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We establish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated with Sierpiński lattices and infinite Sierpiński gaskets and other post-critically finite self-similar sets.

In this note we introduce zeta functions and L -functions for discrete and faithful representations of surface groups in PSL ( d , R ) , for d ≥ 3 . These are natural generalizations of the well known classical Selberg zeta function and L -function for Fuchsian groups, corresponding to the case d = 2 . We show that these complex functions have meromorphic extensions to the entire complex plane C .

We present an elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.

In recent papers, Kenyon et al. (Ergod Theory Dyn Syst 32:1567–1584 2012), and Fan et al. (C R Math Acad Sci Paris 349:961–964 2011, Adv Math 295:271–333 2016) introduced a form of non-linear thermodynamic formalism based on solutions to a non-linear equation using matrices. In this note we consider the more general setting of Hölder continuous functions.

In this article we extend results of Kakutani, Adler–Flatto, Smilansky and others on the classical α -Kakutani equidistribution result for sequences arising from finite partitions of the interval. In particular, we describe a generalization of the equidistribution result to infinite partitions. In addition, we give discrepancy estimates, extending results of Drmota–Infusino [ 8 ].

In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

In this article we study the Lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions. This leads to new explicit formulae for the Lyapunov exponents and to an efficient method for their computation.

The joint spectral radius of a pair of $2\\times 2$ real matrices $(A_{0},A_{1})\\in M_{2}(\\mathbb{R})^{2}$ is defined to be $r(A_{0},A_{1})=\\limsup _{n\\rightarrow \\infty }\\max \\{\\Vert A_{i_{1}}\\cdots A_{i_{n}}\\Vert ^{1/n}:i_{j}\\in \\{0,1\\}\\}$ , the optimal growth rate of the norm of products of these matrices. The Lagarias–Wang finiteness conjecture [Lagarias and Wang. The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl. 214 (1995), 17–42], asserting that $r(A_{0},A_{1})$ is always the $n$ th root of the spectral radius of some length- $n$ product $A_{i_{1}}\\cdots A_{i_{n}}$ , has been refuted by Bousch and Mairesse [Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. J. Amer. Math. Soc. 15 (2002), 77–111], with subsequent counterexamples presented by Blondel et al [An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal. 24 (2003), 963–970], Kozyakin [A dynamical systems construction of a counterexample to the finiteness conjecture. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (Seville, Spain, December 2005). IEEE, Piscataway, NJ, pp. 2338–2343] and Hare et al [An explicit counterexample to the Lagarias–Wang finiteness conjecture. Adv. Math. 226 (2011), 4667–4701]. In this article, we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of $M_{2}(\\mathbb{R})^{2}$ with the property that each member $(A_{0},A_{1})$ of the subset generates uncountably many counterexamples of the form $(A_{0},tA_{1})$ . Our methods employ ergodic theory; in particular, the analysis of Sturmian invariant measures. This approach allows a short proof that the relationship between the parameter $t$ and the Sturmian parameter ${\\mathcal{P}}(t)$ is a devil’s staircase.

In this paper we study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of the Hausdorff dimension of the survivor set.

A recent paper of Chen and Volkmer estimated a quantity related to the spectral radius of a transfer operator and with significance in the study of Fourier multipliers. We provide a rigorous proof of their conjectured numerical value.