Non-ergodic maps in the tangent family

Acta Mathematica Hungarica, 1996

Mathematical Proceedings of the Cambridge …, 2005

We study the dynamics of non-entire transcendental meromorphic functions with a finite asymptotic value mapped after some iterations onto a pole. This situation does not appear in the case of rational or entire functions. We consider the family of non-entire functions

Topology and its Applications, 2000

For a rational f : C → C with a conformal measure µ we show that if there is a subset of the Julia set J (f ) of positive µ-measure whose points are not eventual preimages of critical or parabolic points and have limit sets not contained in the union of the limit sets of recurrent critical points, then µ is non-atomic, µ(J (f )) = 1, ω(x) = J (f ) for µ-a.e. point x ∈ J (f ) and f is conservative, ergodic and exact. The proof uses a version of the Lebesgue Density Theorem valid for Borel measures and conformal balls.

2008

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure. arXiv:0805.2099v3 [math.DS]

Pacific Journal of Mathematics, 1990

We study some classes of totally ergodic functions on locally compact Abelian groups. Among other things, we establish the following result: If R is a locally compact commutative ring, 3ί is the additive group of R, χ is a continuous character of 3$ , and p is the function from 3l n (n e N) into 3% induced by a polynomial of n variables with coefficients in R, then the function χ o p either is a trigonometric polynomial on 3ί n or all of its Fourier-Bohr coefficients with respect to any Banach mean on L°°{^n) vanish.

Encyclopaedia of Mathematical Sciences, 1989

Ergodic Theory and Dynamical Systems, 2013

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

Mathematische Annalen, 2011

We consider volume-preserving flows (Φ f t ) t∈R on S × R, where S is a closed connected surface of genus g ≥ 2 and (Φ f t ) t∈R has the form Φ f t (x, y) = φtx, y + t 0 f (φsx) ds where (φt) t∈R is a locally Hamiltonian flow of hyperbolic periodic type on S and f is a smooth real valued function on S. We investigate ergodic properties of these infinite measure-preserving flows and prove that if f belongs to a space of finite codimension in C 2+ǫ (S), then the following dynamical dichotomy holds: if there is a fixed point of (φt) t∈R on which f does not vanish, then (Φ f t ) t∈R is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension (Φ 0 t ) t∈R . The proof of this result exploits the reduction of (Φ f t ) t∈R to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (φt) t∈R on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.

Fundamenta Mathematicae, 2005

We prove that for Ω being an immediate basin of attraction to an attracting fixed point for a rational mapping of the Riemann sphere, and for an ergodic invariant measure µ on the boundary Fr Ω, with positive Lyapunov exponent, there is an invariant subset of Fr Ω which is an expanding repeller of Hausdorff dimension arbitrarily close to the Hausdorff dimension of µ. We also prove generalizations and a geometric coding tree abstract version. The paper is a continuation of a paper in Fund. Math. 145 (1994) by the author and Anna Zdunik, where the density of periodic orbits in Fr Ω was proved. 1. Introduction. Let Ω be a simply connected domain in C and f be a holomorphic map defined on a neighbourhood W of Fr Ω to C. Assume f (W ∩ Ω) ⊂ Ω, f (Fr Ω) ⊂ Fr Ω and Fr Ω repells to the side of Ω, that is, ∞ n=0 f −n (W ∩ Ω) = Fr Ω. An important special case is where Ω is an immediate basin of attraction of an attracting fixed point for a rational function. This covers also the case of a component of the immediate basin of attraction to a periodic attracting orbit, as one can consider an iterate of f mapping the component to itself. Distances and derivatives are considered in the Riemann spherical metric on C. Let R : D → Ω be a Riemann mapping from the unit disc onto Ω and let g be a holomorphic extension of R −1 • f • R to a neighbourhood of the unit circle ∂D. It exists and it is expanding on ∂D (see [P2, Section 7]). We prove the following Theorem A. Let ν be an ergodic g-invariant probability measure on ∂D such that for ν-a.e. ζ ∈ ∂D the radial limit R(ζ) := lim rր1 R(rζ) exists. Assume that the measure µ := R * (ν) has positive Lyapunov exponent χ µ (f).

The Quarterly Journal of Mathematics, 1988