Topological conjugation of Lorenz maps by β-transformations

1990 ◽  
Vol 107 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul Glendinning
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Necessary And Sufficient ◽  
Constant Slope ◽  
Lorenz Maps ◽  
Basic Sets ◽  
Topological Conjugation

AbstractNecessary and sufficient conditions for a Lorenz map to be topologically conjugate to a piecewise linear map with constant slope (a β-transformation) are given, first in terms of kneading invariants of the maps and then in terms of the topological entropy restricted to basic sets. The dynamics of β-transformations is also described.

PIECEWISE LINEAR MODEL FOR TREE MAPS

2001 ◽  
Vol 11 (12) ◽  
pp. 3163-3169 ◽  
Author(s):  
MATHIEU BAILLIF ◽  
ANDRÉ DE CARVALHO
Keyword(s):  
Linear Model ◽  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Monotone Map ◽  
Piecewise Linear Model ◽  
Constant Slope

We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.

scholarly journals Isomorphisms between positive and negative -transformations

2012 ◽  
Vol 34 (1) ◽  
pp. 153-170 ◽  
Author(s):  
CHARLENE KALLE
Keyword(s):  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Constant Slope

AbstractWe compare a piecewise linear map with constant slope $\beta \gt 1$ and a piecewise linear map with constant slope $-\beta $. These maps are called the positive and negative $\beta $-transformations. We show that for a certain set of $\beta $s, the multinacci numbers, there exists a measurable isomorphism between these two maps. We further show that for all other values of $\beta $between 1 and 2 the two maps cannot be isomorphic.

One-Dimensional Semiconjugacy Revisited

2003 ◽  
Vol 13 (07) ◽  
pp. 1657-1663 ◽  
Author(s):  
J. F. Alves ◽  
J. Sousa Ramos
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Linear Map ◽  
One Dimensional ◽  
Piecewise Linear Map ◽  
Piecewise Monotone ◽  
Positive Topological Entropy ◽  
Interval Map

Let f be a piecewise monotone interval map with positive topological entropy h(f)= log (s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.

scholarly journals Embedding the symbolic dynamics of Lorenz maps

2014 ◽  
Vol 156 (3) ◽  
pp. 505-519 ◽  
Author(s):  
TONY SAMUEL ◽  
NINA SNIGIREVA ◽  
ANDREW VINCE
Keyword(s):  
Topological Entropy ◽  
Symbolic Dynamics ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Interval Map ◽  
Piecewise Continuous ◽  
Necessary And Sufficient ◽  
Lorenz Maps

AbstractNecessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

scholarly journals The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions

2015 ◽  
Vol 25 (13) ◽  
pp. 1550184 ◽  
Author(s):  
Carlos Lopesino ◽  
Francisco Balibrea-Iniesta ◽  
Stephen Wiggins ◽  
Ana M. Mancho
Keyword(s):  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
The Third ◽  
Autonomous Version ◽  
Area Preserving ◽  
Chaotic Saddle

In this paper, we prove the existence of a chaotic saddle for a piecewise-linear map of the plane, referred to as the Lozi map. We study the Lozi map in its orientation and area preserving version. First, we consider the autonomous version of the Lozi map to which we apply the Conley–Moser conditions to obtain the proof of a chaotic saddle. Then we generalize the Lozi map on a nonautonomous version and we prove that the first and the third Conley–Moser conditions are satisfied, which imply the existence of a chaotic saddle. Finally, we numerically demonstrate how the structure of this nonautonomous chaotic saddle varies as parameters are varied.

DIGITAL CONTROL AS SOURCE OF CHAOTIC BEHAVIOR

2010 ◽  
Vol 20 (05) ◽  
pp. 1365-1378 ◽  
Author(s):  
GÁBOR CSERNÁK ◽  
GÁBOR STÉPÁN
Keyword(s):  
Mechanical System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Linear Control ◽  
Control Loop ◽  
Digital Implementation ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Chaotic Vibrations ◽  
Processing Delay

In the present paper, we introduce and analyze a mechanical system, in which the digital implementation of a linear control loop may lead to chaotic behavior. The amplitude of such oscillations is usually very small, this is why these are called micro-chaotic vibrations. As a consequence of the digital effects, i.e. the sampling, the processing delay and the round-off error, the behavior of the system can be described by a piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that the map is chaotic.

The Dynamics of a Piecewise Linear Map and its Smooth Approximation

1997 ◽  
Vol 07 (02) ◽  
pp. 351-372 ◽  
Author(s):  
D. Aharonov ◽  
R. L. Devaney ◽  
U. Elias
Keyword(s):  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Invariant Curves ◽  
Smooth Approximation ◽  
Linear Map ◽  
Island Structure ◽  
The Real ◽  
Piecewise Linear Map ◽  
Real Analytic ◽  
Area Preserving

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.

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