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Viviane Baladi

Publications and source records attributed to Viviane Baladi.

At least 19 recordsLinked to original sources
arXiv2024

A parameter ASIP for the quadratic family

Consider the quadratic family $T_a(x) = a x (1 - x)$, for $x \in [0, 1]$ and mixing Collet--Eckmann (CE) parameters $a \in (2,4)$. For bounded $φ$, set $\tilde φ_{a} := φ- \int φ\, dμ_a$, with $μ_a$ the unique acim of $T_a$, and put $(σ_a (φ))^2 := \int \tilde φ_{a}^2 \, dμ_a + 2 \sum_{i>0} \int \tilde φ_{a} (\tilde φ_{a} \circ T^i_{a}) \, dμ_a$. For any transversal mixing Misiurewicz parameter $a_*$, we find a positive measure set $Ω_*$ of mixing CE parameters, containing $a_*$ as a Lebesgue density point, such that for any Hölder $φ$ with $σ_{a_*}(φ)\ne 0$, there exists $ε_φ>0$ such that, for normalised Lebesgue measure on $Ω_*\cap [a_*-ε_φ, a_*+ε_φ]$, the functions $ξ_i(a)=\tilde φ_a(T_a^{i+1}(1/2))/σ_a (φ)$ satisfy an almost sure invariance principle (ASIP) for any error exponent $γ>2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann's proof for piecewise expanding maps. We need to introduce a variant of Benedicks-Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from a previous work of Baladi, Benedicks, and Schnellmann.

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arXiv2024

Thermodynamic formalism for piecewise expanding maps in finite dimension

We study weighted transfer operators associated to a piecewise expanding map on a compact manifold, and a piecewise Holder weight, acting on Sobolev spaces. We bound the essential spectral radius in terms of a topological pressure for a subadditive potential. Under a new small boundary pressure condition, we improve the estimate by establishing a variational principle for piecewise expanding maps and subadditive potentials. Version v2 is the electronic copy of the version to appear in DCDS

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arXiv2024

Measure of maximal entropy for finite horizon Sinai billiard flows

Using recent work of Carrand on equilibrium states for the billiard map, and bootstrapping via a "leapfrogging" method from a previous article of Baladi and Demers, we construct the unique measure of maximal entropy for two-dimensional finite horizon Sinai (dispersive) billiard flows (and show it is Bernoulli), assuming that the topological entropy of the flow is strictly larger than s_0 log 2 where 0<s_0<1 quantifies the recurrence to singularities. This bound holds in many examples (it is expected to hold generically).

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arXiv2023

Fractional susceptibility functions for the quadratic family: Misiurewicz-Thurston parameters

For the quadratic family, we define the two-variable ($η$ and $z$) fractional susceptibility function associated to a C^1 observable at a stochastic map. We also define an approximate, "frozen" fractional susceptibility function. If the parameter is Misiurewicz-Thurston, we show that the frozen susceptibility function has a pole at $z=1$ for generic observables if a "one-half" transversality condition holds. We introduce "Whitney" fractional integrals and derivatives on suitable sets $Ω$. We formulate conjectures supported by our results on the frozen susceptibility function and numerical experiments. In particular, we expect that the fractional susceptibility function for $η=1/2$ is singular at $z=1$ for Collet-Eckmann maps and generic observables. We view this work as a step towards the resolution of the paradox that the classical susceptibility function is holomorphic at $z=1$ for Misiurewicz-Thurston maps, despite lack of linear response.

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arXiv2023

On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure $μ_*$ of maximal entropy for $T$ (i.e., $h_{μ_*}(T)=h_*$), we show that $μ_*$ has full support and is Bernoulli, and we prove that $μ_*$ is the unique measure of maximal entropy, and that it is different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to $h_*$. Second, $h_*$ is equal to the Bowen--Pesin--Pitskel topological entropy of the restriction of $T$ to a non-compact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map $T$ has at least $C e^{nh_*}$ periodic points of period $n$ for all large enough $n \in \mathbb{N}$.

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arXiv2023

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

We consider C^2 families of C^4 unimodal maps f_t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of f_t depends differentiably on t, as a distribution of order 1. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of the acim for a Benedicks-Carleson map f_t, in terms of a single smooth function and the inverse branches of f_t along the postcritical orbit. Along the way, we prove that the twisted cohomological equation v(x)=α(f (x)) - f'(x) α(x) has a continuous solution α, if f is Benedicks-Carleson and v is horizontal for f. In v3 we added a note containing 3 comments regarding minor typos.

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arXiv2022

Horocycle averages on closed manifolds and transfer operators

We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$ discrete-time hyperbolic dynamics ($r>1$), obtaining anisotropic Banach or Hilbert spaces on which the resolvent of the generator of weighted transfer operators for the flow is quasi-compact. We apply this to study the ergodic integrals of the horocycle flows $h_ρ$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov flow with orientable strong-stable distribution, we establish power-law convergence of the ergodic average. We thereby implement the program of Giulietti-Liverani in the "real-life setting" of geodesic flows in variable negative curvature, where nontrivial resonances exist.

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arXiv2022

Thermodynamic formalism for dispersing billiards

For any finite horizon Sinai billiard map T on the two-torus, we find t_*>1 such that for each t in (0,t_*) there exists a unique equilibrium state $μ_t$ for $- t\log J^uT$, and $μ_t$ is T-adapted. (In particular, the SRB measure is the unique equilibrium state for $- \log J^uT$.) We show that $μ_t$ is exponentially mixing for Holder observables, and the pressure function $P(t)=\sup_μ\{h_μ-\int t\log J^uT d μ\}$ is analytic on (0,t_*). In addition, P(t) is strictly convex if and only if $\log J^uT$ is not $μ_t$ a.e. cohomologous to a constant, while, if there exist $t_a\ne t_b$ with $μ_{t_a}= μ_{t_b}$, then P(t) is affine on (0,t_*). An additional sparse recurrence condition gives $\lim_{t\to 0} P(t)=P(0)$.

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arXiv2020

There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus

We show that the ergodic averages for the horocycle flow on the two-torus associated by Giulietti and Liverani to an Anosov diffeomorphism either grow linearly or are bounded, in other words there are no deviations. For this, we use topological invariance of the Artin-Mazur zeta function to exclude resonances outside of the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools in the proof. As a bonus, we show that for any smooth Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and smooth observables decay with a rate strictly smaller than exp(-h_top(F)). We compare our results with related work of Forni.

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arXiv2020

Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows

We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group L_t of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions. (Revised after referees' comments.)

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arXiv2018

Characteristic functions as bounded multipliers on anisotropic spaces

We show that characteristic functions of domains with boundaries transversal to stable cones are bounded multipliers on a recently introduced scale $U^{t,s}_p$ of anisotropic Banach spaces, under the conditions -1+1/p<s<-t<0 and -(r-1)+t<s, with 1<p<infty. (Amended after comments from the referee and M. Jézéquel, January 10, 2018)

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arXiv2018

The quest for the ultimate anisotropic Banach space

We present a new scale $U^{t,s}_p$ (with $s<-t<0$ and $1 \le p <\infty$) of anisotropic Banach spaces, defined via Paley-Littlewood, on which the transfer operator associated to a hyperbolic dynamical system has good spectral properties. When $p=1$ and $t$ is an integer, the spaces are analogous to the "geometric" spaces considered by Gouëzel and Liverani. When $p>1$ and $-1+1/p t-(r-1) is needed in Section 4.

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arXiv2017

Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables

We consider a smooth one-parameter family $t \to f_t$ of diffeomorphisms with compact transitive Axiom A attractors. Our first result (corrected) is that for any function $G$ in the Sobolev space $H^r_p$, with $p>1$ and $0<r<1/p$, the map $R(t)$ sending $t$ to the average of $G$ with respect to the SRB measure of $f_t$ is $α$-Hölder continuous for all $α<r- |log \mathcal J|/(p|log ν_s|$) where $\mathcal J\le 1$ is the strongest volume contraction and $ν_s<1$ is the weakest contraction. This applies to $θ(x)=h(x)Θ(g(x)-a)$ (for all $α<1- |log \mathcal J|/|log ν_s|$) for $h$ and $g$ smooth and $Θ$ the Heaviside function, if $a$ is not a critical value of $g$. Our second result says that for any such function so that, in addition, the intersection of the set of points $x$ so that $g(x)=a$ with the support of $h$ is foliated by "admissible stable leaves" of $f_t$, the map $R(t)$ is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables is motivated by extreme-value theory. --- Second version, following the referee's comments: We explain better the cone choices around (2.4). Appendix A contains information on the Banach spaces. We added the paragraph containing (2.6) in the proof of Theorem 2.1. In the proof of Theorem 3.3, we do not need to introduce mollifiers. However, the new argument around (2.6) is not available here, so we must replace the pair $(u-1, |s-1|)$ by $(u-2, |s-2|)$. This is why we now assume that $h$ is $C^3$ and that $g$ and the foliations are $C^4$. --- Third version: We have added a corrigendum modifying the first result (Theorem 2.1).

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arXiv2014

Linear response, or else

Consider a smooth one-parameter family t -> f_t of dynamical systems f_t, with |t| m_t is differentiable at t=0 (possibly in the sense of Whitney), and if its derivative can be expressed as a function of f_0, m_0, and d_t f_t|_(t=0). The goal of this note is to present to a general mathematical audience recent results and open problems in the theory of linear response for chaotic dynamical systems, possibly with bifurcations.

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arXiv2013

Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps

We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set D of CE parameters, and, for each s in D, a subset D0 of D of polynomially recurrent parameters containing s as a Lebesgue density point, and constants C>1, G >4, so that, for every 1/2-Holder function A (of 1/2-Holder norm |A|) and all t in D0, |\int A dm_t -\int A dm_s| < C |A| |t-s|^{1/2} |log|t-s||^G (If f_t(x)=tx(1-x), the set D contains almost all CE parameters.) Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter s, we find a set of CE parameters D' accumulating at s, a constant C >1, and an infinitely differentiable function B, so that for all t in D' C |t-s|^{1/2} > |\int B dm_t -\int B dm_s| > |t-s|^{1/2}/C

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arXiv2012

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(z), associated to precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the nontangential limit of the Psi(z) as z tends to 1 exists and coincides with the derivative of the acim with respect to the map (linear response formula). Applying the Wiener-Wintner theorem, we study the singularity type of nontangential limits as z tends to e^{iω}. An additional LIL typicality assumption on the postcritical orbit gives stronger results.

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arXiv2012

Exponential decay of correlations for piecewise cone hyperbolic contact flows

We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author's version of Dolgopyat's estimates for contact flows and the first author's work with Gouëzel on piecewise hyperbolic discrete-time dynamics. (Presentation revised.)

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