Permutations and topological entropy for interval maps

Nonlinearity ◽  
2003 ◽  
Vol 16 (3) ◽  
pp. 971-976 ◽  
Author(s):  
Micha  Misiurewicz
Keyword(s):  
Topological Entropy ◽  
Interval Maps

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.

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

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 The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

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.

CHAOTIC BEHAVIOR OF INTERVAL MAPS AND TOTAL VARIATIONS OF ITERATES

2004 ◽  
Vol 14 (07) ◽  
pp. 2161-2186 ◽  
Author(s):  
GOONG CHEN ◽  
TINGWEN HUANG ◽  
YU HUANG
Keyword(s):  
Nonlinear Systems ◽  
Initial Data ◽  
Topological Entropy ◽  
Chaotic Behavior ◽  
Homoclinic Orbits ◽  
Oscillatory Behavior ◽  
Exponential Rate ◽  
Interval Maps ◽  
Interval Map ◽  
Sensitive Dependence

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fnwill display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fnon subintervals. There are four distinctive cases of the growth of total variations of fnas n→∞:(i) the total variations of fnon I remain bounded;(ii) they grow unbounded, but not exponentially with respect to n;(iii) they grow with an exponential rate with respect to n;(iv) they grow unbounded on every subinterval of I.We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fnare more extreme, the more complex is the behavior of the system.

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.

Stability of the maximal measure for piecewise monotonic interval maps

1997 ◽  
Vol 17 (6) ◽  
pp. 1419-1436 ◽  
Author(s):  
PETER RAITH
Keyword(s):  
Topological Entropy ◽  
Stability Condition ◽  
Finite Union ◽  
Small Perturbations ◽  
Interval Maps ◽  
Positive Topological Entropy ◽  
Maximal Measure ◽  
Closed Intervals ◽  
Piecewise Monotonic

Let $T:X\to{\Bbb R}$ be a piecewise monotonic map, where $X$ is a finite union of closed intervals. Define $R(T)=\bigcap_{n=0}^{\infty} \overline{T^{-n}X}$, and suppose that $(R(T),T)$ has a unique maximal measure $\mu$. The influence of small perturbations of $T$ on the maximal measure is investigated. If $(R(T),T)$ has positive topological entropy, and if a certain stability condition is satisfied, then every piecewise monotonic map $\tilde{T}$, which is contained in a sufficiently small neighbourhood of $T$, has a unique maximal measure $\tilde{\mu}$, and the map $\tilde{T}\mapsto\tilde{\mu}$ is continuous at $T$.

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.

Sign in / Sign up

Export Citation Format

Share Document