Spectral theory for transfer operators on non-standard symbolic spaces arising from transcendental dynamics

Irene Inoquio-Renteria; Rodolfo Viera; Irene Inoquio-Renteria; Rodolfo Viera

doi:10.3934/math.2026410

AIMS Mathematics

2026, Volume 11, Issue 4: 9910-9941. doi: 10.3934/math.2026410

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Spectral theory for transfer operators on non-standard symbolic spaces arising from transcendental dynamics

Irene Inoquio-Renteria ^{1
,
,},
Rodolfo Viera ²

1.
Departamento de Matemáticas, Universidad de La Serena, Avda. Juan Cisternas 1200, La Serena, Chile
2.
Facultad de Ingeniería y Arquitectura, Universidad Central de Chile, Avda. Francisco de Aguirre 0405, La Serena, Chile

Received: 20 November 2025 Revised: 03 March 2026 Accepted: 17 March 2026 Published: 13 April 2026
MSC : 37A30, 37B10, 37D35

We study the shift map on a non-compact symbolic space $ X $ endowed with a metric $ \rho $ that is not compatible with the usual product topology. While abstract, this setting is motivated by symbolic codings arising from transcendental entire maps with Cantor bouquet Julia sets, where the natural metric differs from the standard product topology. We study a spectral theory of Ionescu-Tulcea and Marinescu type for transfer operators acting on Banach spaces of locally Hölder functions adapted to this non-standard metric. We prove quasi-compactness of the transfer operator and establish the existence of a spectral gap, showing that the spectral radius is isolated and corresponds to a simple leading eigenvalue equal to the topological pressure. We identify the associated eigenfunction, conformal measure, and Gibbs state and establish a variational principle. The arguments rely on general symbolic assumptions and do not rely on the product topology. As an application, we consider symbolic models arising from hyperbolic transcendental entire maps of finite order.
- transcendental entire maps,
- symbolic dynamics,
- spectral theory
Citation: Irene Inoquio-Renteria, Rodolfo Viera. Spectral theory for transfer operators on non-standard symbolic spaces arising from transcendental dynamics[J]. AIMS Mathematics, 2026, 11(4): 9910-9941. doi: 10.3934/math.2026410

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Abstract

We study the shift map on a non-compact symbolic space $ X $ endowed with a metric $ \rho $ that is not compatible with the usual product topology. While abstract, this setting is motivated by symbolic codings arising from transcendental entire maps with Cantor bouquet Julia sets, where the natural metric differs from the standard product topology. We study a spectral theory of Ionescu-Tulcea and Marinescu type for transfer operators acting on Banach spaces of locally Hölder functions adapted to this non-standard metric. We prove quasi-compactness of the transfer operator and establish the existence of a spectral gap, showing that the spectral radius is isolated and corresponds to a simple leading eigenvalue equal to the topological pressure. We identify the associated eigenfunction, conformal measure, and Gibbs state and establish a variational principle. The arguments rely on general symbolic assumptions and do not rely on the product topology. As an application, we consider symbolic models arising from hyperbolic transcendental entire maps of finite order.

References

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