Eigenvalues of Transfer Operators for Dynamical Systems with Holes

2014 ◽  
pp. 31-39
Author(s):  
Oscar F. Bandtlow ◽  
Oliver Jenkinson
Keyword(s):  
Dynamical Systems ◽  
Transfer Operators

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.

scholarly journals Locating Ruelle–Pollicott resonances*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 513-566
Author(s):  
Oliver Butterley ◽  
Niloofar Kiamari ◽  
Carlangelo Liverani
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Precise Information ◽  
Transfer Operators ◽  
Hyperbolic Diffeomorphisms ◽  
New Information ◽  
Monotone Maps ◽  
Unitary Approach

Abstract We study the spectrum of transfer operators associated to various dynamical systems. Our aim is to obtain precise information on the discrete spectrum. To this end we propose a unitary approach. We consider various settings where new information can be obtained following different branches along the proposed path. These settings include affine expanding Markov maps, uniformly expanding Markov maps, non-uniformly expanding or simply monotone maps, hyperbolic diffeomorphisms. We believe this approach could be greatly generalised.

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 t-Entropy formulae for concrete classes of transfer operators

2019 ◽  
pp. 122-128
Author(s):  
Krzysztof Bardadyn ◽  
Bartosz Kosma Kwasniewski ◽  
Kirill S. Kurnosenko ◽  
Andrei V. Lebedev
Keyword(s):  
Dynamical Systems ◽  
Spectral Theory ◽  
Invariant Measures ◽  
Weighted Shift ◽  
Legendre Transform ◽  
Shift Operators ◽  
Transfer Operators ◽  
First Case ◽  
Explicit Formulae ◽  
Spectral Theory Of Operators

t-Entropy is a principal object of the spectral theory of operators, generated by dynamical systems, namely, weighted shift operators and transfer operators. In essence t-entropy is the Fenchel – Legendre transform of the spectral potential of an operator in question and derivation of explicit formulae for its calculation is a rather nontrivial problem. In the article explicit formulae for t-entropy for two the most exploited in applications classes of transfer operators are obtained. Namely, we consider transfer operators generated by reversible mappings (i. e. weighted shift operators) and transfer operators generated by local homeomorphisms (i. e. Perron – Frobenius operators). In the first case t-entropy is computed by means of integrals with respect to invariant measures, while in the second case it is computed in terms of integrals with respect to invariant measures and Kolmogorov – Sinai entropy.

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