scholarly journals On the number of invariant measures for random expanding maps in higher dimensions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fawwaz Batayneh ◽  
Cecilia González-Tokman
Keyword(s):  
Invariant Measures ◽  
Higher Dimensions ◽  
Expanding Maps

scholarly journals Invariant measures for random expanding on average Saussol maps

2021 ◽  
Author(s):  
Fawwaz Batayneh ◽  
Cecilia González-Tokman
Keyword(s):  
Upper Bound ◽  
General Class ◽  
Invariant Measures ◽  
Higher Dimensions ◽  
Skew Product ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota–Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and will provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in [F. Batayneh and C. González-Tokman, On the number of invariant measures for random expanding maps in higher dimensions, Discrete Contin. Dyn. Syst. 41 (2021) 5887–5914] on admissible random Jabłoński maps to a more general class of higher-dimensional random 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.

RELATIVE EQUILIBRIUM STATES AND DIMENSIONS OF FIBERWISE INVARIANT MEASURES FOR RANDOM DISTANCE EXPANDING MAPS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350015 ◽  
Author(s):  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Gibbs State ◽  
Relative Equilibrium ◽  
Equilibrium States ◽  
Expanding Maps ◽  
Hölder Continuous ◽  
Random Maps ◽  
Full Dimension ◽  
Relative Metric

We show that the Gibbs states (known from [10] to be unique) of Hölder continuous potentials and random distance expanding maps coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique. In the realm of conformal expanding random maps, we prove that given an ergodic (globally) invariant measure with a given marginal, for almost every fiber the corresponding conditional measure has dimension equal to the ratio of the relative metric entropy and the Lyapunov exponent. Finally we show that there is exactly one invariant measure whose conditional measures are of full dimension. It is the canonical Gibbs state.

scholarly journals On the finite-dimensional marginals of shift-invariant measures

2011 ◽  
Vol 32 (5) ◽  
pp. 1485-1500 ◽  
Author(s):  
J.-R. CHAZOTTES ◽  
J.-M. GAMBAUDO ◽  
M. HOCHMAN ◽  
E. UGALDE
Keyword(s):  
Finite Type ◽  
Invariant Measures ◽  
Convex Set ◽  
Extreme Points ◽  
Higher Dimensions ◽  
Finite Alphabet ◽  
Finite Dimensional ◽  
Shift Action ◽  
Unique Invariant Measure ◽  
Dimension One

AbstractLet Σ be a finite alphabet, Ω=Σℤdequipped with the shift action, and ℐ the simplex of shift-invariant measures on Ω. We study the relation between the restriction ℐnof ℐ to the finite cubes {−n,…,n}d⊂ℤd, and the polytope of ‘locally invariant’ measures ℐlocn. We are especially interested in the geometry of the convex set ℐn, which turns out to be strikingly different whend=1 and whend≥2 . A major role is played by shifts of finite type which are naturally identified with faces of ℐn, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of ℐn, although in dimensiond≥2 there are also extreme points which arise in other ways. We show that ℐn=ℐlocnwhend=1 , but in higher dimensions they differ fornlarge enough. We also show that while in dimension one ℐnare polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of ℐnfor all large enoughn.

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.

Invariant measures of full dimension for some expanding maps

1997 ◽  
Vol 17 (1) ◽  
pp. 147-167 ◽  
Author(s):  
DIMITRIOS GATZOURAS ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Large Class ◽  
Open Problem ◽  
Invariant Measures ◽  
Thermodynamic Formalism ◽  
Invariant Set ◽  
Invariant Sets ◽  
Expanding Maps ◽  
Full Dimension

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

scholarly journals Invariant densities for random systems of the interval

2020 ◽  
pp. 1-39
Author(s):  
CHARLENE KALLE ◽  
MARTA MAGGIONI
Keyword(s):  
Linear Systems ◽  
Invariant Measures ◽  
Piecewise Linear ◽  
Random Systems ◽  
Density Functions ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Random System ◽  
Piecewise Linear Systems ◽  
Invariant Densities

Abstract For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $ -transformations and random Lüroth maps with a hole.

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