Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$. In this paper, we state that

$$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$

$J\subset \overline{{\mathcal{O}}}$

$$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$

$J\subset \overline{{\mathcal{O}}}$

$K$

${\mathcal{H}}^{{\it\alpha}}$

${\it\alpha}$

${\mathcal{P}}^{{\it\alpha}}$

${\it\alpha}$

$K$

${\mathcal{H}}^{{\it\alpha}}(K)$

${\mathcal{P}}^{{\it\alpha}}(K)$

$\mathbb{R}$

Abért and Weiss have shown that the Bernoulli shift ${s}_{\Gamma } $ of a countably infinite group $\Gamma $ is weakly contained in any free measure preserving action $\boldsymbol{a}$ of $\Gamma $. Proving a conjecture of Ioana, we establish a strong version of this result by showing that ${\boldsymbol{s}}_{\Gamma } \times \boldsymbol{a}$ is weakly equivalent to $\boldsymbol{a}$. Using random Bernoulli shifts introduced by Abért, Glasner, and Virag, we generalize this to non-free actions, replacing ${\boldsymbol{s}}_{\Gamma } $ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that, on the weak equivalence class of a free measure preserving action, the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence are not classifiable by countable structures. This in particular shows that there are no free weakly rigid actions, that is, actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group $\Gamma $ is said to have property ${\text{EMD} }^{\ast } $ if the action ${\boldsymbol{p}}_{\Gamma } $ of $\Gamma $ on its profinite completion weakly contains all ergodic measure preserving actions of $\Gamma $, and $\Gamma $ is said to have property $\text{MD} $ if $\boldsymbol{\iota} \times {\boldsymbol{p}}_{\Gamma } $ weakly contains all measure preserving actions of $\Gamma $, where $\boldsymbol{\iota} $ denotes the identity action on a standard non-atomic probability space. Kechris has shown that ${\text{EMD} }^{\ast } $ implies $\text{MD} $ and asked if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.

We investigate under which circumstances the projective subdynamics of multidimensional shifts of finite type can be non-sofic. In particular, we give a sufficient condition ensuring the one-dimensional projective subdynamics of such $\mathbb{Z}^{d}$ systems to be sofic and we show that this condition is already met (along certain, respectively all, sublattices) by most of the commonly used uniform mixing conditions. (Examples of the different situations are given.) Complementary to this we are able to prove a characterization of one-dimensional projective subdynamics for strongly irreducible $\mathbb{Z}^{d}$ shifts of finite type for every $d\geq 2$: in this setting the class of possible subdynamics coincides exactly with the class of mixing $\mathbb{Z}$ sofics. This stands in stark contrast to the much more diverse situation in merely topologically mixing multidimensional shifts of finite type.

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n,1\leq p<(n/2)-1$) on $\mathbb{S}^{2}$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group of $F_{n,p}$. We also establish that $F_{n,p}$ and $F_{n^{\prime },p^{\prime }}$ are quasisymmetrically equivalent if and only if $(n,p)=(n^{\prime },p^{\prime })$.

We consider Hölder continuous fiber bunched $\text{GL}(d,\mathbb{R})$-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of a diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question of whether an Anosov diffeomorphism is smoothly conjugate to a $C^{1}$-small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of $\text{GL}(d,\mathbb{R})$, for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.

For any ${\it\theta},{\it\omega}>1/2$, we prove that, if any $d$-pseudotrajectory of length ${\sim}1/d^{{\it\omega}}$ of a diffeomorphism $f\in C^{2}$ can be $d^{{\it\theta}}$-shadowed by an exact trajectory, then $f$ is structurally stable. Previously it was conjectured [S. M. Hammel et al. Do numerical orbits of chaotic dynamical processes represent true orbits. J. Complexity3 (1987), 136–145; S. M. Hammel et al. Numerical orbits of chaotic processes represent true orbits. Bull. Amer. Math. Soc.19 (1988), 465–469] that for ${\it\theta}={\it\omega}=1/2$, this property holds for a wide class of non-uniformly hyperbolic diffeomorphisms. In the proof, we introduce the notion of a sublinear growth property for inhomogeneous linear equations and prove that it implies exponential dichotomy.

We present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows we obtain absolute continuity under assumptions weaker than those to be found in the literature, and for skew products over translations and Furstenberg transformations we obtain countable Lebesgue spectrum under assumptions not previously covered in the literature.

The enumeration of combinatorial classes of the complex polynomial vector fields in $ \mathbb{C} $ presented by K. Dias [Enumerating combinatorial classes of the complex polynomial vector fields in $ \mathbb{C} $. Ergod. Th. & Dynam. Sys. 33 (2013), 416–440] is extended here to a closed form enumeration of combinatorial classes for degree $d$ polynomial vector fields up to rotations of the $2(d- 1)\mathrm{th} $ roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V. A. Liskovets [Reductive enumeration under mutually orthogonal group actions. Acta Appl. Math. 52 (1998), 91–120].

Let ($M$, ${\rm\Omega}$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^{l}$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^{1}$-submanifold ${\rm\Delta}$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_{N}M$. We suppose that ${\rm\Delta}$ transversely intersects the stable manifold of $N$. Then, we prove that for all ${\it\varepsilon}>0$, and for $n\in \mathbb{N}$ large enough, there exists $x_{n}\in N$ such that $f^{n}({\rm\Delta})$ is ${\it\varepsilon}$-close, in the $C^{1}$ topology, to the strongly unstable manifold of $x_{n}$. As an application of this ${\it\lambda}$-lemma, we prove the existence of shadowing orbits for a finite family of invariant minimal sets (for which we do not assume any regularity) contained in a normally hyperbolic manifold and having heteroclinic connections. As a particular case, we recover classical results on the existence of diffusion orbits (Arnold’s example).

In this article, we prove that for all pairs of primitive Pisot or uniform substitutions with the same dominating eigenvalue, there exists a finite set of block maps such that every block map between the corresponding subshifts is an element of this set, up to a shift.

We obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given.