scholarly journals Birkhoff spectrum for piecewise monotone interval maps

2021 ◽  
Vol 252 (2) ◽  
pp. 203-223
Author(s):  
Thomas Jordan ◽  
Michał Rams
Keyword(s):  
Piecewise Monotone ◽  
Interval Maps

"CONNECT-THE-DOTS" TREE MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1427-1431
Author(s):  
LLUÍS ALSEDÀ ◽  
JOHN GUASCHI ◽  
JÉRÔME LOS ◽  
FRANCESC MAÑOSAS ◽  
PERE MUMBRÚ
Keyword(s):  
Piecewise Monotone ◽  
Interval Maps ◽  
Work In Progress ◽  
Monotone Map ◽  
The Given

We announce the main results of work in progress on piecewise monotone models for patterns of tree maps. More precisely, we define a notion of pattern for tree maps, and given such a pattern, we construct a tree and a piecewise monotone map on this tree with the same pattern. This piecewise monotone model has the least entropy among all models exhibiting the given pattern and has "minimal dynamics". We also give a formula to compute this minimal entropy directly from the pattern. These results generalize the known results for interval maps and the results from Li & Ye [1993].

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).

scholarly journals Topological conjugacies of piecewise monotone interval maps

2001 ◽  
Vol 25 (2) ◽  
pp. 119-127 ◽  
Author(s):  
Nikos A. Fotiades ◽  
Moses A. Boudourides
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Topological Conjugacy ◽  
Markov Condition ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Topological Conjugacies

Our aim is to establish the topological conjugacy between piecewise monotone expansive interval maps and piecewise linear maps. First, we are concerned with maps satisfying a Markov condition and next with those admitting a certain countable partition. Finally, we compute the topological entropy in the Markov case.

ENTROPY OF MAPS WITH HORIZONTAL GAPS

2004 ◽  
Vol 14 (04) ◽  
pp. 1489-1492 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Topological Entropy ◽  
Entropy Function ◽  
Devil's Staircase ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Continuous Map ◽  
Devil’S Staircase ◽  
Staircase Structure

We study the behavior of topological entropy in one-parameter families of interval maps obtained from a continuous map f by truncating it at the level depending on the parameter. When f is piecewise monotone, the entropy function has the devil's staircase structure.

An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps

1994 ◽  
Vol 14 (4) ◽  
pp. 621-632 ◽  
Author(s):  
V. Baladi ◽  
D. Ruelle
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Constant Weight ◽  
Piecewise Constant ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Interval Map ◽  
Piecewise Continuous

AbstractWe consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor—Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.

scholarly journals Spaces of transitive interval maps

2014 ◽  
Vol 35 (7) ◽  
pp. 2151-2170 ◽  
Author(s):  
SERGIĬ KOLYADA ◽  
MICHAŁ MISIUREWICZ ◽  
L’UBOMÍR SNOHA
Keyword(s):  
Piecewise Linear ◽  
Real Interval ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Arcwise Connected

On a compact real interval, the spaces of all transitive maps, all piecewise monotone transitive maps and all piecewise linear transitive maps are considered with the uniform metric. It is proved that they are contractible and uniformly locally arcwise connected. Then the spaces of all piecewise monotone transitive maps with given number of pieces as well as various unions of such spaces are considered and their connectedness properties are studied.

scholarly journals Distributions of Order Patterns of Interval Maps

2013 ◽  
Vol 22 (3) ◽  
pp. 319-341 ◽  
Author(s):  
AARON ABRAMS ◽  
ERIC BABSON ◽  
HENRY LANDAU ◽  
ZEPH LANDAU ◽  
JAMES POMMERSHEIM
Keyword(s):  
Probability Distribution ◽  
Lebesgue Measure ◽  
Compatibility Condition ◽  
Necessary Condition ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Piecewise Continuous ◽  
Order Pattern ◽  
Exact Description

A permutation σ describing the relative orders of the first n iterates of a point x under a self-map f of the interval I=[0,1] is called an order pattern. For fixed f and n, measuring the points x ∈ I (according to Lebesgue measure) that generate the order pattern σ gives a probability distribution μn(f) on the set of length n permutations. We study the distributions that arise this way for various classes of functions f.Our main results treat the class of measure-preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each n this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general f, apart from an obvious compatibility condition, there is no restriction on the sequence {μn(f)}n=1,2,. . ..In addition, we give a necessary condition for f to have finite exclusion type, that is, for there to be finitely many order patterns that generate all order patterns not realized by f. Using entropy we show that if f is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then f cannot have finite exclusion type. This generalizes results of S. Elizalde.

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