One-Dimensional Hyperbolic Attractors with Low Topological Entropy

Attractors play a central role in the theory of dynamical systems, and the topological entropy of an attractor $\Lambda$ yields an important numerical invariant of $\Lambda$. Here, we consider the dynamics defined by a diffeomorphism $f: M \to M$ of a $C^{1}$ manifold $M$ and the corresponding $1$-dimensional hyperbolic attractors. For attractors $\Lambda$ of this kind, one can measure, in a quite natural way, the topological complexity by a positive integer $c(\Lambda )$. It is shown in Theorem A that attractors with topological entropy close to $0$ must have high complexity. The possible values of the topological entropy for $1$-dimensional hyperbolic attractors are logarithms of certain positive algebraic integers, and these values are dense in the set of all positive real numbers. This fact is presented in Theorem B.