ABSOLUTELY CONTINUOUS INVARIANT MEASURES FOR GENERIC MULTI-DIMENSIONAL PIECEWISE AFFINE EXPANDING MAPS

1999 ◽  
Vol 09 (09) ◽  
pp. 1743-1750 ◽  
Author(s):  
J. BUZZI
Keyword(s):  
Open Problem ◽  
Invariant Measures ◽  
Probability Measures ◽  
Higher Dimension ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Piecewise Affine ◽  
Almost All ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

By a well-known result of Lasota and Yorke, any self-map f of the interval which is piecewise smooth and uniformly expanding, i.e. such that inf |f′|>1, admits absolutely continuous invariant probability measures (or a.c.i.m.'s for short). The generalization of this statement to higher dimension remains an open problem. Currently known results only apply to "sufficiently expanding maps". Here we present a different approach which can deal with almost all piecewise expanding maps. Here, we consider both continuous and discontinuous piecewise affine expanding maps.

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.

scholarly journals Absolutely continuous invariant measures for some piecewise hyperbolic affine maps

2008 ◽  
Vol 28 (1) ◽  
pp. 211-228 ◽  
Author(s):  
TOMAS PERSSON
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Bounded Subset ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractA class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

ORDER AND CHAOS FOR A CLASS OF PIECEWISE LINEAR MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1379-1394 ◽  
Author(s):  
VÍCTOR JIMÉNEZ LÓPEZ
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Absolutely Continuous ◽  
Linear Maps ◽  
Sensitive Dependence ◽  
Piecewise Linear Maps ◽  
Almost All ◽  
Absolutely Continuous Invariant

For a class of piecewise linear maps f: I → I from a compact interval I into itself, we describe the asymptotic behavior of the sequence [Formula: see text] for almost all x ∈ I. We also study in this setting the relations among sensitive dependence on initial conditions, existence of scrambled sets of positive measure and existence of absolutely continuous invariant measures.

Non-ergodicity for C1 expanding maps and g-measures

1996 ◽  
Vol 16 (3) ◽  
pp. 531-543 ◽  
Author(s):  
Anthony N. Quasf
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Probability Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractWe introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

scholarly journals On the fractional susceptibility function of piecewise expanding maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Magnus Aspenberg ◽  
Viviane Baladi ◽  
Juho Leppänen ◽  
Tomas Persson
Keyword(s):  
Probability Measure ◽  
Fractional Derivative ◽  
Invariant Probability Measure ◽  
Holomorphic Extension ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Unimodal Map ◽  
Susceptibility Function ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

<p style='text-indent:20px;'>We associate to a perturbation <inline-formula><tex-math id="M1">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> of a (stably mixing) piecewise expanding unimodal map <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> a two-variable fractional susceptibility function <inline-formula><tex-math id="M3">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula>, depending also on a bounded observable <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. For fixed <inline-formula><tex-math id="M5">\begin{document}$ \eta \in (0,1) $\end{document}</tex-math></inline-formula>, we show that the function <inline-formula><tex-math id="M6">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> is holomorphic in a disc <inline-formula><tex-math id="M7">\begin{document}$ D_\eta\subset \mathbb{C} $\end{document}</tex-math></inline-formula> centered at zero of radius <inline-formula><tex-math id="M8">\begin{document}$ &gt;1 $\end{document}</tex-math></inline-formula>, and that <inline-formula><tex-math id="M9">\begin{document}$ \Psi_\phi(\eta, 1) $\end{document}</tex-math></inline-formula> is the Marchaud fractional derivative of order <inline-formula><tex-math id="M10">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> of the function <inline-formula><tex-math id="M11">\begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $\end{document}</tex-math></inline-formula>, at <inline-formula><tex-math id="M12">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M13">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is the unique absolutely continuous invariant probability measure of <inline-formula><tex-math id="M14">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>. In addition, we show that <inline-formula><tex-math id="M15">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> admits a holomorphic extension to the domain <inline-formula><tex-math id="M16">\begin{document}$ \{\, (\eta, z) \in \mathbb{C}^2\mid 0&lt;\Re \eta &lt;1, \, z \in D_\eta \,\} $\end{document}</tex-math></inline-formula>. Finally, if the perturbation <inline-formula><tex-math id="M17">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> is horizontal, we prove that <inline-formula><tex-math id="M18">\begin{document}$ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $\end{document}</tex-math></inline-formula>.</p>

scholarly journals THE DIFFUSION COEFFICIENT FOR PIECEWISE EXPANDING MAPS OF THE INTERVAL WITH METASTABLE STATES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150005 ◽  
Author(s):  
DMITRY DOLGOPYAT ◽  
PAUL WRIGHT
Keyword(s):  
Diffusion Coefficient ◽  
Markov Chain ◽  
Bounded Variation ◽  
Continuous Time ◽  
Probability Measures ◽  
Continuous Time Markov Chain ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

Consider a piecewise smooth expanding map of the interval possessing several invariant subintervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the subintervals lose their invariance in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.

scholarly journals Invariant Measures for the n-Dimensional Border Collision Normal Form

2014 ◽  
Vol 24 (12) ◽  
pp. 1450164 ◽  
Author(s):  
Paul Glendinning
Keyword(s):  
Fractal Dimension ◽  
Normal Form ◽  
Bifurcation Theory ◽  
Invariant Measures ◽  
Topological Dimension ◽  
Absolutely Continuous ◽  
Affine Map ◽  
Piecewise Affine ◽  
Open Set ◽  
Absolutely Continuous Invariant

The border collision normal form is a continuous piecewise affine map of ℝn with applications in piecewise smooth bifurcation theory. We show that these maps have absolutely continuous invariant measures for an open set of parameter space and hence the attractors have Hausdorff (fractal) dimension n. If n = 2 the attractors have topological dimension two, i.e. they contain open sets, and if n > 2 then they have topological dimension n generically.

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