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Mean dimension explosion of induced homeomorphisms

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  • Lacerda, Gabriel
  • Romaña, Sergio

Abstract

Let X be a compact metric space and T:X→X a continuous function. The induced hyperspace map TK acts on the hyperspace K(X) of closed and nonempty subsets of X, and on the continuum hyperspace C(X)⊂K(X) of connected sets. This work studies the mean dimension explosion phenomenon: when the base system T has zero topological entropy, but the mean dimension of the induced map TK is infinite. In particular, this phenomenon occurs for Morse–Smale diffeomorphisms. Furthermore, for a circle homeomorphism H, the mean dimension explosion does not occur if and only if H is conjugate to a rotation. For the metric mean dimension, a different result is obtained: we establish sufficient conditions for the induced hyperspace map to have zero or infinite metric mean dimension.

Suggested Citation

  • Lacerda, Gabriel & Romaña, Sergio, 2026. "Mean dimension explosion of induced homeomorphisms," Chaos, Solitons & Fractals, Elsevier, vol. 208(P1).
    • Handle: RePEc:eee:chsofr:v:208:y:2026:i:p1:s0960077926002286
    DOI: 10.1016/j.chaos.2026.118087
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