Almost everywhere convergence of entangled ergodic averages

We study pointwise convergence of entangled averages of the form $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f,$$ where $f\in L^2(X,\mu)$, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$, and the $T_i$ are ergodic measure preserving transformations on the standard probability space $(X,\mu)$. We show that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for all $f\in L^2$. We also present results for the general $L^p$ case ($1\leq p<\infty$) and for polynomial powers, in addition to continuous versions concerning ergodic flows.Comment: 16 pages, to appear in Integral Equations and Operator Theor

The Single Euro Payments Area

The objective of the pan-European regulatory and self-regulatory work related to the creation of the Single Euro Payments Area (SEPA) is to improve the efficiency of low-value payments (i.e. retail payments) by establishing a common legal framework and the application of standards. Among the elements of the SEPA project, our focus will be on schemes, in particular the SEPA Credit Transfer Scheme (SCT) already essentially implemented in the EU and Hungary, which we will discuss in detail, stressing the fact that SEPA is not a concrete payments system, but rather a set of procedural and legal rules regulating payment methods, in addition to the entire infrastructure supporting the clearing and settlement of transactions. In the longer term, the broad implementation of the schemes can potentially lead to a substantial reduction in costs for banks, which may ultimately be passed on to customers. We will present the practical experience gained in connection with the Credit Transfer Scheme, in light of the information compiled in the course of the central bank’s informative meetings conducted at the affected Hungarian banks. Currently, 12 banks now apply the Credit Transfer Scheme in Hungary. Due to the lack of a domestic euro infrastructure, domestic euro payments are carried out through international channels. Consequently, a much larger portion of euro transfers made by Hungarian banks have already shifted to the new SEPA format than what has been characteristic of retail payments within the euro area thus far. This phenomenon stems from the fact that initially, the SEPA Credit Transfer Scheme offered a competitive alternative most simply from the perspective of cross-border transaction fees.SEPA, SEPA payment schemes, SEPA payment methods, SEPA Credit Transfer Scheme, Payment Services Directive, PSD.