We prove that a ${C}^{1} $ generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that ${C}^{1} $ generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the ${C}^{1} $ topology.

For a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to ‘see’ sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

Given an n-dimensional substitution Φ whose associated linear expansion Λ is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the tiling space ΩΦ to construct a finite-to-one semi-conjugacy G:ΩΦ→𝕋D, called a geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If Λ satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of Λ, G is surjective and coincides with the map onto the maximal equicontinuous factor of the ℝn-action on ΩΦ. We are led to formulate a higher-dimensional generalization of the Pisot substitution conjecture: if Λ satisfies the Pisot family condition and the rank of the one-dimensional cohomology of ΩΦ equals the generalized degree of Λ, then the ℝn-action on ΩΦhas pure discrete spectrum.

We prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

We define new isomorphism invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaos DC2 and its slightly stronger version (which we denote by $\text {DC}1\frac 12$). We prove that: (1) if a topological system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 $(\text {DC}1\frac 12)$ chaotic; (2) every ergodic system with positive Kolmogorov–Sinai entropy is measure-theoretically$^+$ chaotic (even in a slightly stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, that is, of a system of entropy zero with uniform measure-theoretic$^+$chaos.

Mixing (of all orders) rank-one actions $T$ of the Heisenberg group ${H}_{3} ( \mathbb{R} )$ are constructed. The restriction of $T$ to the center of ${H}_{3} ( \mathbb{R} )$ is simple and commutes only with $T$. Mixing Poisson and mixing Gaussian actions of ${H}_{3} ( \mathbb{R} )$ are also constructed. A rigid weakly mixing rank-one action $T$ is constructed such that the restriction of $T$ to the center of ${H}_{3} ( \mathbb{R} )$ is not isomorphic to its inverse.

For a piecewise expanding unimodal interval map $f$ with unique absolutely continuous invariant probability measure $\mu $, a perturbation $X$, and an observable $\varphi $, the susceptibility function is $\Psi _\varphi (z)= \sum _{k=0}^\infty z^k \int X(x) \varphi '( f^k)(x) (f^k)'(x) \, d\mu $. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps. Comm. Math. Phys.275 (2007), 839–859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps. Nonlinearity21 (2008), 677–711] (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer–Simon [Natural boundaries and spectral theory. Adv. Math.226 (2011), 4902–4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les séries de Dirichlet. Ann. Sci. Éc. Norm. Supér. (3) 66 (1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that $\Psi _\varphi (z)$ has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of $\Psi _\varphi (z)$, associated with precritical orbits. If the perturbation $X$ is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of $\Psi _\varphi (z)$ as $z\to 1$ exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map (‘linear response formula’). Applying the Wiener–Wintner theorem, we study the singularity type of non-tangential limits of $\Psi _\varphi (z)$ as $z\to e^{i\omega }$ for real $\omega $. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.

Bowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then there is a unique measure of maximal entropy.

The rate of recurrence to measurable subsets in a conservative, ergodic infinite-measure-preserving system is quantified by generic divergence or convergence of certain sums given by a function $\omega (n)$. In the context of skew products over transformations of a probability space, we relate this notion to the more frequently studied question of the growth rate of ergodic sums (including Lyapunov exponents). We study in particular skew products over an irrational rotation given by bounded variation $ \mathbb{Z} $-valued functions: first the generic situation is studied and recurrence quantified, and then certain specific skew products over rotations are shown to violate this generic rate of recurrence.

The term special overlapping refers to a certain simple type of piecewise continuous function from the unit interval to itself and also to a simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used (1) to find a necessary and sufficient condition for a fractal transformation from the attractor of one special overlapping IFS to the attractor of another special overlapping IFS to be a homeomorphism and (2) to find a formula for the topological entropy of the dynamical system associated with a special overlapping function.

We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space $X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets $\{n_{k}\}$ such that $n_{k+1}/n_{k}\to +\infty $, or such that $n_{k}$ divides $n_{k+1}$ for each $k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each $r\ge 1$, of $r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

We study the Lyapunov exponents of the geodesic flow of a Hilbert geometry. We prove that all of the information is contained in the shape of the boundary at the endpoint of the chosen orbit. We have to introduce a regularity property of convex functions to make this link precise. As a consequence, Lyapunov manifolds tangent to the Lyapunov splitting appear very easily. All of this work can be seen as a consequence of convexity and the flatness of Hilbert geometries.

We give a sharp polynomial bound on the number of Pollicott–Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure and Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer and Sjöstrand in scattering theory and by Faure and Sjöstrand in the theory of Anosov flows.

We construct a lamination of the space of unimodal maps with critical points of fixed degree $d\geq 2$ by the hybrid classes. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps and allows us to transfer a priori bounds in the phase space to the parameter space. This implies that almost every map in such a family is either regular or stochastic.

We give a new proof of Moeckel’s result that for any finite index subgroup of the modular group, almost every real number has its regular continued fraction approximants equidistributed into the cusps of the subgroup according to the weighted cusp widths. Our proof uses a skew product over a cross-section for the geodesic flow on the modular surface. Our techniques show that the same result holds true for approximants found by Nakada’s $\alpha $-continued fractions, and also that the analogous result holds for approximants that are algebraic numbers given by any of Rosen’s $\lambda $-continued fractions, related to the infinite family of Hecke triangle Fuchsian groups.

We construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic and rationally ergodic along a subsequence of iterates. In particular, they exhibit a law of large numbers. This is accomplished by explicitly calculating, for a subsequence of iterates, the number of visits to zero, and it is shown that such number has a Gaussian distribution.

Under suitable hypotheses we establish a quantitative pointwise ergodic theorem which applies to trimmed Birkhoff sums of weakly integrable functions.

We provide a strengthened version of the famous Jakobson's theorem. Consider an interval map $f$ satisfying a summability condition. For a generic one-parameter family ${f}_{t} $ of maps with ${f}_{0} = f$, we prove that $t= 0$ is a Lebesgue density point of the set of parameters for which ${f}_{t} $ satisfies both the Collet–Eckmann condition and a strong polynomial recurrence condition.

In this paper we consider a filtered chain complex $C$ and its differential given by a connection matrix $\Delta $ which determines an associated spectral sequence $({E}^{r} , {d}^{r} )$. We present an algorithm which sweeps the connection matrix in order to span the modules ${E}^{r} $ in terms of bases of $C$ and gives the differentials ${d}^{r} $. In this process a sequence of similar connection matrices and associated transition matrices are produced. This algebraic procedure can be viewed as a continuation, where the transition matrices give information about the bifurcation behavior. We introduce directed graphs, called flow and bifurcation schematics, that depict bifurcations that could occur if the sequence of connection matrices and transition matrices were realized in a continuation of a Morse decomposition, and we present a dynamic interpretation theorem that provides conditions on a parameterized family of flows under which such a continuation could occur.

We compare a piecewise linear map with constant slope $\beta \gt 1$ and a piecewise linear map with constant slope $-\beta $. These maps are called the positive and negative $\beta $-transformations. We show that for a certain set of $\beta $s, the multinacci numbers, there exists a measurable isomorphism between these two maps. We further show that for all other values of $\beta $between 1 and 2 the two maps cannot be isomorphic.