We give explicit bounds for the Hausdorff dimension of the unique invariant measure of C 3 multicritical circle maps without periodic points. These bounds depend only on the arithmetic properties of the rotation number, the number of critical points and their criticalities.

<p style='text-indent:20px;'>Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.</p>

A zero-entropy system is said to be loosely Bernoulli if it can be induced from an irrational rotation of the circle. We provide a criterion for zero-entropy systems to be loosely Bernoulli that is compatible with mixing. Using this criterion, we show the existence of smooth mixing zero-entropy loosely Bernoulli transformations whose Cartesian square is loosely Bernoulli.

For a fixed frequency vector ω ∈ R 2 {0} obeying ω 1 ω 2 < 0, we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate analytic Hamiltonian, having a Lyapunov unstable elliptic equilibrium with frequency ω. In particular, the elliptic fixed points thus constructed will be KAM stable, i.e., accumulated by invariant tori whose Lebesgue density tend to one in the neighborhood of the point and whose frequencies cover a set of positive measure.Similar examples for near-integrable Hamiltonians in action-angle coordinates, in the neighborhood of a Lagrangian invariant torus with arbitrary frequency vector, are also given in this work.