scholarly journals Properties of invariant measures in dynamical systems with the shadowing property

2017 ◽  
Vol 38 (6) ◽  
pp. 2257-2294 ◽  
Author(s):  
JIAN LI ◽  
PIOTR OPROCHA
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Entropy Function ◽  
Shadowing Property ◽  
Topologically Transitive ◽  
Ergodic Measures ◽  
One To One ◽  
Transitive System

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$, ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.

On Devaney chaotic generalized shift dynamical systems

2013 ◽  
Vol 50 (4) ◽  
pp. 509-522 ◽  
Author(s):  
Fatemah Shirazi ◽  
Javad Sarkooh ◽  
Bahman Taherkhani
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Point ◽  
Periodic Points ◽  
List Type ◽  
Topologically Transitive ◽  
Generalized Shift ◽  
One To One ◽  
Shift Dynamical System ◽  
Infinite Sequences

In the following text we prove that in a generalized shift dynamical system (XГ, σφ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent: the dynamical system (XГ, σφ) is chaotic in the sense of Devaneythe dynamical system (XГ, σφ) is topologically transitivethe map φ: Г → Г is one to one without any periodic point.Also for infinite countable Г and finite discrete X with at least two elements (XГ, σφ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

Using Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds

1997 ◽  
Vol 07 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
Rabbijah Guder ◽  
Edwin Kreuzer
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Nonlinear Dynamical Systems ◽  
Powerful Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Long Term Behavior ◽  
Mapping Theory

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.

Tail pressure and the tail entropy function

2011 ◽  
Vol 32 (4) ◽  
pp. 1400-1417 ◽  
Author(s):  
YUAN LI ◽  
ERCAI CHEN ◽  
WEN-CHIAO CHENG
Keyword(s):  
Variational Principle ◽  
Invariant Measures ◽  
Direct Proof ◽  
Equilibrium States ◽  
Entropy Function ◽  
Topological Pressure

AbstractBurguet [A direct proof of the tail variational principle and its extension to maps. Ergod. Th. & Dynam. Sys.29 (2009), 357–369] presented a direct proof of the variational principle of tail entropy and extended Downarowicz’s results to a non-invertible case. This paper defines and discusses tail pressure, which is an extension of tail entropy for continuous transformations. This study reveals analogs of many known results of topological pressure. Specifically, a variational principle is provided and some applications of tail pressure, such as the investigation of invariant measures and equilibrium states, are also obtained.

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