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

Eslami, Peyman; Góra, Pawel

doi:10.1090/S0002-9939-2013-11676-X

Stronger Lasota-Yorke inequality for one-dimensional piecewise expanding transformations
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by Peyman Eslami and Pawel Góra PDF

Proc. Amer. Math. Soc. 141 (2013), 4249-4260 Request permission

Abstract:

For a large class of piecewise expanding $\mathcal {C}^{1,1}$ maps of the interval we prove the Lasota-Yorke inequality with a constant smaller than the previously known $2/\inf |\tau ’|$. Consequently, the stability results of Keller-Liverani apply to this class and in particular to maps with periodic turning points. One of the applications is the stability of acim’s for a class of W-shaped maps. Another application is an affirmative answer to a conjecture of Eslami-Misiurewicz regarding acim-stability of a family of unimodal maps.

References

Abraham Boyarsky and PawełGóra, Laws of chaos, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. Invariant measures and dynamical systems in one dimension. MR 1461536, DOI 10.1007/978-1-4612-2024-4
Peyman Eslami and MichałMisiurewicz, Singular limits of absolutely continuous invariant measures for families of transitive maps, J. Difference Equ. Appl. 18 (2012), no. 4, 739–750. MR 2905294, DOI 10.1080/10236198.2011.590480
Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
PawełGóra, On small stochastic perturbations of mappings of the unit interval, Colloq. Math. 49 (1984), no. 1, 73–85. MR 774854, DOI 10.4064/cm-49-1-73-85
P. Góra and B. Schmitt, Un exemple de transformation dilatante et $C^1$ par morceaux de l’intervalle, sans probabilité absolument continue invariante, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 101–113 (French, with English summary). MR 991491, DOI 10.1017/S0143385700004831
Gerhard Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math. 94 (1982), no. 4, 313–333. MR 685377, DOI 10.1007/BF01667385
Gerhard Keller and Carlangelo Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 1, 141–152. MR 1679080
Z. S. Kowalski, Continuity of invariant measures for piecewise monotonic transformations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 5, 519–524 (English, with Russian summary). MR 382593
Carlangelo Liverani, Multidimensional expanding maps with singularities: a pedestrian approach, Ergodic Theory Dynam. Systems 33 (2013), no. 1, 168–182. MR 3009108, DOI 10.1017/S0143385711000939
A. Lasota and James A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488 (1974). MR 335758, DOI 10.1090/S0002-9947-1973-0335758-1
Zhenyang Li, PawełGóra, Abraham Boyarsky, Harald Proppe, and Peyman Eslami, Family of piecewise expanding maps having singular measure as a limit of ACIMs, Ergodic Theory Dynam. Systems 33 (2013), no. 1, 158–167. MR 3009107, DOI 10.1017/S0143385711000836
Cecilia González-Tokman, Brian R. Hunt, and Paul Wright, Approximating invariant densities of metastable systems, Ergodic Theory Dynam. Systems 31 (2011), no. 5, 1345–1361. MR 2832249, DOI 10.1017/S0143385710000337
Zhenyang Li, W-like maps with various instabilities of acim’s, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013).