What does this research mean for the field?

The compactification of the space of invariant probability measures for a transitive countable Markov shift is affine homeomorphic to the Poulsen simplex, and its space of ergodic probability measures is homeomorphic to ℓ2. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To characterize the topological and geometric structure of the space of invariant and ergodic probability measures for transitive countable Markov shifts.

September 5, 2026Transactions of the American Mathematical SocietyOpen Access

Non-compact spaces of invariant measures

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To characterize the topological and geometric structure of the space of invariant and ergodic probability measures for transitive countable Markov shifts.
Analyzed the compactification of the space of invariant probability measures on transitive countable Markov shifts using combinatorial and topological frameworks.
Evaluated thermodynamic formalism to establish dual variational principles for shifts equipped with uniformly continuous potentials.
Demonstrated that the compactification of the invariant measure space is affine homeomorphic to the Poulsen simplex.
Established that the compactification space contains either a single new ergodic measure or a dense set of them based on the combinatorial properties of the shift.
Proved that the space of ergodic probability measures is homeomorphic to the Hilbert space ℓ₂.

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Implication

Theoretical analysis reveals that invariant measure compactifications in transitive countable Markov shifts form Poulsen simplices, extending ergodic properties to non-compact settings.

Cite This Study

Iommi et al. (2026) studied this question.

synapsesocial.com/papers/6a9bd3726b95aff0620eaadc https://doi.org/10.1090/tran/9692

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Implication

Theoretical analysis reveals that invariant measure compactifications in transitive countable Markov shifts form Poulsen simplices, extending ergodic properties to non-compact settings.

Non-compact spaces of invariant measures

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Discussion

Implication

Cite This Study

Non-compact spaces of invariant measures

Key Points

Discussion

Implication

Cite This Study

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