A ratio ergodic theorem on Borel actions of Zd and Rd
| dc.contributor.author | Holt, Eric Norman, author | |
| dc.contributor.author | Rudolph, Daniel, advisor | |
| dc.date.accessioned | 2024-03-13T19:53:51Z | |
| dc.date.available | 2024-03-13T19:53:51Z | |
| dc.date.issued | 2009 | |
| dc.description.abstract | We prove a ratio ergodic theorem for free Borel actions of Zd and Rd on a standard Borel probability space. The proof employs an extension of the Besicovitch Covering Lemma, as well as a notion of coarse dimension that originates in an upcoming paper of Hochman. Due to possible singularity of the measure, we cannot use functional analytic arguments and therefore diffuse the measure onto the orbits of the action. This diffused measure is denoted μx, and our averages are of the form 1/μx(Bn) ∫ Bn f o T-v(x)dμx. A Følner condition on the orbits of the action is shown, which is the main tool used in the proof of the ergodic theorem. Also, an extension of a known example of divergence of a ratio average is presented for which the action is both conservative and free. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | ETDF_Holt_2009_3385158.pdf | |
| dc.identifier.uri | https://hdl.handle.net/10217/237785 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.rights.license | Per the terms of a contractual agreement, all use of this item is limited to the non-commercial use of Colorado State University and its authorized users. | |
| dc.subject | Besicovitch Covering Lemma | |
| dc.subject | Borel actions | |
| dc.subject | coarse dimension | |
| dc.subject | ergodic theorem | |
| dc.subject | Hochman | |
| dc.subject | mathematics | |
| dc.title | A ratio ergodic theorem on Borel actions of Zd and Rd | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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