Measurable-conjugacy invariance of the Kolmogorov–Sinai entropy of a system

Two measure-preserving systems (α, T, μ) and (β, S, ν) that are measurably conjugate — i.e. there is a measurable isomorphism e : α ≃ᵐ β that is measure-preserving (e_* μ = ν) and intertwines the dynamics (e ∘ T = S ∘ e) — have equal Kolmogorov–Sinai entropies:

h(T) = h(S) (Oseledets.Entropy.ksEntropy_congr_of_conjugacy).

This is the entropy-side companion of the index-reindexing invariance ksEntropyPartition_reindex (which only permutes the index type of a single partition): here the whole space is transported.

Proof

e is a factor map from (α, T, μ) onto (β, S, ν) and e.symm is a factor map the other way. For a factor map the partition-relative entropies of a pulled-back partition agree with those of the original (factor_relative_eq). Hence:

• every partition R of β pulls back through e to a partition e⁻¹R of α with h(e⁻¹R, T) = h(R, S), so h(R, S) ≤ h(T) and therefore h(S) ≤ h(T);
• symmetrically, pulling back through e.symm gives h(T) ≤ h(S).

Both pullbacks preserve the index type, so the pulled-back partitions land directly in the Fin n-indexed family realising ksEntropy; no reindexing is needed. le_antisymm finishes.

Main results

• Oseledets.Entropy.ksEntropy_congr_of_conjugacy: measurable conjugacy ⇒ equal KS entropy.

References

• Peter Walters, An Introduction to Ergodic Theory, GTM 79, Springer (1982), Ch. 4.

theorem Oseledets.Entropy.ksEntropy_congr_of_conjugacy {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {T : α → α} {S : β → β} (hT : MeasureTheory.MeasurePreserving T μ μ) (hS : MeasureTheory.MeasurePreserving S ν ν) (e : α ≃ᵐ β) (he : MeasureTheory.MeasurePreserving (⇑e) μ ν) (hconj : ⇑e ∘ T = S ∘ ⇑e) : ksEntropy hT = ksEntropy hS

Measurable-conjugacy invariance of the Kolmogorov–Sinai entropy of a system. If a measurable isomorphism e : α ≃ᵐ β is measure-preserving and intertwines the two dynamics (e ∘ T = S ∘ e), then the systems (α, T, μ) and (β, S, ν) have equal Kolmogorov–Sinai entropies h(T) = h(S).

Both e and its inverse e.symm are factor maps, so the factor-relative entropy invariance factor_relative_eq transports partition entropies in either direction; pulling partitions back through e.symm gives h(T) ≤ h(S) and through e gives h(S) ≤ h(T).