We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give examples of maps not semiconjugate to a map of constant slope via a non-decreasing map. Our examples are continuous and transitive.

We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.

To any positive contraction $Q$ on $\ell ^{2}(W)$, there is associated a determinantal probability measure $\mathbf{P}^{Q}$ on $2^{W}$, where $W$ is a denumerable set. Let ${\rm\Gamma}$ be a countable sofic finitely generated group and $G=({\rm\Gamma},\mathsf{E})$ be a Cayley graph of ${\rm\Gamma}$. We show that if $Q_{1}$ and $Q_{2}$ are two ${\rm\Gamma}$-equivariant positive contractions on $\ell ^{2}({\rm\Gamma})$ or on $\ell ^{2}(\mathsf{E})$ with $Q_{1}\leq Q_{2}$, then there exists a ${\rm\Gamma}$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination $\mathbf{P}^{Q_{1}}\preccurlyeq \mathbf{P}^{Q_{2}}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when ${\rm\Gamma}$ is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures $\mathbf{P}^{Q}$ as above are $\bar{d}$-limits of finitely dependent processes. Thus, when ${\rm\Gamma}$ is amenable, $\mathbf{P}^{Q}$ is isomorphic to a Bernoulli shift, which was known before only when ${\rm\Gamma}$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.

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.

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

In this paper we apply proof mining techniques to compute, in the setting of CAT$(\unicode[STIX]{x1D705})$ spaces (with $\unicode[STIX]{x1D705}>0$), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.

Let $M$ be a connected $1$-manifold, and let $G$ be a finitely-generated nilpotent group of homeomorphisms of $M$. Our main result is that one can find a collection $\{I_{i,j},M_{i,j}\}$ of open disjoint intervals with dense union in $M$, such that the intervals are permuted by the action of $G$, and the restriction of the action to any $I_{i,j}$ is trivial, while the restriction of the action to any $M_{i,j}$ is minimal and abelian. It is a classical result that if $G$ is a finitely-generated, torsion-free nilpotent group, then there exist faithful continuous actions of $G$ on $M$. Farb and Franks [Groups of homeomorphisms of one-manifolds, III: Nilpotent subgroups. Ergod. Th. & Dynam. Sys.23 (2003), 1467–1484] showed that for such $G$, there always exists a faithful $C^{1}$ action on $M$. As an application of our main result, we show that every continuous action of $G$ on $M$ can be conjugated to a $C^{1+\unicode[STIX]{x1D6FC}}$ action for any $\unicode[STIX]{x1D6FC}<1/d(G)$, where $d(G)$ is the degree of polynomial growth of $G$.

It is known that a polynomial on $\mathbb{C}$ is holomorphically conjugate to its term of highest degree near infinity. By assigning suitable weights, we generalize this fact to polynomial skew products on $\mathbb{C}^{2}$.

We consider some non-uniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs–Markov–Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball $B(x,r)$ converges to a Poisson distribution as the radius $r\rightarrow 0$ and after suitable normalization.

We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity ‘slowly’, and which have Hausdorff dimension equal to $1$. We prove these results by using the idea of an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.

Let $(X,{\it\sigma}_{X}),(Y,{\it\sigma}_{Y})$ be one-sided subshifts and ${\it\pi}:X\rightarrow Y$ a factor map. Suppose that $X$ has the specification property. Let ${\it\mu}$ be a unique invariant Gibbs measure for a sequence of continuous functions ${\mathcal{F}}=\{\log f_{n}\}_{n=1}^{\infty }$ on $X$, which is an almost additive potential with bounded variation. We show that ${\it\pi}{\it\mu}$ is a unique invariant Gibbs measure for a sequence of continuous functions ${\mathcal{G}}=\{\log g_{n}\}_{n=1}^{\infty }$ on $Y$. When $(X,{\it\sigma}_{X})$ is a full shift, we characterize ${\mathcal{G}}$ and ${\it\mu}$ by using relative pressure. This ${\mathcal{G}}$ is a generalization of a continuous function found by Pollicott and Kempton in their work on factors of Gibbs measures for continuous functions. We also consider the following question: given a unique invariant Gibbs measure ${\it\nu}$ for a sequence of continuous functions ${\mathcal{F}}_{2}$ on $Y$, can we find an invariant Gibbs measure ${\it\mu}$ for a sequence of continuous functions ${\mathcal{F}}_{1}$ on $X$ such that ${\it\pi}{\it\mu}={\it\nu}$? We show that such a measure exists under a certain condition. In particular, if $(X,{\it\sigma}_{X})$ is a full shift and ${\it\nu}$ is a unique invariant Gibbs measure for a function in the Bowen class, then there exists a preimage ${\it\mu}$ of ${\it\nu}$ which is a unique invariant Gibbs measure for a function in the Bowen class.

Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.

We consider the dynamical system given by an $\text{Ad}$ -diagonalizable element $a$ of the $\mathbb{Q}_{p}$ -points $G$ of a unimodular linear algebraic group acting by translation on a finite volume quotient $X$ . Assuming that this action is exponentially mixing (e.g. if $G$ is simple) we give an effective version (in terms of $K$ -finite vectors of the regular representation) of the following statement: If ${\it\mu}$ is an $a$ -invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$ , then ${\it\mu}$ is close to the unique $G$ -invariant probability measure of $X$ .

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 apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarník–Besicovitch theorem, localized Jarník–Besicovitch theorem and its several generalizations. As is well known, the classic Jarník–Besicovitch sets, expressed in terms of continued fractions, can be written as

$$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(\log |T^{\prime }x|+\cdots +\log |T^{\prime }(T^{n-1}x)|)}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$

$T$

$a_{n}(x)$

$n$

$x$

$$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(x)(f(x)+\cdots +f(T^{n-1}x))}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$

${\it\tau}(x)$

$f(x)$

$[0,1]$

We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

The structures of the enveloping semigroups of certain elementary finite- and infinite-dimensional distal dynamical systems are given, answering open problems posed in 1982 by Namioka [Ellis groups and compact right topological groups. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, 295–300]. The universal minimal system with (topological) quasi-discrete spectrum is obtained from the infinite-dimensional case. It is proved that, on the one hand, a minimal system is a factor of this universal system if and only if its enveloping semigroup has quasi-discrete spectrum and that, on the other hand, such a factor need not have quasi-discrete spectrum in itself. This leads to a natural generalization of the property of having quasi-discrete spectrum, which is named the ${\mathcal{W}}$ -property.

In this paper, we consider a $\mathbb{Z}^{d}$ extension of the well known fact that subshifts with only finitely many follower sets are sofic. As in Kass and Madden [A sufficient condition for non-soficness of higher-dimensional subshifts. Proc. Amer. Math. Soc.141 (2013), 3803–3816], we adopt a natural $\mathbb{Z}^{d}$ analogue of a follower set called an extender set. The extender set of a finite word $w$ in a $\mathbb{Z}^{d}$ subshift $X$ is the set of all configurations of symbols on the rest of $\mathbb{Z}^{d}$ which form a point of $X$ when concatenated with $w$. As our main result, we show that for any $d\geq 1$ and any $\mathbb{Z}^{d}$ subshift $X$, if there exists $n$ so that the number of extender sets of words on a $d$-dimensional hypercube of side length $n$ is less than or equal to $n$, then $X$ is sofic. We also give an example of a non-sofic system for which this number of extender sets is $n+1$ for every $n$. We prove this theorem in two parts. First we show that if the number of extender sets of words on a $d$-dimensional hypercube of side length $n$ is less than or equal to $n$ for some $n$, then there is a uniform bound on the number of extender sets for words on any sufficiently large rectangular prism; to our knowledge, this result is new even for $d=1$. We then show that such a uniform bound implies soficity. Our main result is reminiscent of the classical Morse–Hedlund theorem, which says that if $X$ is a $\mathbb{Z}$ subshift and there exists an $n$ such that the number of words of length $n$ is less than or equal to $n$, then $X$ consists entirely of periodic points. However, most proofs of that result use the fact that the number of words of length $n$ in a $\mathbb{Z}$ subshift is non-decreasing in $n$, and we present an example (due to Martin Delacourt) which shows that this monotonicity does not hold for numbers of extender sets (or follower sets) of words of length $n$.

Concerning hyperbolic diffeomorphisms, one expects a better smoothness of linearization, but it may be confined by resonance among eigenvalues. Hartman gave a three-dimensional analytic mapping with resonance which cannot be linearized by a Lipschitz conjugacy. Since then, efforts have been made to give the ${\it\alpha}$-Hölder continuity of the conjugacy and hope the exponent ${\it\alpha}<1$ can be as large as possible. Recently, it was proved for some weakly resonant hyperbolic diffeomorphisms that ${\it\alpha}$ can be as large as we expect. In this paper we prove that this result holds for all $C^{\infty }$ weakly resonant hyperbolic diffeomorphisms.

We study the Euler–Lagrange flow of a Tonelli Lagrangian on the 2-torus $\mathbb{T}^{2}$ at a fixed energy level ${\mathcal{E}}\subset T\mathbb{T}^{2}$ strictly above Mañé’s strict critical value. We prove that, if for some rational direction ${\it\zeta}\in S^{1}$ there is no invariant graph ${\mathcal{T}}\subset {\mathcal{E}}$ over $\mathbb{T}^{2}$ for the Euler–Lagrange flow with the property that all orbits on ${\mathcal{T}}$ have an asymptotic direction equal to ${\it\zeta}$, then there are chaotic dynamics in ${\mathcal{E}}$. This implies that, if the topological entropy of the Euler–Lagrange flow in ${\mathcal{E}}$ vanishes, then in ${\mathcal{E}}$ there are invariant graphs for all asymptotic directions ${\it\zeta}\in S^{1}$ and integrable-like behavior on a large scale.