A dynamical system with oscillating time average
(2020) In Master's Theses in Mathematical Sciences FMAM05 20201Mathematics (Faculty of Engineering)
- Abstract
- Given a discrete dynamical system T, one can ask what the time average of the system will be, that is, what is the average position of T^n(x) for large n? Birkhoff’s ergodic theorem, one of the most important results in ergodic theory, says that for an ergodic system on a finite measure space, the time average will in the limit as n -> infinity be equal to the space average for almost all initial values x. In this thesis we study time averages of a dynamical system T : [0, 1] -> [0, 1] that depends on a parameter alpha. We show that there are values of alpha for which the points x, T(x), T^2(x), ... are equally often in the right half of [0, 1] as on the left half. We also show that for other values of alpha, the time average never... (More)
- Given a discrete dynamical system T, one can ask what the time average of the system will be, that is, what is the average position of T^n(x) for large n? Birkhoff’s ergodic theorem, one of the most important results in ergodic theory, says that for an ergodic system on a finite measure space, the time average will in the limit as n -> infinity be equal to the space average for almost all initial values x. In this thesis we study time averages of a dynamical system T : [0, 1] -> [0, 1] that depends on a parameter alpha. We show that there are values of alpha for which the points x, T(x), T^2(x), ... are equally often in the right half of [0, 1] as on the left half. We also show that for other values of alpha, the time average never converges, but instead oscillates between being concentrated on the left and right halves of the unit interval. In the process, we also prove the existence of an absolutely continuous invariant ergodic measure for T. (Less)
- Popular Abstract
- Dynamical systems is the study of how complex systems evolve in time according to deterministic rules. In this thesis we investigate a connection between a certain dynamical system and the so-called ergodic hypothesis, a more than 100 years old hypothesis from statistical physics.
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9015929
- author
- Nilsson, Joel LU
- supervisor
- organization
- course
- FMAM05 20201
- year
- 2020
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- dynamical system, time average, ergodic theorem, ergodic measure
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUTFMA-3407-2020
- ISSN
- 1404-6342
- other publication id
- 2020:E25
- language
- English
- id
- 9015929
- date added to LUP
- 2020-06-24 13:59:26
- date last changed
- 2020-06-24 13:59:26
@misc{9015929,
abstract = {{Given a discrete dynamical system T, one can ask what the time average of the system will be, that is, what is the average position of T^n(x) for large n? Birkhoff’s ergodic theorem, one of the most important results in ergodic theory, says that for an ergodic system on a finite measure space, the time average will in the limit as n -> infinity be equal to the space average for almost all initial values x. In this thesis we study time averages of a dynamical system T : [0, 1] -> [0, 1] that depends on a parameter alpha. We show that there are values of alpha for which the points x, T(x), T^2(x), ... are equally often in the right half of [0, 1] as on the left half. We also show that for other values of alpha, the time average never converges, but instead oscillates between being concentrated on the left and right halves of the unit interval. In the process, we also prove the existence of an absolutely continuous invariant ergodic measure for T.}},
author = {{Nilsson, Joel}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{A dynamical system with oscillating time average}},
year = {{2020}},
}