CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures

Dimension conservation for almost every projection has been well-established by the work of Marstrand, Mattila and Hunt and Kaloshin. More recently, Hochman and Shmerkin used CP-chains, a tool first introduced by Furstenberg, to prove all projections preserve dimension of measures on [0,1]2 that are...

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Detalles Bibliográficos
Publicado: 2017
Materias:
Dynamical systems
Ergodicity
Fractal
Gibbs measures
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v304_n_p227_Almarza
http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p227_Almarza
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00018708_v304_n_p227_Almarza
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spelling paper:paper_00018708_v304_n_p227_Almarza2023-06-08T14:21:49Z CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures Dynamical systems Ergodicity Fractal Gibbs measures Dimension conservation for almost every projection has been well-established by the work of Marstrand, Mattila and Hunt and Kaloshin. More recently, Hochman and Shmerkin used CP-chains, a tool first introduced by Furstenberg, to prove all projections preserve dimension of measures on [0,1]2 that are the product of a ×m-invariant and a ×n-invariant measure (for m, n multiplicatively independent). Using these tools, Ferguson, Fraser and Sahlsten extended that conservation result to (×m,×n)-invariant measures that are the push-forward of a Bernoulli scheme under the (m,n)-adic symbolic encoding. Their proof relied on a parametrization of conditional measures which could not be extended beyond the Bernoulli case. In this work, we extend their result from Bernoulli measures to Gibbs measures on any transitive SFT. Rather than attempt a similar parametrization, the proof is achieved by reducing the problem to that of the pointwise convergence of a double ergodic average which is known to hold when the system is exact. © 2016 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v304_n_p227_Almarza http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p227_Almarza
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Dynamical systems
Ergodicity
Fractal
Gibbs measures
spellingShingle Dynamical systems
Ergodicity
Fractal
Gibbs measures
CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures
topic_facet Dynamical systems
Ergodicity
Fractal
Gibbs measures
description Dimension conservation for almost every projection has been well-established by the work of Marstrand, Mattila and Hunt and Kaloshin. More recently, Hochman and Shmerkin used CP-chains, a tool first introduced by Furstenberg, to prove all projections preserve dimension of measures on [0,1]2 that are the product of a ×m-invariant and a ×n-invariant measure (for m, n multiplicatively independent). Using these tools, Ferguson, Fraser and Sahlsten extended that conservation result to (×m,×n)-invariant measures that are the push-forward of a Bernoulli scheme under the (m,n)-adic symbolic encoding. Their proof relied on a parametrization of conditional measures which could not be extended beyond the Bernoulli case. In this work, we extend their result from Bernoulli measures to Gibbs measures on any transitive SFT. Rather than attempt a similar parametrization, the proof is achieved by reducing the problem to that of the pointwise convergence of a double ergodic average which is known to hold when the system is exact. © 2016
title CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures
title_short CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures
title_full CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures
title_fullStr CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures
title_full_unstemmed CP-chains and dimension preservation for projections of (×m,×n)-invariant Gibbs measures
title_sort cp-chains and dimension preservation for projections of (×m,×n)-invariant gibbs measures
publishDate 2017
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v304_n_p227_Almarza
http://hdl.handle.net/20.500.12110/paper_00018708_v304_n_p227_Almarza
_version_ 1768544435290767360

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