scholarly journals On the fractional susceptibility function of piecewise expanding maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Magnus Aspenberg ◽  
Viviane Baladi ◽  
Juho Leppänen ◽  
Tomas Persson
Keyword(s):  
Probability Measure ◽  
Fractional Derivative ◽  
Invariant Probability Measure ◽  
Holomorphic Extension ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Unimodal Map ◽  
Susceptibility Function ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

<p style='text-indent:20px;'>We associate to a perturbation <inline-formula><tex-math id="M1">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> of a (stably mixing) piecewise expanding unimodal map <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> a two-variable fractional susceptibility function <inline-formula><tex-math id="M3">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula>, depending also on a bounded observable <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. For fixed <inline-formula><tex-math id="M5">\begin{document}$ \eta \in (0,1) $\end{document}</tex-math></inline-formula>, we show that the function <inline-formula><tex-math id="M6">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> is holomorphic in a disc <inline-formula><tex-math id="M7">\begin{document}$ D_\eta\subset \mathbb{C} $\end{document}</tex-math></inline-formula> centered at zero of radius <inline-formula><tex-math id="M8">\begin{document}$ &gt;1 $\end{document}</tex-math></inline-formula>, and that <inline-formula><tex-math id="M9">\begin{document}$ \Psi_\phi(\eta, 1) $\end{document}</tex-math></inline-formula> is the Marchaud fractional derivative of order <inline-formula><tex-math id="M10">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> of the function <inline-formula><tex-math id="M11">\begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $\end{document}</tex-math></inline-formula>, at <inline-formula><tex-math id="M12">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M13">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is the unique absolutely continuous invariant probability measure of <inline-formula><tex-math id="M14">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>. In addition, we show that <inline-formula><tex-math id="M15">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> admits a holomorphic extension to the domain <inline-formula><tex-math id="M16">\begin{document}$ \{\, (\eta, z) \in \mathbb{C}^2\mid 0&lt;\Re \eta &lt;1, \, z \in D_\eta \,\} $\end{document}</tex-math></inline-formula>. Finally, if the perturbation <inline-formula><tex-math id="M17">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> is horizontal, we prove that <inline-formula><tex-math id="M18">\begin{document}$ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $\end{document}</tex-math></inline-formula>.</p>

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

2013 ◽  
Vol 34 (3) ◽  
pp. 777-800 ◽  
Author(s):  
V. BALADI ◽  
S. MARMI ◽  
D. SAUZIN
Keyword(s):  
Probability Measure ◽  
Spectral Theory ◽  
Linear Response ◽  
Invariant Probability Measure ◽  
Natural Boundary ◽  
Absolutely Continuous ◽  
Unimodal Maps ◽  
Generic Condition ◽  
Susceptibility Function ◽  
Absolutely Continuous Invariant

AbstractFor a piecewise expanding unimodal interval map$f$with unique absolutely continuous invariant probability measure$\mu $, a perturbation$X$, and an observable$\varphi $, the susceptibility function is$\Psi _\varphi (z)= \sum _{k=0}^\infty z^k \int X(x) \varphi '( f^k)(x) (f^k)'(x) \, d\mu $. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps.Comm. Math. Phys.275(2007), 839–859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps.Nonlinearity21(2008), 677–711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer–Simon [Natural boundaries and spectral theory.Adv. Math.226(2011), 4902–4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet.Ann. Sci. Éc. Norm. Supér.(3)66(1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that$\Psi _\varphi (z)$has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of$\Psi _\varphi (z)$, associated with precritical orbits. If the perturbation$X$is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of$\Psi _\varphi (z)$as$z\to 1$exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map (‘linear response formula’). Applying the Wiener–Wintner theorem, we study the singularity type of non-tangential limits of$\Psi _\varphi (z)$as$z\to e^{i\omega }$for real$\omega $. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.

scholarly journals Un exemple de transformation dilatante et C1 par morceaux de l'intervalle, sans probabilité absolument continue invariante

1989 ◽  
Vol 9 (1) ◽  
pp. 101-113 ◽  
Author(s):  
P. Gora ◽  
B. Schmitt
Keyword(s):  
Probability Measure ◽  
Modulus Of Continuity ◽  
Invariant Probability Measure ◽  
Negative Answer ◽  
Continuous Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure ◽  
Absolutely Continuous Invariant

AbstractWe construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..

The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps

1998 ◽  
Vol 18 (3) ◽  
pp. 555-565 ◽  
Author(s):  
HENK BRUIN
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Topological Invariant ◽  
Absolutely Continuous ◽  
Unimodal Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant ◽  
Critical Order

Within the class of S-unimodal maps with fixed critical order it is shown that the existence of an absolutely continuous invariant probability measure is not a topological invariant.

ABSOLUTELY CONTINUOUS INVARIANT MEASURES FOR GENERIC MULTI-DIMENSIONAL PIECEWISE AFFINE EXPANDING MAPS

1999 ◽  
Vol 09 (09) ◽  
pp. 1743-1750 ◽  
Author(s):  
J. BUZZI
Keyword(s):  
Open Problem ◽  
Invariant Measures ◽  
Probability Measures ◽  
Higher Dimension ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Piecewise Affine ◽  
Almost All ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

By a well-known result of Lasota and Yorke, any self-map f of the interval which is piecewise smooth and uniformly expanding, i.e. such that inf |f′|>1, admits absolutely continuous invariant probability measures (or a.c.i.m.'s for short). The generalization of this statement to higher dimension remains an open problem. Currently known results only apply to "sufficiently expanding maps". Here we present a different approach which can deal with almost all piecewise expanding maps. Here, we consider both continuous and discontinuous piecewise affine expanding maps.

scholarly journals STATISTICAL PROPERTIES OF ONE-DIMENSIONAL MAPS WITH CRITICAL POINTS AND SINGULARITIES

2006 ◽  
Vol 06 (04) ◽  
pp. 423-458 ◽  
Author(s):  
K. DÍAZ-ORDAZ ◽  
M. P. HOLLAND ◽  
S. LUZZATTO
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Invariant Probability Measure ◽  
Return Time ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Time Asymptotics ◽  
Exponential Decay Of Correlations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Hölder observations.

Piecewise smooth interval maps with non-vanishing derivative

2000 ◽  
Vol 20 (3) ◽  
pp. 749-773 ◽  
Author(s):  
ALE JAN HOMBURG
Keyword(s):  
Probability Measure ◽  
Periodic Orbits ◽  
Invariant Probability Measure ◽  
Piecewise Smooth ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Periodic Attractors ◽  
Absolutely Continuous Invariant

We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

scholarly journals Ergodic properties of bimodal circle maps

2017 ◽  
Vol 39 (06) ◽  
pp. 1462-1500
Author(s):  
SYLVAIN CROVISIER ◽  
PABLO GUARINO ◽  
LIVIANA PALMISANO
Keyword(s):  
Probability Measure ◽  
Periodic Orbits ◽  
Invariant Probability Measure ◽  
Physical Measure ◽  
Absolutely Continuous ◽  
Irrational Numbers ◽  
Circle Maps ◽  
Rotation Interval ◽  
Ergodic Properties ◽  
Absolutely Continuous Invariant

We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^{2}$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular, they hold for Lebesgue almost every rotation interval. By standard results, the measure obtained is a global physical measure and it is hyperbolic.

scholarly journals On infinitely cohomologous to zero observables

2012 ◽  
Vol 33 (2) ◽  
pp. 375-399
Author(s):  
AMANDA DE LIMA ◽  
DANIEL SMANIA
Keyword(s):  
Large Class ◽  
Infinite Sequence ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Koopman Operator ◽  
Hilbert Basis ◽  
Image Position ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

AbstractWe show that for a large class of piecewise expanding maps T, the bounded p-variation observables u0 that admit an infinite sequence of bounded p-variation observables ui satisfying \[ u_{i}= u_{i+1}\circ T-u_{i+1} \] are constant. The method of the proof consists of finding a suitable Hilbert basis for L2(hm), where hm is the unique absolutely continuous invariant probability of T. On this basis, the action of the Perron–Frobenius and the Koopman operator on L2(hm) can be easily understood. This result generalizes earlier results by Bamón, Kiwi, Rivera-Letelier and Urzúa for the case T(x)=ℓx mod   1 , ℓ∈ℕ∖ {0,1} and Lipschitzian observables u0.

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