Proof of `Norm bounds for iterates and tempered families` (1st statement)

Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.

The $L¹$ orbit-decay theorem for $log⁺ ‖𝓛‖$ , applied to $σ⁻¹$ , which is ergodic because $σ$ is.

Lax Lean Archive