Geodesic
One-Dimensional Hyperbolic Attractors with Low Topological Entropy
Informatics and Automation, Geometric topology and set theory, Tome 247 (2004), pp. 35-40.

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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. 
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Kh. Boti. One-Dimensional Hyperbolic Attractors with Low Topological Entropy. Informatics and Automation, Geometric topology and set theory, Tome 247 (2004), pp. 35-40. http://geodesic.mathdoc.fr/item/TRSPY_2004_247_a3/

[1] Adler R. L., Konheim A. G., McAndrew M. H., “Topological entropy”, Trans. Amer. Math. Soc., 114 (1965), 309–319 | DOI | MR | Zbl

[2] Bothe H. G., “How hyperbolic attractors determine their basins”, Nonlinearity, 9 (1996), 1173–1190 | DOI | MR | Zbl

[3] Bowen R., “Topological entropy and Axiom $\mathrm{A}$”, Global analysis, Proc. Symp. Pure Math., 14, Amer. Math. Soc., Providence, RI, 1970, 23–41 | MR

[4] Smale S., “Differential dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747–817 | DOI | MR

[5] Williams R. F., “One-dimensional non-wandering sets”, Topology, 6 (1967), 473–487 | DOI | MR | Zbl