## Abstract

A Markov operator P on a probability space (S, Σ μ) with μ invariant, is called hyperbounded if for some 1 ≤ p≤ q ≤ ∞ it maps (continuously) L^{p} into L^{q}. We deduce from a recent result of Glück that a hyperbounded P is quasi-compact, hence uniformly ergodic, in all L^{r}(S, μ), 1 < r < ∞. We prove, using a method similar to Foguel’s, that a hyperbounded Markov operator has periodic behavior similar to that of Harris recurrent operators, and for the ergodic case obtain conditions for aperiodicity. Given a probability ν on the unit circle, we prove that if the convolution operator P_{ν}f:= ν ⋇ f is hyperbounded, then ν is atomless. We show that there is ν absolutely continuous such that P_{ν} is not hyperbounded, and there is ν with all powers singular such that P_{ν} is hyperbounded. As an application, we prove that if P_{ν} is hyperbounded, then for any sequence (n_{k}) of distinct positive integers with bounded gaps, (n_{k}x) is uniformly distributed mod 1 for ν almost every x (even when ν is singular).

Original language | English |
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Journal | Israel Journal of Mathematics |

DOIs | |

State | Accepted/In press - 1 Jan 2024 |

## ASJC Scopus subject areas

- General Mathematics

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