ON PIECEWISE AFFINE INTERVAL MAPS WITH COUNTABLY MANY LAPS JOZEF BOBOK AND MARTIN SOUKENKA Abstract. We study a special conjugacy class F of continuous piecewise monotone interval maps: with countably many laps, which are locally eventually onto and have common topological entropy log9. We show that F contains a piecewise affine map f λ with a constant slope λ if and only if λ ≥ 9. Our result specifies the known fact that for piecewise affine interval leo maps with count- ably many pieces of monotonicity and a constant slope ±λ, the topological (measure-theoretical) entropy is not determined by λ. We also consider maps from the class F preserving the Lebesgue measure. We show that some of them have a knot point (a point x where D + f (x)= D - f (x)= ∞ and D + f (x)= D - f (x)= −∞) in its fixed point 1/2. 1. Introduction, main results In their interesting article [5] the authors showed, among other re- sults, that laws ruling piecewise monotone interval maps do not work when we admit countably many pieces of monotonicity. They showed Theorem 1.1. [5] For λ> 2 and every α> log 2 there exists a con- tinuous map T λ : [0, 1] → [0, 1] with the following properties: (i) f has countably many turning points. (ii) f is locally eventually onto (leo). (iii) |T ′ λ (x)| = λ for all x ∈ (0, 1), except at the turning points of T . (iv) h top (T λ ) ≤ α. (v) For every ergodic T λ -invariant Borel probability measure the partition into the laps of T λ has finite entropy. Date : August 26, 2009. 2000 Mathematics Subject Classification. 15A48, 37B40, 47B60. Key words and phrases. interval map, knot point, topological conjugacy, topo- logical entropy. The first author was partly supported by the Grant Agency of the Czech Republic contract number 201/09/0854. He also gratefully acknowledges the support of the MYES of the Czech Republic via contract MSM 6840770010. 1