Typical properties of interval maps preserving the Lebesgue measure ^*

Abstract

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.

S-limit shadowing is generic for continuous Lebesgue measure preserving circle maps

“…The particular setting that we are interested in this paper is the family of continuous Lebesgue measure preserving maps of the unit circle C λ(S 1 ) endowed with topology of uniform convergence, which makes it a complete space. Topological and measure theoretical properties of generic Lebesgue measure preserving interval maps were studied in [4,6,5]. We obtain the following two new results: Theorem 1.…”

“…Applying in Subsection 3.4 both perturbations and also the result of Lemma 18 on a sequence {g m } m≥1 dense in C λ,0 (S 1 ), we arrive to new sequences {θ m } m≥1 of maps from C λ(S 1 ), their neighborhoods {U m } m≥1 and also carefully constructed partitions {Q m } m≥1 . In particular, using (6) we can ensure that for some pairs m < n, Q n is a refinement of Q m . The final step consists of the proof that all maps in…”

Chaotic tracial dynamics

“…Various genericity results for noninvertible transformations of (I, µ), where µ is an OU measure on the interval I, are established in [9]. In particular, the following holds.…”

“…In particular, if X is an odd sphere, then the extended action is on an inductive limit of prime dimension drop algebras and this procedure provides another description of the crossed products considered in [30]. Endomorphisms also allow for the inclusion of noninvertible tracial dynamics h : X → X, for example the Pomeau-Manneville-type system described in [56, §5.4] (corrected in § 4 below), and to appeal to and interpret existing results [9] about such systems.…”

S-limit shadowing is generic for continuous Lebesgue measure-preserving circle maps

“…The particular setting that we are interested in this paper is the family of continuous Lebesgue measure-preserving maps of the unit circle C λ(S 1 ) endowed with the topology of uniform convergence, which makes it a complete space. Topological and measure-theoretic properties of generic Lebesgue measure-preserving interval maps were studied in [4][5][6]. We obtain the following two new results.…”