ENTROPY OF EQUILIBRIUM MEASURES OF CONTINUOUS PIECEWISE MONOTONIC MAPS

2004 ◽  
Vol 04 (01) ◽  
pp. 85-94 ◽  
Author(s):  
JÉRÔME BUZZI
Keyword(s):  
Dynamical Systems ◽  
Positive Entropy ◽  
Unique Equilibrium ◽  
Mixing Properties ◽  
Equilibrium Measures ◽  
Thermodynamical Formalism ◽  
Transfer Operators ◽  
Piecewise Monotonic

Considering the thermodynamical formalism of dynamical systems, P. Walters showed that for β-transformations all Lipschitz weights define quasi-compact transfer operators and therefore unique equilibrium measures which additionally have positive entropy and good mixing properties. In this note we generalize this to continuous piecewise monotonic maps of the interval. The case of piecewise monotonic maps with discontinuities remains open.

scholarly journals Lattice reduction in two dimensions: analyses under realistic probabilistic models

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Brigitte Vallée ◽  
Antonio Vera
Keyword(s):  
Dynamical Systems ◽  
Probabilistic Models ◽  
Two Dimensions ◽  
Lattice Reduction ◽  
Lll Algorithm ◽  
Transfer Operators ◽  
International Audience ◽  
Dimension 2

International audience The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm "inside'' the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All the main parameters are studied: execution parameters which describe the behaviour of the algorithm itself as well as output parameters, which describe the geometry of reduced bases.

Entropy of $C^\ast$-dynamical systems defined by bitstreams

1998 ◽  
Vol 18 (4) ◽  
pp. 859-874 ◽  
Author(s):  
V. YA. GOLODETS ◽  
ERLING ST&\Oslash;RMER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Positive Entropy ◽  
Completely Positive ◽  
Car Algebra

We study automorphisms of the CAR-algebra obtained from binary shifts. We consider cases when the $C^\ast$-dynamical system is asymptotically abelian, is proximally asymptotically abelian, is an entropic $K$-system or has completely positive entropy. The entropy is computed in several cases.

ON FUNCTIONS ATTRACTING POSITIVE ENTROPY

2017 ◽  
Vol 97 (1) ◽  
pp. 69-79 ◽  
Author(s):  
ANNA LORANTY ◽  
RYSZARD J. PAWLAK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Positive Entropy ◽  
Density Point

We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$.

scholarly journals THE PRIME DECOMPOSITION OF KNOTTED PERIODIC ORBITS IN DYNAMICAL SYSTEMS

1994 ◽  
Vol 03 (01) ◽  
pp. 83-120 ◽  
Author(s):  
MICHAEL C. SULLIVAN
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Positive Entropy ◽  
Prime Decomposition ◽  
Prime Factors

Templates are used to capture the knotting and linking patterns of periodic orbits of positive entropy flows in 3 dimensions. Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits. We will also see that determining whether two templates are different is highly nontrivial.

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