Inducing schemes for multi-dimensional piecewise expanding maps

<p style='text-indent:20px;'>We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity. As an application we check the conditions for the first return map of a class of multi-dimensional non-Markov, non-conformal intermittent maps.</p>

Induction of the transformation rules of the natural language problem statement into a semantic model for generating solver

The author considers a problem of automatic synthesis (induction) of the rules for transforming the natural language formulation of the problem into a semantic model of the problem. According to this model a program that solves this problem can be generated. The&nbsp; problem is considered in relation to the system of generation, recognition and transformation of programs PGEN ++. Based on the analysis of literary sources, a combined approach was chosen to solve this problem, within which the rules for transforming the natural language formulation into a semantic model of the problem are generated automatically, and the specifications of the generating classes and the rules for generating a program from the model are written manually by a specialist in a specific subject area. Within the framework of object-event models, for the first time, a mechanism for the automatic generation of recognizing scripts and related entities (CSV tables, XPath functions) was proposed. Generation is based on the analysis of the training sample, which includes sentences describing objects in the subject area, in combination with instances of such objects. The analysis is performed by searching for unique keywords and characteristic grammatical relationships, followed by the application of simple eliminative-inducing schemes. A mechanism for the automatic generation of rules for replenishing / completing the primary recognized models to full meaning ones is also proposed. Such generation is performed by analyzing the relations between the objects of the training sample, taking into account information from the specifications of the classes of the subject area. The proposed schemes have been tested on the subject area "Simple vector data processing", the successful transformation of natural language statements (both included in the training set and modified) into semantic models with the subsequent generation of programs solving the assigned tasks is shown.

Lifting measures to inducing schemes

In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y. Pesin and S. Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J.5(3) (2005), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many known examples—any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H. Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys.168(3) (1995), 571–580] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multi-modal maps) and for some multi-dimensional maps.