Logarithmic bounds for ergodic averages of constant type rotation number flows on the torus: a short proof - Archive ouverte HAL Access content directly
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## Logarithmic bounds for ergodic averages of constant type rotation number flows on the torus: a short proof

Jérôme Carrand

#### Abstract

We give a short proof that the ergodic averages of $\mathcal{C}^1$ observables for a $\mathcal{C}^1$ flow on $\mathbb{T}^2$ admitting a closed transversal curve whose Poincar\'e map has constant type rotation number have growth deviating at most logarithmically from a linear one. For this, we relate the latter integral to the Birkhoff sum of a well-chosen observable on the circle and use the Denjoy-Koksma inequality. We also give an example of a nonminimal flow satisfying the above assumptions.

#### Domains

Mathematics [math]

### Dates and versions

hal-03197369 , version 1 (13-04-2021)

### Identifiers

• HAL Id : hal-03197369 , version 1
• ARXIV :

### Cite

Jérôme Carrand. Logarithmic bounds for ergodic averages of constant type rotation number flows on the torus: a short proof. 2021. ⟨hal-03197369⟩

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