On large indecomposable Banach spaces

Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density two to continuum. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly compact (strictly singular)

Monotone maps of GG-like continua with positive topological entropy yield indecomposability

In the previous paper Adv. Math. 304 (2017), pp. 793-808, we proved that if for any graph $G$, a homeomorphism on a $G$-like continuum $X$ has positive topological entropy, then the continuum $X$ contains an indecomposable subcontinuum. Also, if for a tree $G$, a monotone map on a $G$-like continuum $X$ has positive topological entropy, then the continuum $X$ contains an indecomposable subcontinuum. In this note, we extend these results. In fact, we prove that if for any graph $G$, a monotone map on a $G$-like continuum $X$ has positive topological entropy, then the continuum $X$ contains an indecomposable subcontinuum. Also we study topological entropy of monotone maps on Suslinean continua