scholarly journals Strict inequalities for the entropy of transitive piecewise monotone maps

2005 ◽  
Vol 13 (2) ◽  
pp. 451-468 ◽  
Author(s):  
Michał Misiurewicz ◽  
◽  
Peter Raith ◽  
Keyword(s):  
Piecewise Monotone ◽  
Monotone Maps ◽  
Strict Inequalities

ESSENTIAL ENTROPY-CARRYING HORSESHOES AS SET LIMITS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250195 ◽  
Author(s):  
STEVEN M. PEDERSON
Keyword(s):  
Topological Entropy ◽  
Convergent Sequence ◽  
Invariant Set ◽  
Invariant Sets ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Monotone Map ◽  
Monotone Maps

This paper studies the set limit of a sequence of invariant sets corresponding to a convergent sequence of piecewise monotone interval maps. To do this, the notion of essential entropy-carrying set is introduced. A piecewise monotone map f with an essential entropy-carrying horseshoe S(f) and a sequence of piecewise monotone maps [Formula: see text] converging to f is considered. It is proven that if each gi has an invariant set T(gi) with at least as much topological entropy as f, then the set limit of [Formula: see text] contains S(f).

SOME RESULTS ON ENTROPY AND SEQUENCE ENTROPY

1999 ◽  
Vol 09 (09) ◽  
pp. 1731-1742 ◽  
Author(s):  
F. BALIBREA ◽  
V. JIMÉNEZ LÓPEZ ◽  
J. S. CÁNOVAS PEÑA
Keyword(s):  
Topological Entropy ◽  
Elementary Proof ◽  
Type Inequality ◽  
Topological Invariants ◽  
Metric Entropy ◽  
Sequence Entropy ◽  
Piecewise Monotone ◽  
Monotone Maps ◽  
Topological Sequence Entropy

In this paper we study some formulas involving metric and topological entropy and sequence entropy. We summarize some classical formulas satisfied by metric and topological entropy and ask the question whether the same or similar results hold for sequence entropy. In general the answer is negative; still some questions involving these formulas remain open. We make a special emphasis on the commutativity formula for topological entropy h(f ◦ g)=h(g ◦ f) recently proved by Kolyada and Snoha. We give a new elementary proof and use similar ideas to prove commutativity formulas for metric entropy and other topological invariants. Finally we prove a Misiurewicz–Szlenk type inequality for topological sequence entropy for piecewise monotone maps on the interval I=[0, 1]. For this purpose we introduce the notion of upper entropy.

scholarly journals Asymptotic behaviour of iterated piecewise monotone maps

1988 ◽  
Vol 8 (1) ◽  
pp. 111-131 ◽  
Author(s):  
Jürgen Willms
Keyword(s):  
Finite Number ◽  
Asymptotic Behaviour ◽  
Disjoint Union ◽  
Monotone Functions ◽  
Connected Component ◽  
Piecewise Monotone ◽  
Closed Intervals ◽  
Monotone Maps ◽  
Typical Point ◽  
Topological Sense

AbstractIn this paper the asymptotic behaviour of piecewise monotone functions f: I → I with a finite number of discontinuities is studied (where I ⊆ ℝ is a compact interval). It is shown that there is a finite number of f-almost-invariant subsets C1,…, Cr, R1,…, Rs, where each Ci is a disjoint union of closed intervals and each Rj is a Cantor-like subset of I, such that if x is a ‘typical’ point in I (in a topological sense) then exactly one of the following three possibilities will happen:(1) {fn (x)}n ≥ 0 eventually ends up in some Ci.(2) {fn (x)}n ≥ 0 is attracted to some Rj.(3) {fn (x): n ≥ 0} is contained in an open, invariant set Z ⊆ I, which is such that for each n ≥ 1 fn is monotone and continuous on each connected component of Z.Moreover, f acts topologically transitively on each Ci and minimally on each Rj. Furthermore, it is shown how the sets C1,…, Cr, R1,…, Rs can be constructed. Finally, our results are applied to some examples.

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