On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

We prove that if f : G → G is a map on a topological graph G such that the inverse limit lim ← ( G , f ) is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Chéritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if G = [ 0 , 1 ] , then h ( f ) ∈ { 0 , ∞ } , and combined with a result of Ito shows that certain dynamical systems on compact finite-dimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.