We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential $\chi ^2$form.

Let fi∈C2+α(S1∖{ai,bi}), α>0,i=1,2, be circle homeomorphisms with two break points ai,bi, that is, discontinuities in the derivative Dfi, with identical irrational rotation number ρ and μ1([a1,b1])=μ2([a2,b2]), where μi are the invariant measures of fi,i=1,2. Suppose that the products of the jump ratios of Df1 and Df2do not coincide, that is, Df1 (a1 −0)/Df1 (a1 +0)⋅Df1 (b1 −0)/Df1 (b1 +0)≠Df2(a2−0)/Df2(a2+0)⋅Df2(b2−0)/Df2(b2+0) . Then the map ψ conjugating f1 and f2 is a singular function, that is, it is continuous on S1, but Dψ(x)=0 almost everywhere with respect to Lebesgue measure.

Using the works of Mañé [On the topological entropy of the geodesic flows. J. Differential Geom.45 (1989), 74–93] and Paternain [Topological pressure for geodesic flows. Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 121–138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a $\mathcal {C}^{\infty }$Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.

We prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

For a piecewise expanding unimodal interval map $f$ with unique absolutely continuous invariant probability measure $\mu $, a perturbation $X$, and an observable $\varphi $, the susceptibility function is $\Psi _\varphi (z)= \sum _{k=0}^\infty z^k \int X(x) \varphi '( f^k)(x) (f^k)'(x) \, d\mu $. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps. Comm. Math. Phys.275 (2007), 839–859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps. Nonlinearity21 (2008), 677–711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer–Simon [Natural boundaries and spectral theory. Adv. Math.226 (2011), 4902–4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet. Ann. Sci. Éc. Norm. Supér. (3) 66 (1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that $\Psi _\varphi (z)$ has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of $\Psi _\varphi (z)$, associated with precritical orbits. If the perturbation $X$ is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of $\Psi _\varphi (z)$ as $z\to 1$ exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map (‘linear response formula’). Applying the Wiener–Wintner theorem, we study the singularity type of non-tangential limits of $\Psi _\varphi (z)$ as $z\to e^{i\omega }$ for real $\omega $. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.

We construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic and rationally ergodic along a subsequence of iterates. In particular, they exhibit a law of large numbers. This is accomplished by explicitly calculating, for a subsequence of iterates, the number of visits to zero, and it is shown that such number has a Gaussian distribution.

We prove that the flat product metric on ${D}^{n} \times {S}^{1} $ is scattering rigid where ${D}^{n} $ is the unit ball in ${ \mathbb{R} }^{n} $ and $n\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is the map $S: {U}^{+ } \partial M\rightarrow {U}^{- } \partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to ${ \gamma }_{V}^{\prime } ({T}_{0} )$ where ${\gamma }_{V} $ is the unit speed geodesic determined by $V$ and ${T}_{0} $ is the first positive value of $t$ (when it exists) such that ${\gamma }_{V} (t)$ again lies in the boundary. We show that any other Riemannian manifold $(M, \partial M, g)$ with boundary $\partial M$ isometric to $\partial ({D}^{n} \times {S}^{1} )$ and with the same scattering data must be isometric to ${D}^{n} \times {S}^{1} $. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in $(M, \partial M, g)$ have measure zero in the unit tangent bundle.

Let $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

We study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.

Let (Yt:t>0) be a STIT tessellation process and a>1. We prove that the random sequence (anYan:n∈ℤ) is a non-anticipating factor of a Bernoulli shift. We deduce that the continuous time process (atYat:t∈ℝ) is a Bernoulli flow. We use the techniques and results in Martínez and Nagel [Ergodic description of STIT tessellations. Stochastics 84(1) (2012), 113–134]. We also show that the filtration associated to the non-anticipating factor is standard in Vershik’s sense.

We give a unified treatment of decay of correlations for non-uniformly expanding systems with a good inducing scheme. In addition to being more elementary than previous treatments, our results hold for general integrable return time functions under fairly mild conditions on the inducing scheme.

We study the group Diffr0(𝔸) of Cr-diffeomorphisms of the closed annulus that are isotopic to the identity. We show that, for r≠2,3, the linear space of homogeneous quasi-morphisms on the group Diffr0(𝔸) is one-dimensional. Therefore, the commutator length on this group is (stably) unbounded. In particular, this provides an example of a manifold whose diffeomorphism group is unbounded in the sense of Burago, Ivanov and Polterovich.

We prove homotopical rigidity of expanding dynamical systems, by showing that they are determined by a group-theoretic invariant. We use this to show that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules. Moreover, the cut-and-paste rules can be found algorithmically from the algebraic invariant.

Let $G$ be a real algebraic semi-simple Lie group and $\Gamma $ be the fundamental group of a closed negatively curved manifold. In this article we study the limit cone, introduced by Benoist [Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1) (1997), 1–47], and the growth indicator function, introduced by Quint [Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv. 77 (2002), 503–608], for a class of representations $\rho : \Gamma \rightarrow G$ admitting an equivariant map from $\partial \Gamma $ to the Furstenberg boundary of the symmetric space of $G, $ together with a transversality condition. We then study how these objects vary with the representation.

We prove that for a generic measure-preserving transformation $T$, the closed group generated by $T$ is a continuous homomorphic image of a closed linear subspace of $L_0(\lambda , {\mathbb R})$, where $\lambda $ is the Lebesgue measure, and that the closed group generated by $T$contains an increasing sequence of finite-dimensional tori whose union is dense.

We study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

A detailed Ledrappier–Young theory is presented for linear toral dynamics. First, a proof is given for a simplified definition for local entropies which holds in more general settings besides the current linear dynamics. Then it is shown that the transverse dimensions can be defined directly via the Smale structure of the linear dynamical system. A new entropy formula of Ledrappier–Young type is obtained. The conjecture of Ledrappier and Xie [Vanishing transverse entropy in smooth ergodic theory. Ergod. Th. & Dynam. Sys. 31(4) (2011), 1229–1235] is also discussed for such linear dynamics.