A note on circle maps driven by strongly expanding endomorphisms on

ABSTRACT

We investigate the dynamics of a class of smooth maps of the two-torus of the form T(x, y) = (Nx, f_x(y)), where is a monotone family (in x) of orientation preserving circle diffeomorphisms and is large. For our class of maps, we show that the dynamics essentially is the same as that of the projective action of non-uniformly hyperbolic -cocycles. This generalizes a result by L.S. Young [6] to maps T outside the (projective) matrix cocycle case.

Keywords:

• Lyapunov exponents
• synchronization
• forced circle diffeomorphisms

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. That is, a linear skew-product on of the form (x, w) ↦ (Nx, A(x)w), where . Such a map naturally induces a projective action on ; see [6].

2. We could easily work with f where f( ·, y) is monotone and of degree >1; but for simplicity we fix the degree to be 1.

3. In the case of (projective) -cocycles, u(x) would be the direction of the Oseledets’ subspace.

4. There is nothing special with the point y = 0; we could have taken another one.