The theory of dynamical systems has undergone a dramatical revolution in the 20th century. The beauty and power of the theory of dynamical systems is that it links together different areas of mathematics and physics.
In the last 30 years a great deal of attention was dedicated to a statistical description of strange attractors. This led to the development of notions of various dimensions and entropies, which can be associated to the attractor, dynamical system or invariant measure.
In this paper we review these notions and discuss relations between those, among which the most prominent is the so-called multifractal formalism.

A method of proving the existence of a local transversal intersection between two immersed holomorphic curves in $\mathbb{C}^2$ is suggested. It is based on an application of the inverse function theorem, the corresponding inequalities being checked numerically. The method is applied to the problem of interpretation of tips of ferns on the unstable manifold of the semistandard map as complex homoclinic points.

We show that under certain natural conditions the definition of a hyperbolic torus conventional for the general theory of dynamical systems is quite suitable for needs of the KAM-theory.

The problem of four vortex lines with zero total circulation and zero impulse on a unlimited fluid plane, as it is known [1,3,4,16], is reduced to a problem of three point vortices and is integrated in quadratures. In the given work these results are transferred on a case of four vortices in a two-layer rotating liquid. The analysis of phase trajectories of relative motion of vortices is made, and the singularities of absolute motion on an example of a head-on, off-center collision of two two-layer vortex pairs are studied. In particular, the new class of quasistationary solutions for the given type of motions is obtained. The problems of interaction of the distributed (or finite-core) two-layer vortices are discussed.

The topology of an integrable Hamiltonian system with two degrees of freedom, occuring in dynamics of the magnetic heavy body with a fixed point [1], is explored. The equations of critical submanifolds of the supplementary integral $f$, restricted to arbitrary isoenergy surface $Q_h^3$, are obtained. In particular, all the phase trajectories of a stable periodic motion are found. It is proved, that $f$ is a Bottean integral. The bifurcation diagram, full Fomenko–Zieschang invariant and the topology of each regular isoenergy surface $Q_h^3$ are calculated, as well as the topology of phase manifold $M^4$, which has a degenerate peculiarity of the symplectic structure. This peculiarity did not appear in dynamics before. A method of the computer visualization of Liouville tori bifurcations is offering.

We analyse dynamics generated by quadratic complex map at the accumulation point of the period-tripling cascade (see Golberg, Sinai, and Khanin, Usp. Mat. Nauk. V. 38, № 1, 1983, 159; Cvitanovic; and Myrheim, Phys. Lett. A94, № 8, 1983, 329). It is shown that in general this kind of the universal behavior does not survive the translation two-dimensional real maps violating the Cauchy–Riemann equations. In the extended parameter space of the two-dimensional maps the scaling properties are determined by two complex universal constants. One of them corresponds to perturbations retaining the map in the complex-analytic class and equals $\delta_1 \cong 4.6002-8.9812i$ in accordance with the mentioned works. The second constant $\delta_2 \cong 2.5872+1.8067i$ is responsible for violation of the analyticity. Graphical illustrations of scaling properties associated with both these constants are presented. We conclude that in the extended parameter space of the two-dimensional maps the period tripling universal behavior appears as a phenomenon of codimension $4$.

We find the necessary conditions of existence of an additional meromorphic first integral of the Euler's equations on the Lie algebra $so(4)$.