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V. Baladi

Publications and source records attributed to V. Baladi.

8 recordsLinked to original sources
arXiv2019

On the fractional susceptibility function of piecewise expanding maps

We associate to a perturbation $(f_t)$ of a (stably mixing) piecewise expanding unimodal map $f_0$ a two-variable fractional susceptibility function $\Psi_\phi(\eta, z)$, depending also on a bounded observable $\phi$. For fixed $\eta \in (0,1)$, we show that the function $\Psi_\phi(\eta, z)$ is holomorphic in a disc $D_\eta\subset \mathbb{C}$ centered at zero of radius $>1$, and that $\Psi_\phi(\eta, 1)$ is the Marchaud fractional derivative of order $\eta$ of the function $t\mapsto \mathcal{R}_\phi(t):=\int \phi(x)\, d\mu_t$, at $t=0$, where $\mu_t$ is the unique absolutely continuous invariant probability measure of $f_t$. In addition, we show that $\Psi_\phi(\eta, z)$ admits a holomorphic extension to the domain $\{ (\eta, z) \in {\mathbb{C}}^2\mid 0<\Re \eta <1, \, z \in D_\eta \}$. Finally, if the perturbation $(f_t)$ is horizontal, we prove that $\lim_{\eta \to 1}\Psi_\phi(\eta, 1)=\partial_t \mathcal{R}_\phi(t)|_{t=0}$.

math.DS↗
arXiv2017

Correcting the proof of Theorem 3.2 and Corollary 5.2 in Almost sure rates of mixing for i.i.d. unimodal maps by V. Baladi, M. Benedicks, V. Maume-Deschamps, Ann. E.N.S. (2002)

In 2010, Weixiao Shen pointed out to us that the proof of Theorem 3.2 of our 2002 paper in Ann ENS was flawed, and he kindly provided an argument to fix this proof. (We do not make claims on the lower bounds for the stationary density anymore.) In 2017, Wael Bahsoun, Christopher Bose, and Marks Ruziboev pointed out to us that the proof of Corollary 5.2 in the same paper is flawed. We explain how to recover Corollary 5.2 up to replacing the set $M'_q$ on p. 93 by a slightly smaller set (this does not affect the rest of the paper).

math.DS↗
arXiv2016

Dynamical zeta functions

These are notes from a course given in Orsay in 2002 explaining carefully the Milnor-Thurston kneading determinant approach to dynamical zeta functions as interpreted by Baladi and Ruelle (Invent. Math. 1996). We make them available in view of the recent renewed interest in this approach. (see arXiv:1501.00294, The Milnor-Thurston determinant and the Ruelle transfer operator, HH Rugh, Comm. Math. Phys. 342 (2016) 603-614, and arXiv:1407.5313, Kneading with weights, HH Rugh, Lei Tan, J. Fractal Geom. 2 (2015) 339-375)

math.DS↗
arXiv2016

Linear response for intermittent maps

We consider the one parameter family $α\mapsto T_α$ ($α\in [0,1)$) of Pomeau-Manneville type interval maps $T_α(x)=x(1+2^αx^α)$ for $x \in [0,1/2)$ and $T_α(x)=2x-1$ for $x \in [1/2, 1]$, with the associated absolutely continuous invariant probability measure $μ_α$. For $α\in (0,1)$, Sarig and Gouëzel proved that the system mixes only polynomially with rate $n^{1-1/α}$ (in particular, there is no spectral gap). We show that for any $ψ\in L^q$, the map $α\to \int_0^1 ψ\, dμ_α$ is differentiable on $[0,1-1/q)$, and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For $α\ge 1/2$ we need the $n^{-1/α}$ decorrelation obtained by Gouëzel under additional conditions.

math.DS↗
arXiv2007

Linear response formula for piecewise expanding unimodal maps

The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then R(t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture.

math.DS↗
arXiv2004

Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case

Transfer operators M_k acting on k-forms in R^n are associated to smooth transversal local diffeomorphisms and compactly supported weight functions. A formal trace is defined by summing the product of the weight and the Lefschetz sign over all fixed points of all the diffeos. This yields a formal Ruelle-Lefschetz determinant Det^#(1-zM). We use the Milnor-Ruelle-Kitaev equality (recently proved by Baillif), which expressed Det^#(1-zM) as an alternated product of determinants of kneading operators,Det(1+D_k(z)), to relate zeroes and poles of the Ruelle-Lefschetz determinant to the spectra of the transfer operators M_k. As an application, we get a new proof of a theorem of Ruelle on smooth expanding dynamics.

math.DS↗