Decay of correlations for piecewise expanding maps

1995 ◽  
Vol 78 (3-4) ◽  
pp. 1111-1129 ◽  
Author(s):  
Carlangelo Liverani
Keyword(s):  
Decay Of Correlations ◽  
Expanding Maps ◽  
Piecewise Expanding Maps

scholarly journals Shrinking targets in parametrised families

2017 ◽  
Vol 166 (2) ◽  
pp. 265-295 ◽  
Author(s):  
MAGNUS ASPENBERG ◽  
TOMAS PERSSON
Keyword(s):  
Fixed Point ◽  
Hausdorff Dimension ◽  
Expanding Maps ◽  
Target Property ◽  
Non Linear ◽  
Piecewise Expanding Maps

AbstractWe consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for β-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

scholarly journals On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

2016 ◽  
Vol 16 (03) ◽  
pp. 1660013 ◽  
Author(s):  
Nigel P. Byott ◽  
Congping Lin ◽  
Yiwei Zhang
Keyword(s):  
Piecewise Linear ◽  
Unit Interval ◽  
Interval Exchange ◽  
Expanding Maps ◽  
Mixing Properties ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Fold Map ◽  
Constant Slope ◽  
Piecewise Expanding Maps

For an integer [Formula: see text], let [Formula: see text] be the partition of the unit interval [Formula: see text] into [Formula: see text] equal subintervals, and let [Formula: see text] be the class of piecewise linear maps on [Formula: see text] with constant slope [Formula: see text] on each element of [Formula: see text]. We investigate the effect on mixing properties when [Formula: see text] is composed with the interval exchange map given by a permutation [Formula: see text] interchanging the [Formula: see text] subintervals of [Formula: see text]. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365–3390], where we considered only the “stretch-and-fold” map [Formula: see text].

scholarly journals Piecewise expanding maps on the plane with singular ergodic properties

2000 ◽  
Vol 20 (6) ◽  
pp. 1851-1857 ◽  
Author(s):  
MASATO TSUJII
Keyword(s):  
Empirical Measure ◽  
Expanding Maps ◽  
Ergodic Properties ◽  
Expanding Map ◽  
Open Set ◽  
Point Measure ◽  
Piecewise Expanding Maps

For $1\le r<\infty$, we construct a piecewise $C^{r}$ expanding map $F:D\to D$ on the domain $D=(0,1)\times (-1,1)\subset\mathbb{R}^{2}$ with the following property: there exists an open set $B$ in $D$ such that the diameter of $F^{n}(B)$ converges to $0$ as $n\to\infty$ and the empirical measure $n^{-1}\sum_{k=0}^{n-1}\delta_{F^{k}(x)}$ converges to the point measure $\delta_{p}$ at $p=(0,0)$ as $n \to\infty$ for any point $x\in B$.

scholarly journals Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

2012 ◽  
Vol 146 (4) ◽  
pp. 850-863 ◽  
Author(s):  
Paweł Góra ◽  
Zhenyang Li ◽  
Abraham Boyarsky ◽  
Harald Proppe
Keyword(s):  
Expanding Maps ◽  
Invariant Densities ◽  
Piecewise Expanding Maps

scholarly journals The attractor of piecewise expanding maps of the interval

2019 ◽  
Vol 20 (02) ◽  
pp. 2050009 ◽  
Author(s):  
Gianluigi Del Magno ◽  
João Lopes Dias ◽  
Pedro Duarte ◽  
José Pedro Gaivão
Keyword(s):  
Periodic Orbits ◽  
Elementary Proof ◽  
Expanding Maps ◽  
Small Perturbations ◽  
Topological Description ◽  
Piecewise Expanding Maps

We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e. the number of ergodic attractors and their corresponding mixing components do not change under small perturbations of the map. Our methods provide a topological description of the attractor and give an elementary proof of the density of periodic orbits.

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