We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an independent and identically distributed (i.i.d.) process with countable alphabet; (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson–Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.

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 $G$.

In this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

We show that if $(X, T)$ is an extension of an aperiodic subshift (a subsystem of $(\mathop{\{ 1, 2, \ldots , l\} }\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$ for some $l\in \mathbb{N} $) and has mean dimension $\mathrm{mdim} (X, T)\lt (D/ 2), D\in \mathbb{N} $, then it can be equivariantly embedded in $(\mathop{(\mathop{[0, 1] }\nolimits ^{D} )}\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$. The result is sharp. If $(X, T)$ is an extension of an aperiodic zero-dimensional system then it can be equivariantly embedded in $(\mathop{(\mathop{[0, 1] }\nolimits ^{D+ 1} )}\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$.

Let ${ \mathcal{P} }_{G} $ be the family of all topologically mixing, but not exact self-maps of a topological graph $G$. It is proved that the infimum of topological entropies of maps from ${ \mathcal{P} }_{G} $ is bounded from below by $\log 3/ \Lambda (G)$, where $\Lambda (G)$ is a constant depending on the combinatorial structure of $G$. The exact value of the infimum on ${ \mathcal{P} }_{G} $ is calculated for some families of graphs. The main tool is a refined version of the structure theorem for mixing graph maps. It also yields new proofs of some known results, including Blokh’s theorem (topological mixing implies the specification property for maps on graphs).

We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair $({C}_{b} (E), \mathscr{M} (E))$ all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.

We give a definition for a rank-one homeomorphism of a zero-dimensional Polish space X. We show that if a rank-one homeomorphism of X satisfies a certain non-degeneracy condition, then it has trivial centralizer in the group of all homeomorphisms of X, i.e. it commutes only with its integral powers.

Let $f: X\longrightarrow X$ be a map of a continuum. In this paper we examine the following dynamical conditions on $f$: (1) $f$ is continuum-wise fully expansive; (2) $f$ is weakly continuum-wise fully expansive; (3) $f$ is mixing; (4) $f$ is weakly mixing. We first show that (1) implies (2), (2) implies (3) and (3) implies (4). Then we investigate what topological conditions will force the reverse implications to hold and give examples of when the reverse conditions do not hold. In particular, a map of the universal dendrite is given that is weakly mixing but not mixing.

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.

In this paper we show that for every $n\geq 2$ there are minimal systems with perfect weakly mixing sets of order $n$ and all weakly mixing sets of order $n+ 1$ trivial. We present some relations between weakly mixing sets and topological sequence entropy; in particular, we prove that invertible minimal systems with non-trivial weakly mixing sets of order three always have positive topological sequence entropy. We also study relations between weak mixing of sets and other well-established notions from qualitative theory of dynamical systems like (regional) proximality, chaos and equicontinuity in a broad sense.

This paper is devoted to the study of $P$-regularity of viscosity solutions $u(x,P)$, $P\in {\Bbb R}^n$, of a smooth Tonelli Lagrangian $L:T {\Bbb T}^n \rightarrow {\Bbb R}$ characterized by the cell equation $H(x,P+D_xu(x,P))=\overline {H}(P)$, where $H: T^* {\Bbb T}^n\rightarrow {\Bbb R}$ denotes the Hamiltonian associated with $L$ and $\overline {H}$ is the effective Hamiltonian. We show that if $P_0$ corresponds to a quasi-periodic invariant torus with a non-resonant frequency, then $D_xu(x,P)$ is uniformly Hölder continuous in $P$ at $P_0$ with Hölder exponent arbitrarily close to $1$, and if both $H$ and the torus are real analytic and the frequency vector of the torus is Diophantine, then $D_xu(x,P)$ is uniformly Lipschitz continuous in $P$ at $P_0$, i.e., there is a constant $C\gt 0$ such that $\|D_xu(\cdot ,P)-D_xu(\cdot ,P_0)\|_{\infty }\le C\|P-P_0\|$ for $\|P-P_0\|\ll 1$. Similar P-regularity of the Peierls barriers associated with $L(x,v)- \langle P,v \rangle $is also obtained.

Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum. We also give an application of the theorem to random compositions of perturbations of an expanding map in higher dimensions.

We consider the motion of a classical particle under the influence of a random potential on ℝd, in particular the distribution of asymptotic velocities and the question of ergodicity of time evolution.

Consider a random cocycle Φ on a separable infinite-dimensional Banach space preserving a probability measure, which is supported on a random compact set. We show that if Φ satisfies some mild integrability conditions on the differentials, then Ruelle’s inequality relating entropy and positive Lyapunov exponents holds.

We construct templates for geodesic flows on an infinite family of Hecke triangle groups. Our results generalize those of E. Ghys [Knots and dynamics. Proc. Int. Congress of Mathematicians. Vol. 1. International Congress of Mathematicians, Zürich, 2007], who constructed a template for the modular flow in the complement of the trefoil knot in $S^3$. A significant difficulty that arises in any attempt to go beyond the modular flow is the fact that for other Hecke triangles the geodesic flow cannot be viewed as a flow in $S^3$, and one is led to consider embeddings into lens spaces. Our final result is an explicit description of a single ‘Hecke template’ which contains all other templates we construct, allowing a topological study of the periodic orbits of different Hecke triangle groups all at once.

We consider low-dimensional systems with the shadowing property and we study the problem of existence of periodic orbits. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $\epsilon $-transitive class, and in contrast we provide an example of a ${C}^{\infty } $ Kupka–Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class. Finally, we consider the case of transitive endomorphisms of the circle, and we prove that the $\alpha $-Hölder shadowing property with $\alpha \gt 1/ 2$ implies that the system is conjugate to an expanding map.

We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (respectively, lightlike) manifold.

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 prove that every compact manifold without boundary admits a pair of diffeomorphisms that generates ${C}^{1} $ robustly minimal dynamics. We apply the results to the construction of blenders and robustly transitive skew product diffeomorphisms.