A C¹ counterexample to the entropy conjecture

The explanation

Take a point on the unit circle in the complex plane and square it, again and again. Each squaring doubles its angle. Start two points 0.001 of a turn apart. After 5 squarings they are 0.032 of a turn apart. After 9 squarings the gap is 0.512 of a turn. The two points now sit almost opposite each other. A tiny difference at the start has grown into a large one.

Topological entropy measures this spreading. It is the growth rate of the number of orbits you can tell apart at a fixed precision. An orbit is the path a point traces as the map repeats. If the number of distinguishable orbits doubles at each step, the entropy is log 2, about 0.693. Zero entropy means the count grows slower than any exponential rate.

Squaring also leaves a mark on shape. Add a point at infinity to the complex plane and it becomes a sphere, the Riemann sphere. Squaring wraps this sphere around itself twice. The point 1 has exactly two sources, 1 and −1. Both wrap with the same orientation, so the wrapping number, called the degree, is 2. Homology (an algebraic count of the holes and surfaces in a space) turns this into a number. On the sphere's surface, squaring acts as multiplication by 2.

In 1974, Michael Shub proposed a link between these two views. His entropy conjecture says that growth seen in homology should force orbit complexity. Precisely, entropy should be at least the logarithm of the spectral radius of the homology action. The spectral radius is the largest stretching factor, or eigenvalue, of that action. An eigenvalue of 2 would therefore force entropy of at least log 2. The general version asks this of every C¹ map of a compact smooth manifold without boundary. A C¹ map is one whose first derivatives exist and vary continuously. A compact manifold without boundary is a closed, finite space like a sphere.

Many cases were already settled in favour of the conjecture. Manning's inequality handles first homology for every continuous map. A bound by Misiurewicz and Przytycki handles the top-dimensional part, through the degree. Yomdin proved the full inequality in 1987 for infinitely smooth maps. Saghin and Xia, and later Liao, Viana and Yang, proved it for restricted classes of invertible C¹ maps. Counterexamples were known only for rougher maps, called Lipschitz maps. The paper notes that no positive result covers one combinationSource 1. That combination is finite smoothness, maps that are not invertible, and homology in a middle dimension.

This paper builds a map in exactly that gap. Its space is a circle times several copies of the sphere, of dimension at least 5Source 2. The map is C¹ and not invertible. Its entropy is zeroSource 3. Yet on second homology it has eigenvalue 2Source 3. So the conjecture's lower bound, at least log 2, fails for it. The first-homology and top-dimensional results remain untouched, because the eigenvalue sits in the middleSource 2.

The obstacle is plain. One sphere coordinate must look like squaring to homology. But squaring spreads orbits and creates entropy. The map has to keep the doubling in shape while removing it from motion. The paper uses a circle coordinate as a clock that can slow down almost to a haltSource 4. Extra sphere coordinates, called registers, store small signals and later feed them into the squaring coordinateSource 5. As a result, every orbit drifts toward one of two simple sets, and both carry zero entropySource 6. The hardest step is an infinite sum over all possible waiting timesSource 7. A finite version would miss the push needed after a very long delay. The paper calls the link to squaring homological, not dynamicalSource 8. Homology still sees the doubling, but orbits no longer follow it.

The gap in the result is just as specific. It concerns C¹ maps only, so Yomdin's theorem for infinitely smooth maps still stands. The map is not invertible, so the question for invertible maps is not addressed here. The theorem says some number of registers works, but its statement does not give that numberSource 9.

How sure is this? A computer proof checker called Lean has checked a formal version of the main statement. That statement says a noninvertible C¹ map exists with entropy zero and an eigenvalue 2 on second homology. A Comparator statement is listed for it. No specialist review of the paper has been published.

What it doesn't show. This does not disprove the conjecture for infinitely smooth maps, which Yomdin settled in 1987, or for invertible maps.