scholarly journals Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

Nonlinearity ◽  
2016 ◽  
Vol 29 (9) ◽  
pp. 2775-2797 ◽  
Author(s):  
Fabián Contreras ◽  
Dmitry Dolgopyat
Keyword(s):  
Invariant Measures ◽  
Absolutely Continuous ◽  
Unimodal Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant

The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps

1998 ◽  
Vol 18 (3) ◽  
pp. 555-565 ◽  
Author(s):  
HENK BRUIN
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Topological Invariant ◽  
Absolutely Continuous ◽  
Unimodal Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant ◽  
Critical Order

Within the class of S-unimodal maps with fixed critical order it is shown that the existence of an absolutely continuous invariant probability measure is not a topological invariant.

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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