In this paper we show that the natural action of the symmetric group acting on the product space $\{0,1\}^{\mathbb{N}}$ endowed with a Bernoulli measure is approximately transitive. We also extend the result to a larger class of probability measures.

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called $f$-maximizing if the time average of the real-valued function $f$ along the orbit is larger than along all other orbits, and an invariant probability measure is called $f$-maximizing if it gives $f$ a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

We prove that there exists a dense set of analytic expanding maps of the circle for which the Ruelle eigenvalues enjoy exponential lower bounds. The proof combines potential theoretic techniques and explicit calculations for the spectrum of expanding Blaschke products.

In this work, we consider two class $P$-homeomorphisms, $f$ and $g$, of the circle with break point singularities, that are differentiable maps except at some singular points where the derivative has a jump. Assume that they have the same irrational rotation number of bounded type and that the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$, respectively. We show that if $f$ and $g$ are not break-equivalent, then any topological conjugating $h$ between $f$ and $g$ is a singular function, i.e., it is continuous on the circle, but $\text{Dh}(x)=0$ almost everywhere (a.e.) with respect to the Lebesgue measure. In particular, this result holds under some combinatorial assumptions on the jumps at break points. It also generalizes previous results obtained for one and two break points and complements that of Cunha–Smania which was established for break equivalence.

We prove a von Neumann-type ergodic theorem for averages of unitary operators arising from the Furstenberg–Poisson boundary representation (the quasi-regular representation) of any lattice in a non-compact connected semisimple Lie group with finite center.

We prove statistical stability for a family of Lorenz attractors with a $C^{1+\unicode[STIX]{x1D6FC}}$ stable foliation.

Let $F\subseteq \mathbb{R}^{2}$ be a Bedford–McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between zero and one. We prove that any similarity $g$ such that $g(F)\subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by ‘zooming in’ on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self-similar sets in the line.

We consider two independent and stationary measures over $\unicode[STIX]{x1D712}^{\mathbb{N}}$, where $\unicode[STIX]{x1D712}$ is a finite or countable alphabet. For each pair of $n$-strings in the product space we define $T_{n}^{(2)}$ as the length of the shortest path connecting one of them to the other. Here the paths are generated by the underlying dynamic of the measures. If they are ergodic and have positive entropy we prove that, for almost every pair of realizations $(\mathbf{x},\mathbf{y})$, $T_{n}^{(2)}/n$ is concentrated in one, as $n$ diverges. Under mild extra conditions we prove a large-deviation principle. We also show that the fluctuations of $T_{n}^{(2)}$ converge (only) in distribution to a non-degenerate distribution. These results are all linked to a quantity that computes the similarity between those two measures. This is the so-called divergence between two measures, which is also introduced. Several examples are provided.

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this ‘expected formula’ does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $\text{SO}(d)$ for $d\geq 3$. We also obtain new upper bounds, derived using sumsets, for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the weak separation property is satisfied, that is, the overlaps are controllable, then the ‘expected formula’ does hold.

Given a factor code $\unicode[STIX]{x1D70B}$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $\unicode[STIX]{x1D708}$ on $Y$, and a function $V$ on $X$ with sufficient regularity, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\unicode[STIX]{x1D708}$ and maximize the measure pressure $h(\unicode[STIX]{x1D707})+\int V\,d\unicode[STIX]{x1D707}$ among all measures in the fiber $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D708})$. If $\unicode[STIX]{x1D708}$ is fully supported, this bound is the class degree of $\unicode[STIX]{x1D70B}$. This generalizes a previous result for the special case of $V=0$ and thus settles a conjecture raised by Allahbakhshi and Quas.

The Oseledets multiplicative ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures at summer schools in Brazil, France, and Russia.

The orbit closure of any translation surface under the horocycle flow in almost any direction equals its $\text{SL}_{2}(\mathbb{R})$ orbit closure. This result gives rise to new characterizations of lattice surfaces in terms of the horocycle flow.

Let $\unicode[STIX]{x1D70B}$ be a non-degenerate permutation on at least four symbols. We show that the set of uniquely ergodic interval exchange transformations with permutation $\unicode[STIX]{x1D70B}$ is path-connected.

In this paper we investigate maps of the two-torus $\mathbb{T}^{2}$ of the form $T(x,y)=(x+\unicode[STIX]{x1D714},g(x)+f(y))$ for Diophantine $\unicode[STIX]{x1D714}\in \mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\rightarrow \mathbb{T}$, where each $g$ is strictly monotone and of degree 2 and each $f$ is an orientation-preserving circle homeomorphism. For our class of $f$ and $g$, we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^{2}$. Only a low-regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.

We show that $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum _{n\leq K}g(n)$, then for every ergodic dynamical system $(X,{\mathcal{A}},\unicode[STIX]{x1D707},\unicode[STIX]{x1D70F})$ and every $f\in L^{1}(X)$,

$$\begin{eqnarray}\lim _{K\rightarrow \infty }\frac{1}{S_{g,K}}\mathop{\sum }_{n=1}^{K}g(n)f(\unicode[STIX]{x1D70F}^{n}x)=\int _{X}f\,d\unicode[STIX]{x1D707}\quad \text{for }\unicode[STIX]{x1D707}\text{ almost every }x\in X.\end{eqnarray}$$

$L^{p}$

$p>1$

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary effectively closed $\mathbb{Z}^{d}$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^{2}$. Let $H$ be a finitely generated group and $G=\mathbb{Z}^{2}\rtimes _{\unicode[STIX]{x1D711}}H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,T)$ where $Y\subset \{0,1\}^{\mathbb{N}}$, there exists a $G$-subshift of finite type $(X,\unicode[STIX]{x1D70E})$ such that the $H$-subaction of $(X,\unicode[STIX]{x1D70E})$ is an extension of $(Y,T)$. In the case where $T$ is an expansive action, a subshift conjugated to $(Y,T)$ can be obtained as the $H$-projective subdynamics of a sofic $G$-subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.

We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^{2}$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular, they hold for Lebesgue almost every rotation interval. By standard results, the measure obtained is a global physical measure and it is hyperbolic.

A set $R\subset \mathbb{N}$ is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every $\unicode[STIX]{x1D716}>0$ there exists a set $B=\bigcup _{i=1}^{r}a_{i}\mathbb{N}+b_{i}$, where $a_{1},\ldots ,a_{r},b_{1},\ldots ,b_{r}\in \mathbb{N}$, such that

$$\begin{eqnarray}\overline{d}(R\triangle B):=\limsup _{N\rightarrow \infty }\frac{|(R\triangle B)\cap \{1,\ldots ,N\}|}{N}<\unicode[STIX]{x1D716}.\end{eqnarray}$$

$\unicode[STIX]{x1D6F7}_{x}:=\{n\in \mathbb{N}:\boldsymbol{\unicode[STIX]{x1D711}}(n)/n<x\}$

$x\in [0,1]$

$\boldsymbol{\unicode[STIX]{x1D711}}$

$R\subset \mathbb{N}$

$\overline{d}(R)>0$

(a) $R$ is divisible, i.e. $\overline{d}(R\cap u\mathbb{N})>0$ for all $u\in \mathbb{N}$;

(b) $R$ is an averaging set of polynomial single recurrence;

(c) $R$ is an averaging set of polynomial multiple recurrence.

As an application, we show that if $R\subset \mathbb{N}$ is rational and divisible, then for any set $E\subset \mathbb{N}$ with $\overline{d}(E)>0$ and any polynomials $p_{i}\in \mathbb{Q}[t]$, $i=1,\ldots ,\ell$, which satisfy $p_{i}(\mathbb{Z})\subset \mathbb{Z}$ and $p_{i}(0)=0$ for all $i\in \{1,\ldots ,\ell \}$, there exists $\unicode[STIX]{x1D6FD}>0$ such that the set

$$\begin{eqnarray}\{n\in R:\overline{d}(E\cap (E-p_{1}(n))\cap \cdots \cap (E-p_{\ell }(n)))>\unicode[STIX]{x1D6FD}\}\end{eqnarray}$$

Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if ${\mathcal{A}}$ is a finite alphabet, $\unicode[STIX]{x1D702}\in {\mathcal{A}}^{\mathbb{N}}$ is rationally almost periodic, $S$ denotes the left-shift on ${\mathcal{A}}^{\mathbb{Z}}$ and

$$\begin{eqnarray}X:=\{y\in {\mathcal{A}}^{\mathbb{Z}}:\text{each word appearing in}~y~\text{appears in}~\unicode[STIX]{x1D702}\},\end{eqnarray}$$

$\unicode[STIX]{x1D702}$

$S$

$\unicode[STIX]{x1D708}$

$X$

$(X,\unicode[STIX]{x1D708},S)$

We define the concept of continuum-wise expansivity for flows, and we prove that continuum-wise expansive flows on compact metric spaces with topological dimension greater than one have positive entropy.