This is the second of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this second paper, we restrict our attention to non-spectrally degenerate random walks and we prove precise asymptotics of the probability $p_n(e,e)$ of going back to the origin at time $n$. We combine techniques adapted from thermodynamic formalism with the rough estimates of the Green function given by part I to show that $p_n(e,e)\sim CR^{-n}n^{-3/2}$, where $R$ is the inverse of the spectral radius of the random walk. This both generalizes results of Woess for free products and results of Gouëzel for hyperbolic groups.

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

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 ${\mathcal{L}}$.

We study fast approximation of integrals with respect to stationary probability measures associated to iterated function systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.

We show that if a unital injective endomorphism of a $C^{\ast }$-algebra admits a transfer operator, then both of them are compressions of mutually inverse automorphisms of a bigger algebra. More generally, every interaction group – in the sense of Exel – extending an action of an Ore semigroup by injective unital endomorphisms of a $C^{\ast }$-algebra, admits a dilation to an action of the corresponding enveloping group on another unital $C^{\ast }$-algebra, of which the former is a $C^{\ast }$-subalgebra: the interaction group is obtained by composing the action with a conditional expectation. The dilation is essentially unique if a certain natural condition of minimality is imposed, and it is faithful if and only if the interaction group is also faithful.

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by exp$({{C}_{k}}{{\left| s \right|}^{\delta }})$ in strips $|\,\text{Re}\,s\,|\,\le \,K$, where $\delta$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott–Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\left\{ \left| \,\operatorname{Re}s \right|\,\le \,{{\left| \,\operatorname{Im}s \right|}^{\alpha }} \right\}$ is given, followed by weaker lower bound estimates in strips $\{\text{Re}\,s\,>\,-C,\,|\,\text{IM}\,s|\,\le \,r\}$, and logarithmic neighbourhoods $\{|\text{Re}s|\le {{}_{\rho }}\log |\operatorname{Im}s|\}$. Recent numerical work of Strain-Zworski suggests the upper bounds in strips are optimal.

The aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

For any non-empty subset $\mathcal I$ of the natural numbers, let $\Lambda_{\mathcal I}$ denote those numbers in the unit interval whose continued fraction digits all lie in $\mathcal I$. Define the corresponding transfer operator

$$ \mathcal L_{\mathcal I, \beta} f(z) = \sum_{n \in \mathcal I} \left( \frac{1}{n + z} \right)^{2 \beta} f\left( \frac{1}{n + z} \right) $$

for $\text{Re} (\beta) > \max (0, \theta_{\mathcal I})$, where $\text{Re} (\beta) = \theta_{\mathcal I}$ is the abscissa of convergence of the series $\sum_{n \in \mathcal I}n^{-2 \beta}$.

When acting on a certain Hilbert space $\mathcal H_{\mathcal I, \beta}$, we show that the operator $\mathcal L_{\mathcal I, \beta}$ is conjugate to an integral operator $\mathcal K_{\mathcal I, \beta}$. If furthermore $\beta$ is real, then $\mathcal K_{\mathcal I, \beta}$ is selfadjoint, so that $\mathcal L_{\mathcal I, \beta} : \mathcal H_{\mathcal I, \beta} \to \mathcal H_{\mathcal I, \beta}$ has purely real spectrum. It is proved that $\mathcal L_{\mathcal I, \beta}$ also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space $C^\omega [0, 1]$, and on the Fréchet space $C^\infty [0, 1]$.

The analytic properties of the map $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ are investigated. For certain alphabets $\mathcal I$ of an arithmetic nature (for example, $\mathcal I = \{\text{primes}\}$, $\mathcal I = \{\text{squares}\}$, $\mathcal I$ an arithmetic progression, $\mathcal I$ the set of sums of two squares) it is shown that $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ admits an analytic continuation beyond the half-plane $\text{Re} (\beta) > \theta_{\mathcal I}$.

We present a spectral theory for a class of operators satisfying a weak“Doeblin–Fortet" condition and apply it to a class of transition operators.This gives the convergence of the series ∑k≥0krPkƒ, $r \in \mathbb{N}$ ,under some regularity assumptions and implies the central limit theoremwith a rate in $n^{- \frac{1}{2} }$ for the corresponding Markov chain.An application to a non uniformly hyperbolic transformation on the interval is also given.