Induced Dynamics of Non-Autonomous Dynamical Systems

The paper proves that for an equi-continuous NDS (X,f0,∞), positive base entropy implies infinite entropy for the hyperspace (K(X), f̄0,∞) and for the fuzzified system (F(X), f̃0,∞) under the endograph metric. It does so by: (i) using the product-entropy rule h(f(m)0,∞)=m h(f0,∞), (ii) constructing the symmetric-power factor Φ:Xm→Km(X) and invoking equality of topological entropies under finite-to-one equi-semiconjugacy, and (iii) lifting to the fuzzified system via a known monotonicity result. The candidate solution follows the same architecture: it establishes the product rule directly from separated/spanning definitions, builds the symmetric-power equi-semiconjugacy Φm with uniformly bounded finite fibers, uses the finite-to-one equi-semiconjugacy principle for equi-continuous NDSs to get h(Km)=m h(f), and then embeds K(X) into F(X) via crisp fuzzy sets to conclude h(f̃0,∞)≥h(f̄0,∞)=+∞. Aside from a minor counting slip in the paper (the fiber bound “≤ m!” is not generally valid but finiteness suffices), the arguments are aligned and correct.