We consider the class of the continuous functions from [0, 1] into itself which preserve the Lebesgue measure. This class endowed with the uniform metric constitutes a complete metric space. We investigate the dynamical properties of typical maps from the space.

Abstract:We study continuous countably-piecewise monotone interval maps and formulate conditions under which these are conjugate to maps of constant slope, particularly when this slope is given by the topological entropy of the map. We confine our investigation to the Markov case and phrase our conditions in the terminology of the Vere-Jones classification of infinite matrices.

Abstract. We answer affirmatively Coven's question [PC]: Suppose f : I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2? 0. Introduction. The aim of this paper is to estimate the topological entropy of an interval map knowing the cardinalities of its level sets.The topological entropy as a numerical measure for the complexity of a dynamical system is deeply studied in various contexts. One direction of possible research concerns the connection between entropy and level sets for a continuous (differentiable) map. Several interesting results have already been found-see for instance [Bo], [Ly], [MP], [MS]. However, many questions remain open even for interval maps. Our goal is to show one very particular result (Theorem 3.3), the proof of which is neither immediate nor easy. The solution is based on known strong results (Theorems 1.1 and 1.3) concerning the topological entropy and symbolic dynamics. We suppose that our proof can partially explain some difficulties that we meet when similar questions are considered in other topological dynamics.Let I = [a, b] be a closed real interval and let L 2 (I) be the set of all continuous functions mapping I into itself that satisfy the conditionWe show the following.

In this article we study dynamical behaviour of generic Lebesgue measure-preserving interval maps. We show that for each k ⩾ 1 the set of periodic points of period at least k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Moreover, we obtain analogous results also in the context of generic Lebesgue measure-preserving circle maps. Furthermore, building on the former results, we show that there is a dense collection of transitive Lebesgue measure-preserving interval maps whose periodic points have full Lebesgue measure and whose periodic points of period k have positive measure for each k ⩾ 1. Finally, we show that the generic continuous maps of the interval which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.

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 log 9. 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 countably 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 Dini's derivatives satisfy D + f (x) = D − f (x) = ∞ and D + f (x) = D − f (x) = −∞) in its fixed point 1/2.