Vol 85, No 2 (2021)

We study a quite natural class of diffeomorphisms $G$ on $\mathbb{T}^{\infty}$, where $\mathbb{T}^{\infty}$ is the infinite-dimensional torus (the direct product of countably many circles endowed with the topology of uniform coordinatewise convergence). The diffeomorphisms under consideration can be represented as the sums of a linear hyperbolic map and a periodic additional term. We find some constructive sufficient conditions, which imply that any $G$ in our class is hyperbolic, that is, an Anosov diffeomorphism on $\mathbb{T}^{\infty}$. Moreover, under these conditions we prove the following properties standard in the hyperbolic theory: the existence of stable and unstable invariant foliations, the topological conjugacy to a linear hyperbolic automorphism of the torus and the structural stability of $G$.

We construct an integrable function whose Fourier series possesses the following property. After an appropriatechoice of signs of the coefficients of this series, the partial sums of the resulting series are dense in $L^p$, $p\in(0,1)$.

We prove that the manifold of non-singular points of a stable real caustic germ of type $E_6$and the manifolds of points of transversal intersection of its smooth branches consist only of contractibleconnected components. We also calculate the number of these components.

Jabbarov [1] obtained the exact value of the exponent of convergence of the singular integral in Tarry's problemfor homogeneous polynomials of degree $2$. We extend this result to the case of polynomials of degree $n$.