scholarly journals C1-Genericity of symplectic diffeomorphisms and lower bounds for topological entropy

2017 ◽  
Vol 32 (4) ◽  
pp. 461-489 ◽  
Author(s):  
Thiago Catalan ◽  
Vanderlei Horita
Keyword(s):  
Topological Entropy ◽  
Lower Bounds ◽  
Symplectic Diffeomorphisms

scholarly journals On topological entropy of triangular maps of the square

1993 ◽  
Vol 48 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Lluís Alsedà ◽  
Sergiĭ F. Kolyada ◽  
Ľubomír Snoha
Keyword(s):  
Topological Entropy ◽  
Lower Bounds ◽  
Lower Semicontinuous ◽  
Continuous Maps ◽  
Triangular Maps

We study the topological entropy of triangular maps of the square. We show that such maps differ from the continuous maps of the interval because there exist triangular maps of the square of “type 2∞” with infinite topological entropy. The set of such maps is dense in the space of triangular maps of “type at most 2∞” and the topological entropy as a function of the triangular maps of the square is not lower semicontinuous. However, we show that for these maps the characterisation of the lower bounds of the topological entropy depending on the set of periods is the same as for the continuous maps of the interval.

Automatized Search for Complex Symbolic Dynamics with Applications in the Analysis of a Simple Memristor Circuit

2014 ◽  
Vol 24 (07) ◽  
pp. 1450104 ◽  
Author(s):  
Zbigniew Galias
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Lower Bounds ◽  
Symbolic Dynamics ◽  
Third Order ◽  
Return Map ◽  
Topological Sense

An automatized method to search for complex symbolic dynamics is proposed. The method can be used to show that a given dynamical system is chaotic in the topological sense. Application of this method in the analysis of a third-order memristor circuit is presented. Several examples of symbolic dynamics are constructed. Positive lower bounds for the topological entropy of an associated return map are found showing that the system is chaotic in the topological sense.

scholarly journals Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation

2019 ◽  
Vol 40 (9) ◽  
pp. 2552-2570
Author(s):  
SCOTT SCHMIEDING
Keyword(s):  
Asymptotic Behavior ◽  
Automorphism Group ◽  
Finite Type ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Shift Of Finite Type ◽  
Dimension Group ◽  
Bounded Below

Let $(X_{A},\unicode[STIX]{x1D70E}_{A})$ be a shift of finite type and $\text{Aut}(\unicode[STIX]{x1D70E}_{A})$ its corresponding automorphism group. Associated to $\unicode[STIX]{x1D719}\in \text{Aut}(\unicode[STIX]{x1D70E}_{A})$ are certain Lyapunov exponents $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$, which describe asymptotic behavior of the sequence of coding ranges of $\unicode[STIX]{x1D719}^{n}$. We give lower bounds on $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$ in terms of the spectral radius of the corresponding action of $\unicode[STIX]{x1D719}$ on the dimension group associated to $(X_{A},\unicode[STIX]{x1D70E}_{A})$. We also give lower bounds on the topological entropy $h_{\text{top}}(\unicode[STIX]{x1D719})$ in terms of a distinguished part of the spectrum of the action of $\unicode[STIX]{x1D719}$ on the dimension group, but show that, in general, $h_{\text{top}}(\unicode[STIX]{x1D719})$ is not bounded below by the logarithm of the spectral radius of the action of $\unicode[STIX]{x1D719}$ on the dimension group.

scholarly journals A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms

2013 ◽  
Vol 34 (5) ◽  
pp. 1503-1524 ◽  
Author(s):  
THIAGO CATALAN ◽  
ALI TAHZIBI
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Topological Entropy ◽  
Upper Semicontinuity ◽  
Periodic Points ◽  
Surface Diffeomorphisms ◽  
Symplectic Diffeomorphisms ◽  
Symbolic Extension ◽  
Symplectic Diffeomorphism ◽  
Volume Preserving

AbstractWe prove that a${C}^{1} $generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that${C}^{1} $generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the${C}^{1} $topology.

Lower bounds of the topological entropy for continuous maps of the circle of degree one

Nonlinearity ◽  
1988 ◽  
Vol 1 (3) ◽  
pp. 463-479 ◽  
Author(s):  
L Alseda ◽  
J Llibre ◽  
F Manosas ◽  
M Misiurewicz
Keyword(s):  
Topological Entropy ◽  
Lower Bounds ◽  
Continuous Maps

scholarly journals METASTABILITY, LYAPUNOV EXPONENTS, ESCAPE RATES, AND TOPOLOGICAL ENTROPY IN RANDOM DYNAMICAL SYSTEMS

2013 ◽  
Vol 13 (04) ◽  
pp. 1350004 ◽  
Author(s):  
GARY FROYLAND ◽  
OGNJEN STANCEVIC
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Topological Entropy ◽  
Lower Bounds ◽  
Random Systems ◽  
Random Dynamical Systems ◽  
Upper And Lower Bounds ◽  
Random System ◽  
Random Maps ◽  
Frobenius Operator

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

scholarly journals Lower bounds for the topological entropy

2005 ◽  
Vol 12 (3) ◽  
pp. 555-565 ◽  
Author(s):  
Katrin Gelfert ◽  
Keyword(s):  
Topological Entropy ◽  
Lower Bounds
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