How Many Inflections are There in the Lyapunov Spectrum?

Iommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Locating Sickness: Disability, Queerness, and Race in a Memoir

This paper re-reads Sick: A Memoir (2018) by Porochista Khakpour, as a transnational feminist and queer text, to investigate how the author locates her disability and queerness with the diaspora, homelessness, and rise of governmental violence. Through the lens of feminist and disability studies, Sick can be read as an outstanding narrative of the queerness, disability, in-between-ness, and of course, resistance of a queer and disabled woman of color. The paper argues that Khakpour’s story should be regarded as an attempt to write complexities of intersectional and multi-layered identities that challenge the discourses of detection and diagnosis; criticize the politics of race among the community of Iranian-diaspora and in America; and highlight the role of home, belonging, and the feeling of homelessness caused by state policies of nation-building and exclusion. Further, Khakpour proposes a new guideline for feminist geography that accommodates female, queer, disabled, and diasporic Iranian-American bodies on the expanding map of excluded and erased subjects.

Positive Lyapunov Exponent for Some Schrödinger Cocycles Over Strongly Expanding Circle Endomorphisms

We show that for a large class of potential functions and big coupling constant $$\lambda $$ λ the Schrödinger cocycle over the expanding map $$x\mapsto bx ~( \text{ mod } 1)$$ x ↦ b x ( mod 1 ) on $$\mathbb {T}$$ T has a Lyapunov exponent $$>(\log \lambda )/4$$ > ( log λ ) / 4 for all energies, provided that the integer $$b\ge \lambda ^3$$ b ≥ λ 3 .

Transfer operators for ultradifferentiable expanding maps of the circle

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$ ) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$ , providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$ .

Central limit theorem for generalized Weierstrass functions

Let [Formula: see text] be a [Formula: see text] expanding map of the circle and let [Formula: see text] be a [Formula: see text] function. Consider the twisted cohomological equation [Formula: see text] which has a unique bounded solution [Formula: see text]. We show that [Formula: see text] is either [Formula: see text] or continuous but nowhere differentiable. If [Formula: see text] is nowhere differentiable then the Newton quotients of [Formula: see text], after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of [Formula: see text]) to the normal distribution. In particular, [Formula: see text] is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

Explicit coupling argument for non-uniformly hyperbolic transformations

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. We verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For non-uniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated with the inducing time. Finally, for non-uniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component $\unicode[STIX]{x1D708}$ in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis is stable under perturbations and means that the dynamics is univalued in a neighborhood of ${\mathcal{K}}$. Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit $\unicode[STIX]{x1D708}\rightarrow \infty$. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.

Family of Smale–Williams Solenoid Attractors as Solutions of Differential Equations: Exact Formula and Conjugacy

We show that the family of the Smale–Williams solenoid attractors parameterized by its contraction rate can be characterized as solutions of a set of differential equations. The exact formula describing the attractor can be obtained by solving the differential equations subject to explicitly given initial conditions. Using the formula, we present in this note a simple and explicit proof of the result that the dynamics on the solenoid is topologically conjugate to the shift on the inverse limit space of the expanding map t ↦ mt mod 1 for some integer m ≥ 2 and to a suspension over the adding machine.