Not having a background in entropy, I need to gain some facility with computing it in LCA. In any case, for a "finite-dimensional" compact connected abelian group G, your/Prodanov's td(G) is fully invariant, and also a(G), the arc component of 0 in G, is fully invariant; hence, their intersection, say c(G), is fully invariant; c(G) turns out to be a dense rational vector space with dim_Q[c(G)] = dim_R[G], so a continuous endomorphism f of such a G is uniquely determined by its action on c(G); but still it is not clear what the f-invariant closed subgroups look like in general for G; a closed subgroup of such a G is profinite or a subprotorus or a product of a profinite subgroups and a subprotorus; such an f would be invariant on a subprotorus H of G if dim_R[H] = dim_R[G], but not in general for a subprotorus of smaller dimension because, for example, factors of a product of subprotori could be permuted by our morphism f; with respect to profinite subgroups of G being f-invariant, it is possible but any interesting endomorphism f will move things around in such a way that profinite subgroups are not f-invariant (in fact, the usual case is the profinite subgroups of the same "nameless size" to be permuted in the large); so, with respect to the addition theorem restricted to the protorus setting, if the topological entropy of an endomorphism of a finite abelian group is 0, then we get a 'yes' for the addition theorem in the protorus setting; for the general setting (revisiting the "invariant-profinite-subgroup"case in the process), I'm guessing the addition theorem is resolved for R^m so in a hypothetical generic LCA group L we assume WLOG there is a compact open subgroup; this L has connected component of 0 a protorus; the part of L "over" L_0 is WLOG topologically torsion (to explain WLOG here is a key part of the argument), so one can focus on LCA groups obtained by extending topologically torsion groups by protori or vice versa; this area is hopelessly complicated in general (constituting the duals of discrete "mixed" abelian groups) but most of the classes of LCA groups in the literature do not involve these ugly monsters (which of course comprise most of the category), so one compromises by focusing on products of topologically torsion groups and protori; once again, the same logic for WLOG above in the "vector-subgroup-free" setting applies here in the setting of "products of topologically torsion groups and protori" in that we can assume WLOG that the topologically torsion factor is periodic; so we study f-invariant closed subgroups of (i) finite-dimensional real vector spaces, (ii) periodic LCA groups, (iii) protori. I'm guessing based on above that (i) and (iii) are positively resolved so one can reduce one's focus to veracity of the addition theorem in the setting of periodic LCA groups; this reduces a la Braconnier/Villenkin to the truth of the matter for periodic p-groups; that is, to finite products of p-adic integers and/or p-adic numbers and/or finite p-groups; presumably arguing as above we can restrict our attention to the veracity of the addition theorem in the setting of (Z_p)^m x (Q_p)^n; again, making my 11th presumption, I'm guessing the addition theorem is resolved for closed subgroups of (Q_p)^n, which also subsumes the case (Z_p)^m. So, if a miracle just happened and all the above is true, then the addition problem reduces to the question of whether for a "non-splitting" compact abelian group H realized as the extension of a profinite group by a protorus, or vice versa, satisfies the addition theorem; that is, H is homeomorphic to, but not topologically isomorphic to, H_0 x H/H_0 with (WLOG) H_0 a protorus and H/H_0 profinite; so, if one knows that topological entropy is strictly a "topological property" in this setting, then we are done, right?

Thanks for all you do for modern mathematics research!

I am not sure how one replies on this forum, so I am appending to my initial comment.

Is the Addition theorem true for (Q_p)^n, n a positive integer, Q_p the p-adic numbers?

The proposed reduction is not clear to me. As mentioned in the comments to the question, Bowen established the Addition theorem for compact metrisable groups G and this can easily be extended to all compact groups.

In a hypothetical generic LCA group L we assume WLOG there is a compact open subgroup; this L has connected component of 0 a protorus; the part of L "over" L_0 is WLOG periodic (lemma), so in LCA it suffices to study whether "non-splitting" extensions of a periodic group by a protorus (or vice versa) satisfy the Addition theorem, is that right? Do the closed subgroups of (Q_p)^n satisfy the Addition theorem? Aloha.