Stronger Lasota-Yorke inequality for one-dimensional piecewise expanding transformations

Boyarsky

et al. 2012

J Stat Phys

The statistical behavior of families of maps is important in studying the stability properties of chaotic maps. For a piecewise expanding map τ whose slope > 2 in magnitude, much is known about the stability of the associated invariant density. However, when the map has slope magnitude ≤ 2 many different behaviors can occur as shown in (Keller in Monatsh. Math. 94(4): 1982) for W maps. The main results of this note use a harmonic average of slopes condition to obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Since these constants are determined explicitly the results can be extended to families of approximating maps.Keywords Absolutely continuous invariant measures · Piecewise expanding maps of interval · Lower bound for invariant density · Explicit constants for rate of convergence · Harmonic average of slopes

Section: Lower Bound For the Invariant Densitymentioning

confidence: 88%

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

Boyarsky

et al. 2012

J Stat Phys

“…The results of this paper inspired the introduction of the harmonic average of slopes condition. Recently, the Lasota-Yorke inequality has been strengthened [8] by using the harmonic average of the slopes on each side of the partition points rather than the doubled reciprocal of the minimal slope. This allows us to show stability of the acim of the limit map for a larger class of maps.…”

Section: Introductionmentioning

confidence: 99%

“…This allows us to show stability of the acim of the limit map for a larger class of maps. The smoothness assumption in [8] is piecewise C 1+1 .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Harmonic average of slopes and the stability of absolutely continuous invariant measure

Journal of Mathematical Analysis and Applications

Boyarsky

2012

Absolutely continuous invariant measures Stability of acim Piecewise expanding maps of interval a b s t r a c tFor families of piecewise expanding maps which converge to a map with a fixed or periodic turning point touching a branch with slope of modulus equal to or less than 2, the standard Lasota-Yorke argument fails to prove stability. It is the goal of this paper to use instead the harmonic average of slopes condition for a large class of maps satisfying the summable oscillation condition for the reciprocal of the derivative. Using Rychlik's Theorem for a family of perturbations we prove weak compactness in L 1 of the density functions associated with them. From this it follows that we have stability of absolutely continuous invariant measures of the limit map.

“…P τ is the Perron-Frobenius operator induced by τ on the space of functions of bounded variation and by isolated spectrum we mean the part of the spectrum which lies outside the essential spectral radius. Papers [5] and [6] show that such stability takes place if the family {τ a } a≥0 satisfies Lasota-Yorke inequality ( [7] or [2] for strengthened form) with uniform constants. Usual conditions ensuring this are |τ ′ a | > 2 + ε plus the minimal length of subintervals of defining partitions uniformly separated from 0.…”

Section: Introductionmentioning

confidence: 99%

Instability of the isolated spectrum for W-shaped maps

2012

Ergod. Th. Dynam. Sys.

Link to this article: http://journals.cambridge.org/abstract_S0143385712000223 How to cite this article: ZHENYANG LI and PAWEŁ GÓRA (2013). Instability of the isolated spectrum for W-shaped maps.Abstract. In this note we consider the W-shaped map W 0 = W s 1 ,s 2 with 1/s 1 + 1/s 2 = 1 and show that the eigenvalue 1 is not stable. We do this in a constructive way. For each perturbing map W a we show the existence of a 'second' eigenvalue λ a , such that λ a → 1 as a → 0, which proves instability of the isolated spectrum of W 0 . At the same time, the existence of second eigenvalues close to 1 causes the maps W a to behave in a metastable way. There are two almost-invariant sets, and the system spends long periods of consecutive iterations in each of them, with infrequent jumps from one to the other.