Periodic points and topological entropy of one dimensional maps

1980 ◽  
pp. 18-34 ◽  
Author(s):  
Louis Block ◽  
John Guckenheimer ◽  
Michal Misiurewicz ◽  
Lai Sang Young
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
One Dimensional ◽  
One Dimensional 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 Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

Thermodynamic formalism for certain nonhyperbolic maps

1999 ◽  
Vol 19 (5) ◽  
pp. 1365-1378 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Number Theory ◽  
Gibbs Measure ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Periodic Points ◽  
Two Dimensional ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Conformal Measures ◽  
One Dimensional Maps

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS AND STABILITY OF SARKOVSKĬ ORDERING

1999 ◽  
Vol 09 (09) ◽  
pp. 1867-1876 ◽  
Author(s):  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Infinite Number ◽  
Periodic Points ◽  
One Dimensional ◽  
One Dimensional Maps

In this paper we present Sarkovskĭ theorem for multidimensional perturbations of one-dimensional maps. We outline a proof that if the unperturbed one-dimensional map has a point of period n, then sufficiently close multidimensional perturbations of this map have periodic points of all periods which are allowed by Sarkovskĭ theorem. We describe also how the approach from the proof of this theorem is used to show the existence of an infinite number of periodic points for Rössler equations.

TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS

2001 ◽  
Vol 11 (05) ◽  
pp. 1443-1446 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Topological Entropy ◽  
Positive Entropy ◽  
One Dimensional ◽  
Interval Map ◽  
One Dimensional Maps

We prove that if an interval map of positive entropy is perturbed to a compact multidimensional map then the topological entropy cannot drop down considerably if the perturbation is small.

On the full periodicity kernel for one-dimensional maps

1999 ◽  
Vol 19 (1) ◽  
pp. 101-126
Author(s):  
M. CARME LESEDUARTE ◽  
JAUME LLIBRE
Keyword(s):  
Topological Space ◽  
Periodic Points ◽  
One Dimensional ◽  
Fixed Set ◽  
Branching Point ◽  
One Dimensional Maps

Let $\bpropto$ be the topological space obtained by identifying the points 1 and 2 of the segment $[0,3]$ to a point. Let $\binfty$ be the topological space obtained by identifying the points 0, 1 and 2 of the segment $[0,2]$ to a point. An $\bpropto$ (respectively $\binfty$) map is a continuous self-map of $\bpropto$ (respectively $\binfty$) having the branching point fixed. Set $E\in\{\bpropto,\binfty\}$. Let $f$ be an $E$ map. We denote by $\Per(f)$ the set of periods of all periodic points of $f$. The set $K \subset{\mathbb N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) if $f$ is an $E$ map and $K\subset \Per(f)$, then $\Per(f)={\mathbb N}$; (2) for each $k\in K$ there exists an $E$ map $f$ such that $\Per(f)={\mathbb N}\setminus\{ k\}$. In this paper we compute the full periodicity kernel of $\bpropto$ and $\binfty$.

Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps

Nonlinearity ◽  
2008 ◽  
Vol 21 (11) ◽  
pp. 2555-2567 ◽  
Author(s):  
Ming-Chia Li ◽  
Ming-Jiea Lyu ◽  
Piotr Zgliczyński
Keyword(s):  
Topological Entropy ◽  
One Dimensional ◽  
One Dimensional Maps
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