TY - GEN
T1 - Cookie-cutter-like sets with graph-directed construction
AU - Fan, Shen
AU - Liu, Qing Hui
AU - Wen, Zhi Ying
N1 - Publisher Copyright:
© Springer Science+Business Media New York 2013.
PY - 2013
Y1 - 2013
N2 - In this chapter, we extend the cookie-cutter-like construction introduced by Ma, Rao, and Wen to the case having the graph-directed construction which is introduced by Mauldin and Williams and obtain a new class of fractals, which can be used to study the dimensions of the spectrum of discrete Schrödinger operators. Under suitable assumptions we prove that this class of fractals possesses the properties of bounded variation, bounded distortion, bounded covariation, and the existence of Gibbs-like measures. With these properties we give expressions for the Hausdorff dimensions, box dimensions, and packing dimensions of the fractals. We also discuss the continuous dependence of the dimensions on the defining data.
AB - In this chapter, we extend the cookie-cutter-like construction introduced by Ma, Rao, and Wen to the case having the graph-directed construction which is introduced by Mauldin and Williams and obtain a new class of fractals, which can be used to study the dimensions of the spectrum of discrete Schrödinger operators. Under suitable assumptions we prove that this class of fractals possesses the properties of bounded variation, bounded distortion, bounded covariation, and the existence of Gibbs-like measures. With these properties we give expressions for the Hausdorff dimensions, box dimensions, and packing dimensions of the fractals. We also discuss the continuous dependence of the dimensions on the defining data.
UR - http://www.scopus.com/inward/record.url?scp=84975760917&partnerID=8YFLogxK
U2 - 10.1007/978-0-8176-8400-6_12
DO - 10.1007/978-0-8176-8400-6_12
M3 - Conference contribution
AN - SCOPUS:84975760917
SN - 9780817683993
T3 - Trends in Mathematics
SP - 235
EP - 254
BT - Further Developments in Fractals and Related Fields
A2 - Barral, Julien
A2 - Seuret, Stéphane
PB - Springer International Publishing
T2 - International Conference on Fractals and Related Fields, 2011
Y2 - 1 June 2011
ER -