We introduce a notion of sensitivity with respect to a continuous real-valued bounded map which provides a sufficient condition for a continuous transformation, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior (also known as irregular points). The applications of this criterion recover, and extend, several known theorems on the genericity of the irregular set, in addition to yielding a number of new results, including information on the irregular set of geodesic flows, in both negative and non-positive curvature, and semigroup actions.

For a $C^1$ non-conformal repeller, this paper proves that there exists an ergodic measure of full Carathéodory singular dimension. For an average conformal hyperbolic set of a $C^1$ diffeomorphism, this paper constructs a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a $C^{1+\alpha }$ diffeomorphism, this paper shows that there exists an ergodic measure of maximal dimension.

Let $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$. Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$, $n\geqslant 1$. For any $t \in {\mathbb R}$, denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$. By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$. We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

In this paper we show how to apply classical probabilistic tools for partial sums $\sum _{j=0}^{n-1}\varphi \circ \tau ^j$ generated by a skew product $\tau $, built over a sufficiently well-mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $\varphi $, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate-deviations principle, several exponential concentration inequalities and Rosenthal-type moment estimates for skew products with $\alpha $-, $\phi $- or $\psi $-mixing base maps and expanding-on-average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (in contrast to [2]) is that the random maps are not independent, they do not preserve the same measure and the observable $\varphi $ depends also on the base space. For stretched exponentially ${\alpha }$-mixing base maps our proofs are based on multiple correlation estimates, which make the classical method of cumulants applicable. For $\phi $- or $\psi $-mixing base maps, we obtain an ASIP and maximal and concentration inequalities by establishing an $L^\infty $ convergence of the iterates ${\mathcal K}^{\,n}$ of a certain transfer operator ${\mathcal K}$ with respect to a certain sub-${\sigma }$-algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.

Let $ T:[0,1]\to [0,1] $ be an expanding Markov map with a finite partition. Let $ \mu _\phi $ be the invariant Gibbs measure associated with a Hölder continuous potential $ \phi $. For $ x\in [0,1] $ and $ \kappa>0 $, we investigate the size of the uniform approximation set

$$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}.\end{align*} $$

The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $-almost every (a.e.) $ x $ is proven to be $ 1/\alpha _{\max } $, where $ \alpha _{\max }=-\int \phi \,d\mu _{\max }/\int \log |T'|\,d\mu _{\max } $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $. Moreover, when $ \kappa>1/\alpha _{\max } $, we show that for $ \mu _\phi $-a.e. $ x $, the Hausdorff dimension of $ \mathcal U^\kappa (x) $ agrees with the multifractal spectrum of $ \mu _\phi $.

Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $\mu $-sequence entropy tuple, the $\mu $-mean sensitive tuple, and the $\mu $-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.

For typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal linear map. Using this result, we show that suitable assumptions on the periodic orbits have consequences over the entire subshift.

We establish (some directions of) a Ledrappier correspondence between Hölder cocycles, Patterson–Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson–Sullivan measures for $\vartheta $-Anosov representations over a local field and show that these are parameterized by the $\vartheta $-critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the $\vartheta $-limit cone of the representation in question: if ${\mathsf {u}}$ is such a half-line, then the subset of ${\mathsf {u}}$-conical limit points has either total mass if $|\vartheta |\leq 2$ or zero mass if $|\vartheta |\geq 4.$ The case $|\vartheta |=3$ remains unsettled.

We construct measures of maximal u-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.

We show that a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.