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In this short note we describe a simple but remarkably effective method for rigorously estimating Lyapunov exponents for expanding maps of the interval. We illustrate the applicability of this method with some standard examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We relate the classical nineteenth century Schottky–Klein function in complex analysis to a counting problem for pairs of geodesics in hyperbolic geometry studied by Fenchel. We then solve the counting problem using ideas from ergodic theory and thermodynamic formalism. [ABSTRACT FROM AUTHOR]

For real analytic expanding interval maps, a novel method is given for rigorously approximating the diffusion coefficient of real analytic observables. As a theoretical algorithm, our approximation scheme is shown to give quadratic exponential convergence to the diffusion coefficient. The method for converting this rapid convergence into explicit high precision rigorous bounds is illustrated in the setting of Lanford's map $$x\mapsto 2x +\frac{1}{2}x(1-x) \pmod 1 $$ . [ABSTRACT FROM AUTHOR]

In this article we present an approach to describing the geometry and curvature of the configuration spaces of a class of simple idealized planar linkages. This is based on determining the curvature of such configuration spaces canonically embedded into Euclidean space, and then the behaviour of the dynamics of such linkages can be understood via the associated geodesic flow. Our objective is to present a method which, in principle, can be applied to many different examples. [ABSTRACT FROM AUTHOR]

In this paper we study the Blackwell and Furstenberg measures, which play an important role in information theory and the study of Lyapunov exponents. For the Blackwell measure we determine parameter domains of singularity and give upper bounds for the Hausdorff dimension. For the Furstenberg measure, we establish absolute continuity for some parameter values. Our method is to analyze linear fractional iterated function schemes which are contracting on average, have no separation properties (that is, we do not assume that the open set condition holds, see Hutchinson in Indiana Univ. Math. J. 30:713-747, ) and, in the case of the Blackwell measure, have place dependent probabilities. In such a general setting, even an effective upper bound on the dimension of the measure is difficult to achieve. [ABSTRACT FROM AUTHOR]

We show how ideas originating in the theory of dynamical systems inspire a new approach to numerical integration of functions. Any Lebesgue integral can be approximated by a sequence of integrals with respect to equidistributions, i.e. evenly weighted discrete probability measures concentrated on an equidistributed set. We prove that, in the case where the integrand is real analytic, suitable linear combinations of these equidistributions lead to a significant acceleration in the rate of convergence of the approximate integral. In particular, the rate of convergence is faster than that of any Newton-Cotes rule. [ABSTRACT FROM AUTHOR]

We consider a system of “ generalised linear forms” defined on a subset x = ( x ij ) of ℝd by , where d = d 1 + ⋯+ d l and for each pair of integers ( i, j), 1 ≤ i ≤ l, 1 ≤ j ≤ d i the sequence of functions ( g ( x))is differentiable on an interval X ij . Then let , for x in the Cartesian product X = × × X ij ⊂ ℝd. Let R = I 1 × ⋯ × I l be a rectangle in $$ \mathbb{T}^l $$ and for each N ≥ 1 let and then define where the supremum is over all rectangles in $$ \mathbb{T}^l $$ . We show that for almost every x ∈ $$ \mathbb{T}^d $$ we have that for appropriate α. Other related results are also described. [ABSTRACT FROM AUTHOR]

In this note we present an axiomatic approach to the decay of correlations for maps of arbitrary dimension with indifferent periodic points. As applications, we apply our results to the well-known Manneville–Pomeau equation and the inhomogeneous diophantine approximation algorithm. [ABSTRACT FROM AUTHOR]

In this paper we investigate the regularity of the topological entropy h for C perturbations of Anosov flows. We show that the topological entropy varies (almost) as smoothly as the perturbation. The results in this paper, along with several related results, have been announced in [KKPW]. [ABSTRACT FROM AUTHOR]