We prove that if ${\rm\Gamma}$ is a subgroup of $\text{Diff}_{+}^{\,1+{\it\epsilon}}(I)$ and $N$ is a natural number such that every non-identity element of ${\rm\Gamma}$ has at most $N$ fixed points, then ${\rm\Gamma}$ is solvable. If in addition ${\rm\Gamma}$ is a subgroup of $\text{Diff}_{+}^{\,2}(I)$, then we can claim that ${\rm\Gamma}$ is meta-abelian.

We generalize the polynomial Szemerédi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^{m}$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne [Intersective polynomials and the polynomial Szemerédi theorem. Adv. Math.219(1) (2008), 369–388] on polynomials over the integers.

We study the center problem for the class ${\mathcal{E}}_{{\rm\Gamma}}$ of Abel differential equations $dv/dt=a_{1}v^{2}+a_{2}v^{3}$, $a_{1},a_{2}\in L^{\infty }([0,T])$, such that images of Lipschitz paths $\tilde{A}:=(\int _{0}^{\cdot }a_{1}(s)\,ds,\int _{0}^{\cdot }a_{2}(s)\,ds):[0,T]\rightarrow \mathbb{R}^{2}$ belong to a fixed compact rectifiable curve ${\rm\Gamma}$. Such a curve is said to be universal if whenever an equation in ${\mathcal{E}}_{{\rm\Gamma}}$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_{1},a_{2}$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with the product being the composition of series, explicit examples of universal curves, and approximation of Lipschitz triangulable curves by universal ones.

The transfer operator associated to a flow (continuous time dynamical system) is a one-parameter operator semigroup. We consider the operator-valued Laplace transform of this one-parameter semigroup. Estimates on the Laplace transform have been used in various settings in order to show the rate at which the flow mixes. Here we consider the case of exponential mixing and the case of rapid mixing (superpolynomial). We develop the operator theory framework amenable to this setting and show that the same estimates may be used to produce results, in terms of the operators, which go beyond the results for the rate of mixing.

We investigate functorial properties of Putnam’s homology theory for Smale spaces. Our analysis shows that the addition of a conjugacy condition is necessary to ensure functoriality. Several examples are discussed that elucidate the need for our additional hypotheses. Our second main result is a natural generalization of Putnam’s Pullback Lemma from shifts of finite type to non-wandering Smale spaces.

We study the dimension spectrum of Lyapunov exponents for multimodal maps of the interval and their generalizations. We also present related results for rational maps on the Riemann sphere.

In this paper we consider diffeomorphisms preserving hyperbolic Sinaĭ–Ruelle–Bowen (SRB) probability measures ${\it\mu}$ having intersections for almost every pair of the stable and unstable manifolds. In this context, when the dimension of the unstable manifold is constant almost everywhere, we show the ergodicity of ${\it\mu}$. As an application we obtain another proof of the ergodicity of a hyperbolic SRB measure for transitive surface diffeomorphisms, which is shown by Rodriguez Hertz, Rodriguez Hertz, Tahzibi and Ures [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb{R}^{d}$. The result is based on a probabilistic version of the Falconer–Sloan condition $C(s)$ introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contraction factor in the interval $(0,\frac{1}{3}]$.

For continuously orbit equivalent one-sided topological Markov shifts $(X_{A},{\it\sigma}_{A})$ and $(X_{B},{\it\sigma}_{B})$, their eventually periodic points and cocycle functions are studied. As a result, we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^{A},\bar{H}_{+}^{A})$ and $(\bar{H}^{B},\bar{H}_{+}^{B})$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of their ordered cohomology groups, so that the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

We provide a full characterization of relations between the shadowing property and the thick shadowing property. We prove that they are equivalent properties for non-wandering systems, the thick shadowing property is always a consequence of the shadowing property, and the thick shadowing property on the chain-recurrent set and the thick shadowing property are the same properties. We also provide a full characterization of the cases when for any family ${\mathcal{F}}$ with the Ramsey property an arbitrary sequence of points can be ${\it\varepsilon}$-traced over a set from ${\mathcal{F}}$.

We provide a complete characterization of extreme points of the space of sofic representations. We also show that the restriction map $\text{Sof}(G,P^{{\it\omega}})$ to $\text{Sof}(H,P^{{\it\omega}})$, where $H\subset G$, is not always surjective. The first part of the paper is a continuation of Păunescu [A convex structure on sofic embeddings. Ergod. Th. & Dynam. Sys.34(4) (2014), 1343–1352] and follows more closely the plan of Brown [Topological dynamical systems associated to $\text{II}_{1}$-factors. Adv. Math.227(4) (2011), 1665–1699].

Given a periodic point ${\it\omega}$ in a ${\it\psi}$-mixing shift with countable alphabet, the sequence $\{S_{n}\}$ of random variables counting the number of multiple returns in shrinking cylindrical neighborhoods of ${\it\omega}$ is considered. Necessary and sufficient conditions for the convergence in distribution of $\{S_{n}\}$ are obtained, and it is shown that the limit is a Pólya–Aeppli distribution. A global condition on the shift system which guarantees the convergence in distribution of $\{S_{n}\}$ for every periodic point is introduced. This condition is used to derive results for $f$-expansions and Gibbs measures. Results are also obtained concerning the possible limit distribution of sub-sequences $\{S_{n_{k}}\}$. A family of examples in which there is no convergence is presented. We also exhibit an example for which the limit distribution is pure Poissonian.

In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if ${\hat{g}}$ is its lift to the universal covering of $S$ that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) the set of fixed points for ${\hat{g}}$ projects to a closed subset $F$ which contains an essential continuum; (2) $g$ has non-contractible periodic points of every sufficiently large period; or (3) there exists a uniform bound $M>0$ such that, if $\hat{x}$ projects to a contractible periodic point, then the ${\hat{g}}$ orbit of $\hat{x}$ has diameter less than or equal to $M$. Some consequences for homeomorphisms of surfaces whose rotation set is a singleton are derived.

Let $f:M\rightarrow M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with non-dense forward orbit: $E(f,y):=\{z\in M:y\notin \overline{\{f^{k}(z),k\in \mathbb{N}\}}\}$ for some $y\in M$. Define $E_{x}(f,y):=E(f,y)\cap W^{u}(x)$ for any $x\in M$. Following a method of Broderick, Fishman and Kleinbock [Schmidt’s game, fractals, and orbits of toral endomorphisms. Ergod. Th. & Dynam. Sys.31 (2011), 1095–1107], we show that $E_{x}(f,y)$ is a winning set for Schmidt games played on $W^{u}(x)$ which implies that $E_{x}(f,y)$ has Hausdorff dimension equal to $\dim W^{u}(x)$. Furthermore, we show that for any non-empty open set $V\subset M$, $E(f,y)\cap V$ has full Hausdorff dimension equal to $\dim M$, by constructing measures supported on $E(f,y)\cap V$ with lower pointwise dimension converging to $\dim M$ and with conditional measures supported on $E_{x}(f,y)\cap V$. The results can be extended to the set of points with forward orbit staying away from a countable subset of $M$.