We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these. §0 Introduction Let (X,B,m, T ) be an invertible, ergodic measure preserving transformation of a σ-finite measure space, then there are no other σ-finite, m-abso...

As a cornerstone of functional analysis, Hahn–Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications o...

In this note we intend to present basic ergodic theory. We begin with the notion of a measure preserving transformation. We then define ergodicity and provide examples. Finally we sketch Birkhoff’s Ergodic Theorem and elucidate it with some examples.

Using the Perron-Frobenius operator we establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give an extensive specific example using the tent map.

The multiplicative ergodic theorem is valid under an integrability condition on the linearized ow with respect to an invariant measure. We investigate the case were the ow is generated by a stochastic diierential equation and give a criterion in terms of the vector elds and the (generally non-adapted) invariant measure assuring the validity of the integrability condition.

We introduce a stochastic business cycle model and study the underlying stochastic Hopf bifurcations with respect to probability densities at different parameter values. Our analysis is based on the calculate of the largest Lyapunov exponent via multiplicative ergodic theorem and the theory of boundary analysis for quasi-non-integrable Hamiltonian systems. Some numerical simulations of the mode...

We introduce the concepts of rotation numbers and rotation vectors for billiard maps. Our approach is based on the birkhoff ergodic theorem. We anticipate that it will be useful, in particular, for the purpose of establishing the non-ergodicity of billiard in certain domains.

Ergodic theory studies the long-term averaging properties of measurepreserving dynamical systems. In this paper, we state and present a proof of the ergodic theorem due to George Birkhoff, who observed the asymptotic equivalence of the time-average and space-average of a point x in a finite measure space. Then, we examine a number of applications of this theorem in number-theoretic problems, in...

This paper will explore the basics of discrete-time Markov chains used to prove the Ergodic Theorem. Definitions and basic theorems will allow us to prove the Ergodic Theorem without any prior knowledge of Markov chains, although some knowledge about Markov chains will allow the reader better insight about the intuitions behind the provided theorems. Even for those familiar with Markov chains, ...