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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]

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [Comment. Math. Helv. 95 (2020), pp. 593–633]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain–Kontorovich [Ann. of Math. (2) 180 (2014), pp. 137–196], Huang [ An improvement to Zaremba's conjecture , ProQuest LLC, Ann Arbor, MI, 2015] and Kan [Mat. Sb. 210 (2019), pp. 75–130]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [Amer. J. Math. 120 (1998), pp. 691-721]. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in this paper in a way that is straightforward to implement. These estimates apparently cannot be obtained by other known methods. [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 prove that the algorithm of [19] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set E 2 (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25. The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension. [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]

The famous Birkhoff ergodic theorem shows that given an ergodic measure the averages of an integrable function along typical orbits converge to the integral of the function. The multifractal spectra describe the sets of points for which the averages converge to another limit. In this note, we will consider the specific setting of conformal repellers and show how to estimate the Hausdorff dimensions of such sets via approximations to their alternative characterizations as zeros of appropriate determinant functions. [ABSTRACT FROM PUBLISHER]

In this note we give a simple proof that the Hausdorff dimension of the basic set for a real analytic Smale horseshoe map depends analytically on the transformation. This method is based on the use of dynamical zeta functions. We prove analogous statements for the value of the pointwise dimension of the measure of maximal entropy and then use this to address an interesting question raised by Damanik and Gorodetski. [ABSTRACT FROM AUTHOR]

In this paper we address an interesting question on the computation of the dimension of self-affine sets in Euclidean space. A well-known result of Falconer showed that under mild assumptions the Hausdorff dimension of typical self-affine sets is equal to its Singularity dimension. Heuter and Lalley subsequently presented a smaller open family of non-trivial examples for which there is an equality of these two dimensions. In this article we analyse the size of this family and present an efficient algorithm for estimating the dimension. [ABSTRACT FROM AUTHOR]

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure μ λ is the probability measure corresponding to the law of the random variable ξ = ∑ k = 0 ∞ ξ k λ k , where ξ k are i.i.d. random variables assuming values −1 and 1 with equal probability and 1 2 < λ < 1. In particular, for Bernoulli convolutions we give a uniform lower bound dim H ⁡ (μ λ) ≥ 0.96399 for all 1 2 < λ < 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]

This note shows how a spectral result of Peigne for iterated function schemes, with contraction in mean, can be applied to derive statistical properties of the measure (in particular, Berry-Esseen bounds in the central limit theorems with error terms, large deviation results, etc.). [ABSTRACT FROM AUTHOR]

In this paper we consider extensions of Livsic's well-known classical results on rigidity of measurable cocycles. We consider transformations which need not be hyperbolic, and cocycles which take values in certain non-abelian non-compact groups. 1991 Mathematics Subject Classification 58F11, 58F15. [ABSTRACT FROM AUTHOR]