Objectives

To develop parameter classes by finite sets of eigenfunctions in which a particular sets of values that has the properties of limits in the sens of restarting points for the next fractal levels
The studies deal with non-self-similar fractals which are built, level by level, from coarser to finer, by a finite set of construction principles, using the finite ramification property. These kind of fractals, introduced by Hambly, in order to model random fractals, as he used a random process to select the parameters in the construction. We intend to study the restriction sets where the boundary points can be fixed points of the mappings and this means to eliminate some mass effect from fractal
The interval of a fractal has only two isometrics, the identity and the flip x->(1-x). It is necessary to develop another local symmetries for particular cases, which means that the inside values can be categorized in sets which are used for an isomorphism between certain parts of fractal
By defining a harmonic function in order to have basic function of fractal continuous, since the harmonic equation only holds in the inside, we can allow the function to be discontinuous at the boundary. On the interval this distinction is meaningless because any linear function defined on the interior will extend continuously to a boundary. In the area of products fractals we intend to study how to construct by taking products of lines and differential calculus on Euclidean space and how these may be based on partial derivatives which are essentially one-dimensional constructions. From these a theory can be developed by taking products of fractals and suitably lifting the energy and Laplacian from the factors to the product because the theory of differential operators on products of fractals is more like to the theory of ordinary differential equations. This way was a different way to be treated by Strichartz
The study will continue with the solvability of differential equations where we need to state the precise form of the Lipschitz condition, which is actually a local Lipschitz condition in the u-variable alone because to take values from Rn become similar with the statement for ordinary differential equations
The last point of the study consists in convergence of ordinary Fourier series. The classical approach leads to have either assume some sort of smoothness for the function, using a summability method or weaken the notion of convergence. On SG it is virtually free; the requirement is that you choose the partial sums in a sensible way. We intend to study the restrictive case integrating a Dirichlet kernel which does not behave like a nice approximate identity, where we can study if can have uniform convergence along a sequence of circular partial sums and to study the case in which it is possible to obtain convergence results on other post critically finite fractals