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For self-similar sets on $\mathbb{R}$ satisfying the exponential separation condition we show that the natural projections of shift invariant ergodic measures is equal to $\min\{1,\frac{h}{-\chi}\}$, where $h$ and $\chi$ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin's recent result on the $L^{q}$ dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets., Comment: 16 pages. A few details have been clarified and section 5 has been expanded. Now accepted in the Proceedings of the LMS

Working on strongly irreducible planar self-affine sets satisfying the strong open set condition, we calculate the Birkhoff spectrum of continuous potentials and the Lyapunov spectrum., Comment: 32 pages, 1 figure

For piecewise monotone interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In the uniformly hyperbolic case we obtain complete results, in the case with parabolic behaviour we are able to describe the part of the sets where the lower Lyapunov exponent is positive. In addition we give some lower bounds on the full spectrum in this case. This is an extension of work of Hofbauer on the entropy and Lyapunov spectra., Comment: 22 pages

We study equilibrium measures (K\"aenm\"aki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the K\"aenm\"aki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type., Comment: 20 pages. A few minor clarifications made and some typos corrected. To appear in the Journal of Fractal Geometry

Previous study of the Assouad dimension of planar self-affine sets has relied heavily on the underlying IFS having a `grid structure', thus allowing for the use of approximate squares. We study the Assouad dimension of a class of self-affine carpets which do not have an associated grid structure. We find that the Assouad dimension is related to the box and Assouad dimensions of the (self-similar) projection of the self-affine set onto the first coordinate and to the local dimensions of the projection of a natural Bernoulli measure onto the first coordinate. In a special case we relate the Assouad dimension of the Przytycki-Urba\'nski sets to the lower local dimensions of Bernoulli convolutions., Comment: 14 pages, to appear in Proc. Amer. Math. Soc

We study the pointwise perturbations of countable Markov maps with infinitely many inverse branches and establish the following continuity theorem: Let $T_k$ and $T$ be expanding countable Markov maps such that the inverse branches of $T_k$ converge pointwise to the inverse branches of $T$ as $k \to \infty$. Then under suitable regularity assumptions on the maps $T_k$ and $T$ the following limit exists: $$\lim_{k \to \infty} \dim_\mathrm{H} \{x : \theta_k'(x) \neq 0\} = 1,$$ where $\theta_k$ is the topological conjugacy between $T_k$ and $T$ and $\dim_\mathrm{H}$ stands for the Hausdorff dimension. This is in contrast with the fact that other natural quantities measuring the singularity of $\theta_k$ fail to be continuous in this manner under pointwise convergence such as the H\"older exponent of $\theta_k$ or the Hausdorff dimension $\dim_\mathrm{H} (\mu \circ \theta_k)$ for the preimage of the absolutely continuous invariant measure $\mu$ for $T$. As an application we obtain a perturbation theorem in non-uniformly hyperbolic dynamics for conjugacies between intermittent Manneville-Pomeau maps $x \mapsto x + x^{1+\alpha} \mod 1$ when varying the parameter $\alpha$., Comment: 34 pages, 3 figures

In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures., Comment: 26 pages

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| = O(|\xi|^{-\eta}), \quad \text{as } |\xi| \to \infty.$$ We show that this property holds for any Gibbs measure $\mu$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result implies that the Fourier-Stieltjes coefficients of the Minkowski's question mark function decay to $0$ polynomially answering a question of Salem from 1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents., Comment: v3: 29 pages; peer-reviewed version, fixes typos and added more elaborations, and included comments on Salem's problem. To appear in Math. Ann

In this note we consider the Hausdorff dimension of self-affine sets with random perturbations. We extend previous work in this area by allowing the random perturbation to be distributed according to distributions with unbounded support as long as the measure of the tails of the distribution decay super polynomially., Comment: 18 pages. A few typographical errors have been corrected and some clarifications have been made. Now accepted for publication in 'Annales Academi{\ae} Scientiarum Fennic{\ae}, Mathematica'

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit the first example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics., Comment: Some updates following referee suggestions

This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions are also given., Comment: New version, now with 28 papers. Much more detail is now given in section 4. We've also corrected a few typos and added some additional explanation to some of the proofs (e.g. prop 6.1)

We study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the "renewal flow", which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points., Comment: In this version of the paper some typos have been corrected and some references updated. Note that the former title of this paper was "Parabolic suspension flows"

In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension., Comment: 32 pages

This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions., Comment: Example 5.2 expanded. Typos corrected. Section 6.1 superced the note "Thermodynamic formalism for the positive geodesic flow on the modular surface" arXiv:1009.4623

In this paper we investigate multifractal decompositions based on values of Birkhoff averages of functions from a class of symbolically continuous functions. This will be done for an expanding interval map with infinitely many branches and is a generalisation of previous work for expanding maps with finitely many branches. We show that there are substantial differences between this case and the setting where the expanding map has only finitely many branches., Comment: 24 pages

Let $\nu_\lambda^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of $\nu_\lambda^p$ for typical $\lambda$. Namely, we investigate the size of the sets \[ \Delta_{\lambda,p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log \nu_\lambda^p(B(x,r))}{\log r} =\alpha\right\}. \] Our main results highlight the fact that for almost all, and in some cases all, $\lambda$ in an appropriate range, $\Delta_{\lambda,p}(\alpha)$ is nonempty and, moreover, has positive Hausdorff dimension, for many values of $\alpha$. This happens even in parameter regions for which $\nu_\lambda^p$ is typically absolutely continuous., Comment: 24 pages, 2 figures

We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ...$ if $i_n>\Phi(i_{n-1})$ for certain increasing functions $\Phi:\mathbb{N}\rightarrow\mathbb{N}$. We compute both the Hausdorff and packing dimensions of such sets. Our results generalize work of Ramharter which shows that the set of continued fractions with strictly increasing digits has Hausdorff dimension 1/2., Comment: 14 pages

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain regularity assumptions on the potential the Birkhoff spectrum is real analytic. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite., Comment: 27 pages. Substantial changes have been made to Theorem 1.2 and sections 4,5 and 6. Some minor corrections have been made elsewhere

In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford-McMullen. This extends the work of King where the multifractal analysis is computed with strong additional separation assumptions., Comment: 11 pages. We have added more detail to the proofs of lemmas 2,3 and 4 as well as correcting a few errors

We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $\Lambda$ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of $\Lambda$ in a non-principal direction has Hausdorff dimension $\min(\gamma,1)$, where $\gamma$ is the Hausdorff dimension of $\Lambda$. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets., Comment: 20 pages. Some minor errors have been corrected and a few points have been clarified

We study differentiability of topological conjugacies between expanding piecewise $C^{1+\epsilon}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a "rigidity dichotomy" for the type of conjugacies under consideration.

There are well-known examples of dynamical systems for which the Birkhoff averages with respect to a given observable along some or all of the orbits do not converge. It has been suggested that such orbits could be classified using higher order averages. In the case of a bounded observable, we show that a classical result of G.H. Hardy implies that if the Birkhoff averages do not converge, then neither do the higher order averages. If the Birkhoff averages do not converge then we may denote by $[\alpha_k,\beta_k]$ the limit set of the $k$-th order averages. The sequence of intervals thus generated is nested: $[\alpha_{k+1},\beta_{k+1}] \subset [\alpha_k,\beta_k]$. We can thus make a distinction among nonconvergent Birkhoff averages; either: B1. $\cap_{k=1}^\infty [\alpha_k,\beta_k]$ is a point $B_\infty$, or, B2. $\cap_{k=1}^\infty [\alpha_k,\beta_k]$ is a non-trivial interval $[\alpha_\infty,\beta_\infty]$. We give characterizations of the types B1 and B2 in terms of how slowly they oscillate and we give examples that exhibit both behaviours B1 and B2 in the context of full shifts on finite symbols and "Bowen's example". For finite full shifts, we show that the set of orbits with type B2 behaviour has full topological entropy., Comment: 16 pages, 1 figure, revised and corrected version. This manuscript has been accepted for publication in Dynamical Systems

We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems., Comment: New version, main changes: the results hold for continuous potentials (previously: Holder potentials) and C^1 maps (previously: C^1+\alpha), we also prove analyticity of the spectrum

We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville--Pomeau map., Comment: 19 pages. We have corrected some typographical errors and clarified a few points