Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Stefano Galatolo; Rafael Lucena

doi:10.3934/dcds.2020079

Article Contents

2020, Volume 40, Issue 3: 1309-1360. Doi: 10.3934/dcds.2020079

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Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Stefano Galatolo^1, and
Rafael Lucena^2,

1.
Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56100 Pisa, Italy
2.
Instituto de Matemática, Av. Lourival Melo Mota, Tabuleiro do Martins, s/n, Maceió -AL, CEP 57072-900, Brazil

Received: November 2018

Revised: August 2019

Published: December 2019

This work was partially supported by Alagoas Research Foundation-FAPEAL (Brazil) Grants 60030 000587/2016, CNPq (Brazil) Grants 300398/2016-6, CAPES (Brazil) Grants 99999.014021/2013-07 and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE- 2012-IRSES 318999 BREUDS)

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Abstract

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation).

As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $ \delta $, the physical measure varies continuously, with a modulus of continuity $ O(\delta \log \delta ) $, which is asymptotically optimal for this kind of piecewise smooth maps.

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Mathematics Subject Classification: Primary: 37A25, 37A10; Secondary: 37C30, 37D50.

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