For a topological dynamical system, that is, a homeomorphism of a compact metrizable space, passing to a factor (the image under a continuous surjection that commutes with the dynamics) can only decrease topological entropy. Shub and Weiss asked in 1991 when a positive-entropy system has a continuous factor of strictly smaller entropy, in particular one of arbitrarily small positive entropy. Lindenstrauss (1995) proved that for a finite-dimensional system of entropy hh the entropies of its factors fill the entire interval [0,h][0,h], and constructed two infinite-dimensional examples, one of them minimal, in which every nontrivial factor has infinite topological entropy; both examples themselves have infinite entropy, so they do not exhibit a gap below a finite ceiling. This leaves the finite-entropy case open: does there exist a system with positive finite topological entropy and a constant c>0c>0 such that every nontrivial factor has entropy at least cc? Several large classes are known to admit no such gap (finite-dimensional systems; systems with the small boundary property, including all uniquely ergodic systems; extensions of nontrivial minimal systems with zero mean dimension, including all minimal systems of finite entropy; zero-dimensional systems), yet as stated in 2021 it is not even known whether every positive-finite-entropy system has some nontrivial factor of strictly smaller entropy.

Background

A topological dynamical system is a pair (X,T)(X,T) where XX is a compact metrizable space and T ⁣:X→XT\colon X\to X is a homeomorphism (the question below was also posed for non-invertible continuous surjections; passing to the natural extension reduces that setting to this one without changing entropies). The topological entropy htop(X,T)∈[0,∞]h_{\mathrm{top}}(X,T)\in[0,\infty] measures the exponential growth rate of the number of orbit segments distinguishable at a fixed resolution and is the standard numerical measure of orbit complexity. A factor of (X,T)(X,T) is a system (Y,S)(Y,S) together with a continuous surjection π ⁣:X→Y\pi\colon X\to Y satisfying π∘T=S∘π\pi\circ T=S\circ\pi; the factor is nontrivial when YY contains at least two points. Since htop(Y,S)≤htop(X,T)h_{\mathrm{top}}(Y,S)\le h_{\mathrm{top}}(X,T) always holds, factoring can only lower entropy, and the natural question is by how much: complete flexibility would mean every positive entropy level below the original is reachable by a nontrivial factor, while rigidity would mean some positive amount of entropy survives in every nontrivial factor.

Shub and Weiss asked in 1991 when a positive-entropy system has a factor of strictly smaller entropy, in particular one of arbitrarily small positive entropy. Their main positive tool, systematized by Lindenstrauss, is the small boundary property (SBP): every point of XX has arbitrarily small neighborhoods whose topological boundaries are dynamically negligible in a suitable entropy-theoretic sense. Any system with the SBP has, for any two distinct points and any ε>0\varepsilon>0, a factor of topological entropy below ε\varepsilon separating them (this is Theorem 6.7 of Lindenstrauss (1999), credited to Shub and Weiss (1991), where the argument appears for systems without periodic points, and to Lindenstrauss (1995), where it covers all SBP systems). Since every uniquely ergodic system, i.e. one admitting a single invariant probability measure, has the SBP by Lindenstrauss and Weiss (2000), uniquely ergodic systems have nontrivial factors of arbitrarily small entropy. Lindenstrauss (1995) settled the finite-dimensional case completely: if XX has finite Lebesgue covering dimension and htop(X,T)=hh_{\mathrm{top}}(X,T)=h, then for every factor (Y,S)(Y,S) of strictly lower entropy and every h′′∈[htop(Y,S),h]h''\in[h_{\mathrm{top}}(Y,S),h] there is an intermediate factor of entropy exactly h′′h''; in particular the entropies of factors fill [0,h][0,h]. He also constructed two infinite-dimensional examples, the two-sided shift on the Hilbert cube ([0,1]Z,σ)([0,1]^{\mathbb{Z}},\sigma) and a minimal subsystem of it, in which every nontrivial factor has infinite topological entropy. Both examples themselves have infinite entropy (the Hilbert cube shift has entropy ∞\infty and mean dimension 11), so they show only that entropy is rigid in the infinite-entropy world; whether a gap can occur below a finite entropy ceiling was left open, together with a conjecture that minimal systems of finite entropy admit no gap.

The modern structural invariant for this circle of questions is mean topological dimension, introduced by Lindenstrauss and Weiss: a dynamical analogue of covering dimension, defined by averaging the minimal dimension of covers of longer and longer orbit segments, and meaningful precisely for infinite-dimensional spaces. Two facts anchor the theory: a system of finite topological entropy, and a fortiori one of finite topological dimension, has zero mean dimension; and a system with the SBP has zero mean dimension. Lindenstrauss (1999) proved a converse for extensions of nontrivial minimal systems (systems admitting a factor in which every orbit is dense): for these, zero mean dimension implies the SBP and hence separating factors of arbitrarily small entropy, so every extension of a nontrivial minimal system with finite entropy, in particular every minimal system of finite entropy, has nontrivial factors of arbitrarily small entropy; this proves the 1995 conjecture for minimal systems. The same paper identifies periodic-point structure as the main obstruction to removing the minimality hypothesis: if the set of periodic points is not zero-dimensional, the SBP fails. Gutman, Lindenstrauss and Tsukamoto (2016) extended the structure theory to Zk\mathbb{Z}^k-actions with the marker property (a tower-construction hypothesis, satisfied in particular by every system with a completely aperiodic minimal factor), showing that for such systems zero mean dimension is equivalent to being an inverse limit of finite-entropy systems. On the symbolic side, Hong, Jung and Lee (2013) proved that for a factor code between sofic shifts the entropies of intermediate subshifts are dense in the interval between the two entropies, and McGoff and Pavlov (2021) proved that for subshifts over any countable amenable group the entropies of intermediate subshift factors are dense in the admissible interval, while for zero-dimensional systems every value in [h(Y,S),htop(X,T)][h(Y,S),h_{\mathrm{top}}(X,T)] is attained by an intermediate zero-dimensional factor; in particular zero-dimensional systems admit no gap. Jin and Qiao (2025) constructed further systems, of subshift-of-block-type inside the Hilbert cube, for which every nontrivial factor has infinite entropy, equivalently positive mean dimension, but all of these examples again have infinite entropy themselves.

What survives is the general finite-entropy case, with no dimension, minimality, marker, or zero-dimensionality hypothesis. Since finite entropy forces zero mean dimension, the question lives entirely inside the zero-mean-dimensional world, and the known sufficient conditions for small-entropy factors (the SBP and the finite-dimensional, zero-dimensional, uniquely ergodic, and extension-of-minimal structures) do not cover it. As McGoff and Pavlov state in 2021, it is still not known whether every system with positive, finite entropy must have a nontrivial factor of strictly smaller entropy, so a fortiori it is not known whether such systems always admit nontrivial factors of arbitrarily small entropy, or whether a finite-entropy gap system exists. A related but distinct line studies lowering entropy over subsets rather than factors: Huang, Ye and Zhang (2010) proved that every finite-entropy system contains compact subsets of every entropy value up to the entropy of the system, a flexibility statement at the subset level that highlights how selective the factor-level question is.

Problem Statement

Let a topological dynamical system be a pair (X,T)(X,T) where XX is a compact metrizable space and T ⁣:X→XT\colon X\to X is a homeomorphism, and let htop(X,T)∈[0,∞]h_{\mathrm{top}}(X,T)\in[0,\infty] denote its topological entropy. Call a factor (Y,S)(Y,S) of (X,T)(X,T), i.e. a system (Y,S)(Y,S) with at least two points for which some continuous surjection π ⁣:X→Y\pi\colon X\to Y satisfies π∘T=S∘π\pi\circ T=S\circ\pi, nontrivial. Does there exist a topological dynamical system (X,T)(X,T) with 0<htop(X,T)<∞0<h_{\mathrm{top}}(X,T)<\infty and a constant c>0c>0 such that every nontrivial factor (Y,S)(Y,S) of (X,T)(X,T) satisfies htop(Y,S)≥ch_{\mathrm{top}}(Y,S)\ge c? Equivalently, is it false that every topological dynamical system with 0<htop(X,T)<∞0<h_{\mathrm{top}}(X,T)<\infty has, for each ε>0\varepsilon>0, a nontrivial factor of topological entropy less than ε\varepsilon? A positive example must be infinite-dimensional (factor entropies of finite-dimensional systems fill [0,htop(X,T)][0,h_{\mathrm{top}}(X,T)] by Lindenstrauss (1995)), cannot be zero-dimensional (Hong-Jung-Lee 2013; McGoff-Pavlov 2021), cannot be uniquely ergodic, must fail the small boundary property (which yields separating factors of arbitrarily small entropy for every system, by a theorem of Shub-Weiss and Lindenstrauss), and cannot be an extension of a nontrivial minimal system (Lindenstrauss 1999, Corollary 6.9); by Lindenstrauss-Weiss (2000) it automatically has zero mean dimension, so it must be a zero-mean-dimensional system lying outside every known framework that produces small-entropy factors. Even the weakest version, whether every system with positive finite topological entropy has some nontrivial factor of strictly smaller entropy, is stated as open in McGoff-Pavlov (2021).

References

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