On the fractional susceptibility function of 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>

Recognition of unimodal map germs from the plane to the plane by invariants

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to [Formula: see text]-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic [Formula: see text]. On the basis of this characterization, we present an algorithm to compute the type of the unimodal maps from the plane to the plane without computing the normal form and also give its implementation in the computer algebra system Singular.

Effective high-temperature estimates for intermittent maps

Using quantitative perturbation theory for linear operators, we prove a spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (‘high-temperature’ regime). Hölder and bounded $p$-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau–Manneville map, any potential with Lipschitz constant less than 0.0014 has a transfer operator acting on $\operatorname{Lip}([0,1])$ with a spectral gap; and that for any two-to-one unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on $\operatorname{BV}([0,1])$ with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in Giulietti et al [The calculus of thermodynamical formalism. J. Eur. Math. Soc., to appear. Preprint, 2015, arXiv:1508.01297], allowing all results there to be applied under the high-temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.

A Cr unimodal map with an arbitrary fast growth of the number of periodic points

In this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’ of critical points is necessary for the Martens–de Melo–van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.

NUMERICAL APPROXIMATION OF THE PERCENTAGE OF ORDER FOR ONE-DIMENSIONAL MAPS

The percentage of organized motion of the chaotic zone (which shall from now on be referred to as percentage of order) for the logistic, the sine-square and the 4-exponent map, is calculated. The calculations are reached via a sampling method that incorporates the Lyapunov exponent. Although these maps are specially selected examples of one-dimensional ones, the conclusions can also be applied to any other one-dimensional map. Since the metric characteristics of a bifurcation diagram of a unimodal map, such as the referred percentage of order, are dependent on the order of the maximum, this dependence is verified for several maps. Once the chaotic zone can be separated into regions between the sequential band mergings, the percentage of order corresponding to each region is calculated for the logistic map. In each region, the resultant area occupied by order, or the supplementary area occupied by chaos, participates in a sequence similar to Feigenbaum's one, which converges to the same respective Feigenbaum's constant.

SMOOTH SYMMETRIC AND LORENZ MODELS FOR UNIMODAL MAPS

For a given unimodal map F:I→I on the interval I, we consider symmetric unimodal maps (models) so that they are conjugate to F. The question motivated by [Gambaudo & Tresser, 1992] is the following: whether it is possible for symmetric model to preserve smoothness of the initial map F? We construct a symmetric model which is proved to be as smooth as F provided F has a nonflat turning point with sufficient "reserve of local evenness" at the turning point (in terms of one-sided higher derivatives at the turning point, see Definition 2.4 and Theorem 2.7). We also consider from different points of view the relationship between dynamical and ergodic properties of unimodal maps and of symmetric Lorenz maps. In particular, we present a one-to-one correspondence preserving the measure theoretic entropy, between the set of invariant measures of a symmetric unimodal map F and the set of symmetric invariant measures of the Lorenz model of F (Theorem 3.5), where by Lorenz model of F we mean the discontinuous map obtained from F by reversing its decreasing branch. Finally we extend for nonsymmetric unimodal maps, the result of Gambaudo and Tresser [1992] on Ck structural instability of the maps whose rotation interval has irrational end point (answering a question from [Gambaudo & Tresser, 1992]).