scholarly journals A self-consistent dynamical system with multiple absolutely continuous invariant measures

2021 ◽  
Vol 8 (1) ◽  
pp. 9-32
Author(s):  
Fanni M. Sélley ◽  
Keyword(s):  
Dynamical System ◽  
Invariant Measures ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Self Consistent ◽  
Absolutely Continuous Invariant

STOCHASTIC PERTURBATIONS OF POSITION DEPENDENT RANDOM MAPS

2003 ◽  
Vol 03 (04) ◽  
pp. 545-557 ◽  
Author(s):  
WAEL BAHSOUN ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY
Keyword(s):  
Dynamical System ◽  
Invariant Measures ◽  
Invariant Density ◽  
Absolutely Continuous ◽  
Stochastic Perturbations ◽  
Random Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant

A random map is a dynamical system consisting of a collection of maps which are selected randomly by means of fixed probabilities at each iteration. In this note, we consider absolutely continuous invariant measures of random maps with position dependent probabilities and prove that they are stable under small stochastic perturbations. This result depends on a new lemma which handles arbitrarily small extra partition elements that may arise from the perturbation of the random map. For perturbations satisfying additional conditions, we give precise estimates of the error in the invariant density.

scholarly journals Multidimensional expanding maps with singularities: a pedestrian approach

2012 ◽  
Vol 33 (1) ◽  
pp. 168-182 ◽  
Author(s):  
CARLANGELO LIVERANI
Keyword(s):  
Invariant Measures ◽  
Statistical Properties ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Basic Properties ◽  
Bv Functions ◽  
Absolutely Continuous Invariant ◽  
Functions Of Bounded Variations ◽  
Bounded Derivative

AbstractI provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses basic properties of multidimensional BV functions (the space of functions of bounded variations).

scholarly journals Absolutely continuous invariant measures for non-uniformly expanding maps

2009 ◽  
Vol 29 (4) ◽  
pp. 1185-1215 ◽  
Author(s):  
HUYI HU ◽  
SANDRO VAIENTI
Keyword(s):  
Invariant Measure ◽  
Fixed Points ◽  
Large Class ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractFor a large class of non-uniformly expanding maps of ℝm, with indifferent fixed points and unbounded distortion and that are non-necessarily Markovian, we construct an absolutely continuous invariant measure. We extend previously used techniques for expanding maps on quasi-Hölder spaces to our case. We give general conditions and provide examples to which our results apply.

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