Katok’s entropy conjecture

Let $T\colon X\to X$ be a continuous map of a compact metric space, and let $h_{\mathrm{top}}(T)$ denote its topological entropy. For every $\alpha\in[0,h_{\mathrm{top}}(T)]$ , there exists an ergodic $T$ -invariant Borel probability measure $\mu$ such that $h_\mu(T)=\alpha$ . Equivalently, the set of entropies of ergodic invariant measures is exactly $[0,h_{\mathrm{top}}(T)]$ .

References

Primary source

arXiv

Additional references

Ergodic measures of intermediate entropies for amenable group actions with an approximate product property — arXiv — Wenda Zhang, Xiankun Ren

Posted 9 hours ago · edited 9 hours ago

Progress summary

Refreshed 9 hours ago

Claimed progress

A conditional theorem gives intermediate-entropy measures for a broad class of amenable group actions, but it does not settle Katok’s conjecture in full generality.

Katok’s intermediate entropy conjecture concerns realizing every allowed intermediate entropy by an ergodic measure. The newly retrieved work addresses this only for systems with additional structural properties; it should not be conflated with Katok’s separate entropy-rigidity conjecture.

Conditional amenable-action result (date not stated)

Wenda Zhang and Xiankun Ren claim the conjecture for amenable group actions satisfying the specification property and asymptotic entropy-expansiveness. A related summary states the result under an approximate product property. This is substantive conditional progress, not a proof of the unrestricted conjecture, and no independent verification was found.

Current status (as of October 2026): The conjecture is claimed under specification or approximate-product and asymptotic entropy-expansiveness hypotheses, while the general case remains open.

Sources

No solutions have been posted yet.