We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations, and discusses questions and open problems which arise.

We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole at a uniform rate. We show that under suitable conditions the topological entropy varies locally Hölder continuously with the local Hölder exponent depending itself on the value of the topological entropy.

We study polygonal billiards with reflection laws contracting the angle of reflection towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic Sinai–Ruelle–Bowen measures whose basins cover a set of full Lebesgue measure.

We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley–Zehnder index of a fixed point changes. The main result is that when the Conley–Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

We prove that when $f$ is a continuous self-map acting on a compact metric space $(X,d)$ that satisfies the shadowing property, then the set of irregular points (i.e., points with divergent Birkhoff averages) has full entropy. Using this fact, we prove that, in the class of $C^{0}$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbits of such points converges to some Sinai–Ruelle–Bowen-like measure in the weak$^{\ast }$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.

We show that for every ergodic measure-preserving system $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ with commuting transformations $S$ and $T$, the average

$$\begin{eqnarray}\frac{1}{N^{3}}\mathop{\sum }_{i,j,k=0}^{N-1}f_{0}(S^{j}T^{k}x)f_{1}(S^{i+j}T^{k}x)f_{2}(S^{j}T^{i+k}x)\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$N\rightarrow \infty$

$f_{0},f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$

$(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{i=0}^{N-1}f_{1}(S^{i}x)f_{2}(T^{i}x)\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$N\rightarrow \infty$

$f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed non-empty subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.

We show under weak hypotheses that $\unicode[STIX]{x2202}X$, the Roller boundary of a finite-dimensional CAT(0) cube complex $X$ is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group $\unicode[STIX]{x1D6E4}$. In particular, we show that if $\unicode[STIX]{x1D6E4}$ admits a non-elementary proper action on $X$, and $\unicode[STIX]{x1D707}$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\unicode[STIX]{x1D707}$-stationary measure on $\unicode[STIX]{x2202}X$ making it the Furstenberg–Poisson boundary for the $\unicode[STIX]{x1D707}$-random walk on $\unicode[STIX]{x1D6E4}$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.

In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit non-trivial monodromy.

We prove that special flows over an ergodic rotation of the circle under a $C^{1}$ roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$, ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.

Given a finite irreducible set of real $d\times d$ matrices $A_{1},\ldots ,A_{M}$ and a real parameter $s>0$, there exists a unique shift-invariant equilibrium state on $\{1,\ldots ,M\}^{\mathbb{N}}$ associated to $(A_{1},\ldots ,A_{M},s)$. In this paper we characterize the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterize when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by $A_{1},\ldots ,A_{M}$, and when it is a Bernoulli measure. We also give a general sufficient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by Kusuoka are explored in an appendix.

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is the best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

We prove that, for $C^{1}$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class $H(p)$ have all their Lyapunov exponents bounded away from zero, then $H(p)$ must be (uniformly) hyperbolic. This is in the spirit of the works on the stability conjecture, but with a significant difference that the homoclinic class $H(p)$ is not known isolated in advance, hence the ‘weak’ periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an ‘intrinsic’ nature, and the classical proof of the stability conjecture does not work. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.