Pointwise Ergodic Theory: Examples and Entropy

A view on the physical meaning of the so called ergodic hypothesis: its role on the foundations of equilibrium statistical mechanics in mid '800, its interpretations and hints at its relevance for modern nonequilibrium statistical mechanics. Followed by appendices with detailed comments on the original papers.

We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},\nu,\tau)$ and any $f\in L^p(\nu)$, $p>1$, the limit $$ \lim_{n\to \infty}{1\over \sum_{k=1}^{n} d(k)} \sum_{k=1}^{n} d(k)f(\tau^k x)$$ exists $\nu$-almost everywhere. We also obtain similar results for other arithmetical functions, like $\theta(n)$ function counting ...

Let ${\bf X}=(X, \Sigma, m, \tau)$ be a dynamical system. We prove that the bilinear series $\sideset{}{'}\sum_{n=-N}^{N}\frac{f(\tau^nx)g(\tau^{-n}x)}{n}$ converges almost everywhere for each $f,g\in L^{\infty}(X).$ We also give a proof along the same lines of Bourgain's analog result for averages.

In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples: the Stein continuity principle, Bourgain's entropy criteria and Kakutani-Rochlin lemma, most classical device for these questions in ergodic theory. First, we state a $L^1$-version of the continuity principle and give an example of its usefulness by applying it to some famous problem on divergence almost everywhere of Fourier series. Ne...

This paper has been withdrawn by the author due to an error.

This is a survey on Sarnak's Conjecture

The principal results proved in this expository thesis are the IP polynomial Szemer\'edi theorem for nilpotent groups and the multiple term return times theorem with nilsequence weights. It also contains extensions of the convergence theorem for nilpotent polynomial multiple ergodic averages and the return times theorem to locally compact second countable amenable groups.

We consider several fundamental properties of grand variable exponent Lebesgue spaces. Moreover, we discuss Ergodic theorems in these spaces whenever the exponent is invariant under the transformation.