This paper surveys the new, algorithmic theory of moving frames developed by the author and M.Fels. The method is used to classify joint invariants and joint differential invariants of transformation groups, and equivalence and symmetry properties of submanifolds. Applications in classical invariant theory, geometry, and computer vision are indicated.

Let $f$ be a $C^1$-diffeomorphism of the circle $\mathbb{T}^1=\mathbb{R}/\mathbb{Z}$ with an irrational rotation number. We show that, for every real number $s $, there exists a probability measure $\mu _s$, unique if $f$ is $C^2$, that satisfies, for any function $\varphi \in C^0(\mathbb{T}^1)$: $$ \int\limits_{\mathbb{T}^1}\varphi \,d\mu _s=\int\limits_{\mathbb{T}^1}\varphi \circ f\;(Df)^s\,d\mu _s . $$ This measure continuously depends on the pair $( s,f) $ when one considers the weak topology on measures and the $C^1$-topology on diffeomorphisms. Examples are given where uniqueness fails with $f$ of class $C^1$.
These results partially extend to the case of a rational rotation number for non degenerate semi-stable diffeomorphisms of the circle.
We then show that the set of diffeomorphisms that have a given irrational rotation number has a tangent hyperplane at any $C^2$-diffeomorphism, the direction of which is the kernel of $\mu _{-1}$.

We prove structural stability under small perturbations of a family of real analytic Hamiltonian systems of $n+1$ degrees of freedom $(n \geqslant 2)$, comprising an invariant partially hyperbolic $n$-torus with the Kronecker flow on it with a diophantine frequency, and an unstable (stable) exact Lagrangian submanifold (whisker), containing this torus. This is the preservation of the exact Lagrangian properties of the whisker that we focus upon. Hence, we develop a Normal form, which is valid globally in the neighborhood of the perturbed whisker and enables its representation as an exact Lagrangian submanifold in the original coordinates, whose generating function solves the Hamilton–Jacobi equation.

In the paper we obtain the bifurcation sets for a family of Liouville integrable Hamiltonian systems with the additional integral of fourth degree.

We consider non-linear autonomous systems of three differential equations, which coincide (under a special choice of their parameters) with some known models of chaotic dynamics. The conditions on parameters, providing the existence of exact analytical solutions, expressed via solutions of the third Painleve equation are obtained.

The motion of the Lagrangian top, the fixed point of which performs small-amplitude vertical oscillations, is studied. The case, when values of constants $a$ and $b$ of cyclic integrals are related by $|a|=|b|$, is considered. It is known that for the classical Lagrangian top with a fixed point there exists either one regular precession of the top or none at $a=b$ depending on values of parameters of the problem (the parameter $a$ and the parameter, characterizing the position of the center of mass of the top on the axis of symmetry); at $a=-b$ there is no regular precessions.
The existence of periodic motions of the top close to regular precessions (with a period equal to the period of oscillations of the fixed point) is investigated in case of oscillations of the fixed point. In the first part of the present paper periodic motions of the top, which occur from regular precession, are found in case $a=b$ and stability of these motions is determined. In the second part it is additionally assumed that oscillations of the fixed point have a high frequency, and angular velocities of precession and self-rotation of the top are small. A qualitatively new result is obtained which does not have an analogue either in the classical problem or in a problem of motion of the top with small-amplitude oscillations of the fixed point: Motions of the top close to regular precessions are found, these motions exist both at $a=b$ and at $a=-b$, and their number is different (there are two, one or none at $a=b$, and there are two or none at $a=-b$) depending on values of parameters of the problem. The problem of stability of these motions is solved precisely with the help of the KAM-theory.

In this paper in terms of random fractal the description of a growing random fractal is proposed. The refined relations describing the fractal dimension of a random fractal and the volume occupied by a dendritic particle in $d$-dimensional topological space are established. Moreover, the established relations are derived for an arbitrary form of elementary fractal fragments which fit dendritic branches. The calculation results for the parabolic form of the fragment of an elementary fractal which grows with time are given.