Local behavior of smooth functions for the energy Laplacian on the Sierpinski gasket

Strichartz, Robert S.; Tse, Shu Tong

doi:10.1524/anly.2010.1038

Cited by 8 publications

(14 citation statements)

References 18 publications

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“…In respect of RCUK policies on publicly-funded research data, the author notes that no research data were generated in the course of this research. In this appendix we will show that in the case where s = 2 both the pressure and the equilibrium state admit simple closed-form expressions, the latter in terms of the Kusuoka measures defined by S. Kusuoka [24] which have been the subject of recent research [2,20,41].…”

Section: Acknowledgementsmentioning

confidence: 99%

Ergodic properties of matrix equilibrium states

Morris¹

2017

Ergod. Th. Dynam. Sys.

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Abstract. Given a finite irreducible set of real d × d matrices A 1 , . . . , A M and a real parameter s > 0, there exists a unique shift-invariant equilibrium state on {1, . . . , M } N associated to (A 1 , . . . , A M , s). In this article we characterise the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterise when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by A 1 , . . . , A M , and when it is a Bernoulli measure. We also give a general sufficient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by S. Kusuoka are explored in an appendix.

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Section: Acknowledgementsmentioning

confidence: 99%

Ergodic properties of matrix equilibrium states

Morris¹

2017

Ergod. Th. Dynam. Sys.

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“…Thus, the assertion of Theorem 5.6 is still true when we perturb the harmonic structure (D, r) slightly. (iii) With regard to the Radon-Nikodym derivative dν h1 /dν h2 , some numerical computation has been carried out in [18].…”

mentioning

confidence: 99%

“…(iii) With regard to the Radon-Nikodym derivative dν h1 /dν h2 , some numerical computation has been carried out in [18].…”

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confidence: 99%

Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

Hino

2009

Proceedings of the London Mathematical Society

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We introduce the concept of index for regular Dirichlet forms by means of energy measures, and discuss its properties. In particular, it is proved that the index of strong local regular Dirichlet forms is identical with the martingale dimension of the associated diffusion processes. As an application, a class of self-similar fractals is taken up as an underlying space. We prove that first-order derivatives can be defined for functions in the domain of the Dirichlet forms and their total energies are represented as the square integrals of the derivatives.

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“…We can thus view abstract the Kusuoka measure as a natural generalisation of the Bernoulli measure, the difference being that we "multiply matrices instead of numbers". The second example is a brief discussion of a well-studied case, that of the Sierpiński gasket, extensively studied in [1], [2], [15], [16], and in many other works.…”

Section: More Results and The Structure Of The Papermentioning

confidence: 99%

Ergodic theory of Kusuoka measures

Johansson

Öberg²,

Pollicott³

2017

J. Fractal Geom.

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In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role (cf.[13], [7], [2]). Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the Hölder continuous functions, as well as the highly discontinuous g-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpiński gaskets.

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Local behavior of smooth functions for the energy Laplacian on the Sierpinski gasket

Cited by 8 publications

References 18 publications

Ergodic properties of matrix equilibrium states

Ergodic properties of matrix equilibrium states

Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

Ergodic theory of Kusuoka measures

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