The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of Hölder potentials globally, which answers a question of Kucherenko and Quas. In the second part of the work, we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as the number of symbols in the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.

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

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$-measures. Indag. Math. (N.S.)22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

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.

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.

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.

One of the unfulfilled aims of the authors of the preceding paper [W. Parry and M. Pollicott. An analogue of Bauer’s theorem for closed orbits of skew products. Ergod. Th. & Dynam. Sys.28 (2008), 535–546] was to find a dynamical analogue of Artin reciprocity. In this addendum, we present one such version, suggested by work of Sunada.

In this article we prove an analogue of Bauer’s theorem from algebraic number theory in the context of hyperbolic systems.

In this paper we study certain conformal iterated function schemes in two dimensions that are natural generalizations of the Sierpinski carpet construction. In particular, we consider scaling factors for which the open set condition fails. For such ‘fat Sierpinski carpets’ we study the range of parameters for which the dimension of the set is exactly known, or for which the set has positive measure.

We consider the class of $\mathbb{R}^n$ extensions of Anosov diffeomorphisms on infranilmanifolds, and find necessary and sufficient conditions for topological transitivity. In particular, if the fiber is $\mathbb{R}$, the existence of a semi-orbit with the projection on $\mathbb{R}$ unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and stable topological transitivity are equivalent.

We present a powerful approach to computing the Hausdorff dimension of certain conformally self-similar sets. We illustrate this method for the dimension \mathop{\rm dim}\nolimits_H(E_2) of the set E_2, consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking feature of this method is that the successive approximations converge to \mathop{\rm dim}\nolimits_H(E_2) at a super-exponential rate

In this paper we show that strong generalizations of the measurable Livšic theorem for cocycles taking values in connected non-compact linear semisimple Lie groups, a canonical example being SL(2, \mathbb{R}), can be deduced from an elegant approach of Brin and Pesin to the dynamics of partially hyperbolic systems.

In this paper we obtain a polynomial error term for the closed orbit counting function associated to certain hyperbolic flows. In the case of weak-mixing transitive Anosov flows no further conditions are required; for general weak-mixing hyperbolic flows a diophantine condition on the periods of the closed orbits is required.

We introduce non-homogeneous equilibrium states which include the classical equilibrium states for subshifts of finite type, Riesz products and G-measures. We prove a rather precise estimate for ∥Pnf∥∞ where Pn are the averaging operators. Applications are given to estimate ∥Lngf∥∞ (Lg being a transfer operator on a subshift of finite type) and then to prove a central limit theorem for f(Tnx) (T being the shift), to study the almost everywhere convergence of some lacunary series with respect the equilibrium state and then to compute the Hausdorff dimension of the equilibrium state etc.

In this paper we show that the recent work of Dolgopiat and earlier work of the author give a rich class of examples of Axiom A attracting flows which mix exponentially fast. As an immediate corollary we have a proof of a conjecture of Ruelle.