle formulations restricted to invertible maps or other narrower classes. The main result has a Lean formalization, but the supplied material does not specify which supporting claims are covered. The source identifies no remaining open questions and supplies no construction details.

Mini glossary

$C^{1}$ self-map: A map from a space to itself whose first derivatives exist and are continuous.
Noninvertible: A map is noninvertible when it has no inverse map that undoes its action.
Topological entropy: A measure of orbit complexity under repeated application of a map.
Second real homology: An algebraic description of two-dimensional cycle structure in a space, using real coefficients.
Eigenvalue: A scaling factor by which a linear map multiplies some nonzero vector.
Homological spectral radius: The largest absolute value of the eigenvalues of the map’s induced action on homology.

Keywords : Topological entropy · Homological growth · Entropy conjecture

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Constructs a noninvertible C¹ self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C¹ self-maps: homological growth need not force positive orbit complexity.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/151.md

Manuscripts in this family

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A C^1 Counterexample to the Entropy Conjecture

25/09/2026

To be verified

We disprove the general C¹ self-map formulation of Shub's entropy conjecture. We construct a noninvertible C¹ self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.

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