Results on entropy, the clock-register map, attraction and the homological eigenvalue, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Entropy does not depend on the metric — Proposition 1.3. Compatible metrics on a compact space give the same topological entropy.
• Invariant pieces — Lemma 2.1. A restriction has at most the entropy of the map; a union of two compact invariant sets has the larger of their entropies; periodic maps have entropy zero.
• Finite-dimensional homology of compact manifolds — Lemma 2.2. A compact smooth n-manifold without boundary has finite-dimensional real homology, vanishing above n, without orientation or connectedness assumptions. Complexification and the total-homology spectral radius are justified.
• Entropy localization (OpenAI) — Theorem 3.1. If every orbit approaches a compact forward invariant set Z with h(F|Z) = 0, then h(F) = 0; no uniform time of approach is needed.
• Radial register maps have zero entropy (OpenAI) — Lemma 4.1. Products of sphere maps that send the sphere into a disc and move radii by a nondecreasing map have zero entropy.
• The clock barrier — Lemma 2.1. A smooth 100-periodic b with values in [0,1], vanishing only at 50, equal to one away from it, with t + b(t) <= 50 on [0,50].
• The map is C^1 (OpenAI) — Proposition 6.1. The prediction inputs are C^1 with |U| < 1/5, and the clock-register map f is a C^1 self-map of M.
• Attraction (OpenAI) — Theorem 1.1. Every orbit tends to Z_0 (clock at 50, main coordinate 0) or to Z_inf (main coordinate infinite, registers zero).
• Bounded clocks — Lemma 2.1. A bounded lifted clock converges to 50 modulo 100, and then the main coordinate tends to 0.
• Preparation — Lemma 3.1. In a complete passage the registers are zero at the first preparation step, there are one to four preparation steps, a full readout occurs, and |z| > 1/2 at preparation.
• Clock prediction (OpenAI) — Lemma 4.1. Each readout window is met at most once per passage, and the predictions meet it exactly at the actual readout step.
• Registers are linear until read (OpenAI) — Lemma 4.2. At its readout a register has modulus at most 1/2 and equals the explicit sum of its prepared phases.
• Escape in one passage (OpenAI) — Lemma 5.1. Readouts are in phase with z^2, so |z'| >= |z|^2 through the passage, and some step has |z| > 1.
• Zero entropy (OpenAI) — Proposition 1.1. f has zero topological entropy, and so do its restrictions to Z_0 and Z_inf.
• A sphere on which f squares — Lemma 2.1. f(0, z, 0, ..., 0) = (1, z^2, 0, ..., 0), so f composed with i is homotopic to i composed with squaring.
• Squaring has degree two — Proposition 3.1. z -> z^2 multiplies the fundamental class of the Riemann sphere by 2, with real coefficients.
• The eigenvector (OpenAI) — Theorem 4.1. The class of the sphere i(Sigma) is nonzero in H_2(M; R), and f_* multiplies it by 2.
• A C^1 counterexample to the entropy conjecture (OpenAI) — Theorem 5.1. A C^1 self-map of a compact manifold of dimension 2q+3 >= 7, neither injective nor surjective, with zero entropy and the eigenvalue 2 on H_2.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.