scholarly journals Counting preimages

2017 ◽  
Vol 38 (5) ◽  
pp. 1837-1856 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
ANA RODRIGUES
Keyword(s):  
Finite Type ◽  
Topological Entropy ◽  
Equal Probability ◽  
Topological Conjugacy ◽  
Metric Entropy ◽  
Backward Trajectories ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Special Measure ◽  
Natural Measure

For non-invertible maps, subshifts that are mainly of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, which are random in the sense that, at each step, every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It turns out that, instead of the topological entropy, we get the metric entropy of a special measure, which we call the fair measure. In general, this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy.

Fair measure and fair entropy for non-Markov interval maps

2016 ◽  
Vol 17 (05) ◽  
pp. 1750035 ◽  
Author(s):  
Ana Rodrigues ◽  
Yiwei Zhang
Keyword(s):  
Equilibrium State ◽  
Topological Entropy ◽  
Thermodynamic Formalism ◽  
Equal Probability ◽  
Stochastic Mechanics ◽  
Tent Map ◽  
Backward Trajectories ◽  
Interval Maps ◽  
Special Measure

In a recent paper [8] the entropy of a special measure, the fair measure was introduced. The fair entropy is computed following backward trajectories in a way such that at each step every preimage can be chosen with equal probability. In this paper, we continue studying the fair measure and the fair entropy for non-invertible interval maps under the framework of thermodynamic formalism. We extend several results in [8] to the non-Markov setting, and we prove that for each symmetric tent map the fair entropy is equal to the topological entropy if and only if the slope is equal to [Formula: see text]. Moreover, we also show that the fair measure is usually an equilibrium state, which has its own interest in stochastic mechanics.

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.

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).

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.

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.

Common extensions and hyperbolic factor maps for coded systems

1995 ◽  
Vol 15 (3) ◽  
pp. 517-534 ◽  
Author(s):  
Doris Fiebig
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Topological Entropy ◽  
Topological Conjugacy ◽  
Shift Of Finite Type ◽  
Coded Systems ◽  
Factor Maps

AbstractThe classification of dynamical systems by the existence of certain common extensions has been carried out very successfully in the class of shifts of finite type (‘finite equivalence’, ‘almost topological conjugacy‘). We consider generalizations of these notions in the class of coded systems. Topological entropy is shown to be a complete invariant for the existence of a common coded entropy preserving extension. In contrast to the shift of finite type setting, this extension cannot be made bounded-to-1 in general. Common extensions with hyperbolic factor maps lead to a version of almost topological conjugacy for coded systems.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

"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].

scholarly journals Subspaces of interval maps related to the topological entropy

2019 ◽  
pp. 1 ◽  
Author(s):  
Xiaoxin Fan ◽  
Jian Li ◽  
Yini Yang ◽  
Zhongqiang Yang
Keyword(s):  
Topological Entropy ◽  
Interval Maps
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