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Register a new userConjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces
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Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to stationary measures on the sets and may be used to carry well known flows and spectral analysis over from familiar settings to new ones. The focus of this work is on a number of surprising applications including (i) what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, being discontinuous at a countable dense set of points, yet have very good approximation properties; (ii) Lebesgue measure-preserving flows, on polygonal laminas, whose wave-fronts are fractals. The key idea is to exploit fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems. Some of the examples relate to work of Oxtoby and Ulam concerning ergodic flows on regions bounded by polygons.
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By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesins entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedlands entropy for certain $C^{2}$ $mathbb{N}^2$-actions.
For a large class of irreducible shift spaces $XsubsettA^{Z^d}$, with $tA$ a finite alphabet, and for absolutely summable potentials $Phi$, we prove that equilibrium measures for $Phi$ are weak Gibbs measures. In particular, for $d=1$, the result holds for irreducible sofic shifts.
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In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.
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