| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 29, 2014 |
| Latest Amendment Date: | July 29, 2014 |
| Award Number: | 1362838 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Edward Taylor
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2014 |
| End Date: | July 31, 2017 (Estimated) |
| Total Intended Award Amount: | $120,000.00 |
| Total Awarded Amount to Date: | $120,000.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
4300 MARTIN LUTHER KING BLVD HOUSTON TX US 77204-3067 (713)743-5773 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
4800 Calhoun Boulevard Houston TX US 77204-2015 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
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| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
It is natural to ask how much our knowledge of the present tells us about the future. If we study some system that evolves in time according to known rules, what predictions can we make based on observing the present state of the system? It is by now well-understood that even relatively simple systems can display chaotic behavior, in which our ability to make exact predictions decays quite quickly as we look further ahead. In this case one may hope to treat the system as a random process and make statistical predictions using tools from probability theory. This approach has been successfully carried out for uniformly hyperbolic systems, the most strongly chaotic. However, most physically realistic examples fall outside of this class, including important models from meteorology (the Lorenz system), population dynamics (the logistic map), and others. This motivates the study of non-uniformly hyperbolic systems, where the present state of knowledge is much less complete. There has been progress towards understanding certain classes of non-uniformly hyperbolic systems, but many open problems remain, both for systems that have been studied and for broader classes of systems. This research project will give new results for several important classes of non-uniformly hyperbolic systems, and is an important step in developing a more complete understanding of physically relevant systems displaying chaotic behavior.
Key elements of the uniformly hyperbolic theory include existence and uniqueness results for equilibrium states and SRB measures, together with statistical properties for these measures, such as the central limit theorem governing long-term fluctuations of observations around an expected average, and large deviations principles describing the probability of outcomes far from that average. Some of these results, but not all, are known for classes of one-dimensional maps and their perturbations (strongly dissipative maps), partially hyperbolic systems, and geodesic flows. The PI will extend these results to weakly dissipative maps and more general geodesic flows, and will strengthen existing results in all three categories. The key innovation making these extensions possible is the introduction by the PI and his co-authors of new tools for non-uniform hyperbolicity: a notion of "effective hyperbolicity" for the construction of SRB measures, and a notion of "thermodynamic specification" for uniqueness and statistical properties of equilibrium states. These tools have already yielded a number of new results and have clear applicability to broader classes of systems.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The project has led to significant progress in tools, techniques, and applications for studying systems with hyperbolic (chaotic) behavior via their equilibrium states and SRB measures. This progress can largely be classified into three directions.
First, the machinery of "thermodynamic specification" introduced by the PI and Dan Thompson has been extended to its most general form and has been applied to several important classes of examples, including partially hyperbolic systems and geodesic flows in nonpositive curvature; these are the first general results on existence and uniqueness of equilibrium states for these classes of systems. Recent work by the PI has established qualitative decay of correlations for these equilibrium states; further work is required to obtain quantitative estimates. The project has also led to progress on other "weak versions" of the specification property: the PI and Ronnie Pavlov have found a surprising threshold between uniqueness and non-uniqueness associated to the "almost specification" property, and the PI and Van Cyr have found a related threshold that separates systems where equilibrium states may have zero entropy from those for which they must have positive entropy.
Second, the machinery of "effective hyperbolicity" introduced by the PI, Dmitry Dolgopyat, and Yakov Pesin has been used to establish existence of SRB measures for new classes of systems. SRB measures are a specific example of the more general class of equilibrium states, and an extension of this result to general equilibrium states is the core of recent and ongoing work by the PI, Pesin, and Agnieszka Zelerowicz.
Third, the project has begun to clarify the relationship between these techniques and other approaches to non-uniformly hyperbolic systems, including Pesin theory and Young towers. The PI has shown that for symbolic models of dynamical systems, the thermodynamic specification property can be used to construct a Young tower and obtain strong statistical properties. He has also obtained related results in the non-symbolic setting, in work with Stefano Luzzatto and Pesin.
The project's support has also enabled the PI to remain active in dissemination of the research results via conference presentations, to develop young mathematicians by mentoring high school, undergraduate, and graduate students, and to enable beginning researchers to learn fundamental ideas in dynamical systems theory by his role as one of the principal organizers for the Houston Summer School in Dynamical Systems in 2015, 2016, and 2017.
Last Modified: 09/08/2017
Modified by: Vaughn Climenhaga
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