On periodic mapping data of a two-dimensional torus with one saddle orbit

Periodic data of diffeomorphisms with regular dynamics on surfaces were studied using zeta functions in a series of already classical works by such authors as P. Blanchard, J. Franks, S. Narasimhan, S. Batterson and others. The description of periodic data for gradient-like diffeomorphisms of surfaces were given in the work of A. Bezdenezhnykh and V. Grines by means of the classification of periodic surface transformations obtained by J. Nielsen. V. Grines, O. Pochinka, S. Van Strien showed that the topological classification of arbitrary Morse-Smale diffeomorphisms on surfaces is based on the problem of calculating periodic data of diffeomorphisms with a single saddle periodic orbit. Namely, the construction of filtering for Morse-Smale diffeomorphisms makes it possible to reduce the problem of studying periodic surface diffeomorphism data to the problem of calculating periodic diffeomorphism data with a single saddle periodic orbit. T. Medvedev, E. Nozdrinova, O. Pochinka solved this problem in a general formulation, that is, the periods of source orbits are calculated from a known period of the sink and saddle orbits. However, these formulas do not allow to determine the feasibility of the obtained periodic data on the surface of this kind. In an exhaustive way, the realizability problem is solved only on a sphere. In this paper we establish a complete list of periodic data of diffeomorphisms of a two-dimensional torus with one saddle orbit, provided that at least one nodal point of the map is fixed.

Diffeomorphisms with infinitely many irrational invariant curves

For a surface diffeomorphism f∈Diff l(M), with l≥8, we prove that if f exhibits a non-transversal heteroclinic cycle composed of two fixed saddle points Q1 and Q2, one dissipative and the other expansive, then there exists an open set 𝒱⊂Diff l(M) such that $ f \in \overline {\mathcal {V}}$ and there exists a dense set 𝒟⊂𝒱 such that for all g∈𝒟, g exhibits infinitely many invariant periodic curves with irrational rotation numbers. Moreover, these curves are C1 conjugated to an irrational rotation on 𝕊1.

The entropy of C2 surface diffeomorphisms in terms of Hausdorff dimension and a Lyapunov exponent

In this paper we prove that if the entropy of an ergodic measure preserved by a C2 surface diffeomorphism is positive then it is equal to the product of the Hausdorff dimension of the quotient measure defined by the family of stable manifolds and the positive Lyapunov exponent.