Piecewise linear iterated function systems on the line of overlapping construction

Abstract In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

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Exceptional sets for self-similar fractals in Carnot groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004110000083 ◽

2010 ◽

Vol 149 (1) ◽

pp. 147-172 ◽

Cited By ~ 1

Author(s):

ZOLTÁN M. BALOGH ◽

RETO BERGER ◽

ROBERTO MONTI ◽

JEREMY T. TYSON

Keyword(s):

Hausdorff Dimension ◽

Euclidean Space ◽

Iterated Function Systems ◽

Carnot Groups ◽

Exceptional Set ◽

Carathéodory Metric ◽

Exceptional Sets ◽

Iterated Function ◽

Self Similar ◽

Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

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TOPOLOGICAL FULL GROUPS OF -ALGEBRAS ARISING FROM -EXPANSIONS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788714000214 ◽

2014 ◽

Vol 97 (2) ◽

pp. 257-287 ◽

Cited By ~ 1

Author(s):

KENGO MATSUMOTO ◽

HIROKI MATUI

Keyword(s):

Real Line ◽

Piecewise Linear ◽

Unit Interval ◽

Linear Functions ◽

Discrete Groups ◽

Piecewise Linear Functions ◽

Real Numbers ◽

The Real ◽

Theoretical Property ◽

Thompson Groups

AbstractWe introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by $\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the $\beta $-expansion of $1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of $\beta $.

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Infinite iterated function systems with overlaps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.86 ◽

2014 ◽

Vol 36 (3) ◽

pp. 890-907 ◽

Cited By ~ 1

Author(s):

SZE-MAN NGAI ◽

JI-XI TONG

Keyword(s):

Hausdorff Dimension ◽

Iterated Function Systems ◽

Topological Pressure ◽

Separation Condition ◽

Limit Sets ◽

Weak Separation Condition ◽

Iterated Function ◽

Weak Separation ◽

Separation Conditions

We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

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Maximal orders on groups of piecewise linear functions of the real line

Ordered Algebraic Structures ◽

10.1201/9781482283150-8 ◽

2001 ◽

pp. 85-98

Keyword(s):

Real Line ◽

Piecewise Linear ◽

Linear Functions ◽

Piecewise Linear Functions ◽

The Real ◽

Maximal Orders

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Multiplier-less realisation of piecewise linear functions

Proceedings of 1997 IEEE International Symposium on Circuits and Systems. Circuits and Systems in the Information Age ISCAS '97 ◽

10.1109/iscas.1997.612856 ◽

2002 ◽

Author(s):

D.H. Horrocks ◽

S.J. Conder

Keyword(s):

Piecewise Linear ◽

Linear Functions ◽

Piecewise Linear Functions

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Modelling the single-electron transistor with piecewise linear functions

10.1109/ecctd.2009.5274932 ◽

2009 ◽

Cited By ~ 1

Author(s):

Arturo Sarmiento-Reyes ◽

Luis Hernandez-Martinez ◽

Miguel Angel Gutierrez de Anda ◽

Francisco Javier Castro Gonzalez

Keyword(s):

Piecewise Linear ◽

Linear Functions ◽

Single Electron ◽

Single Electron Transistor ◽

Piecewise Linear Functions

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Mesh duality and Legendre duality

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1990.0039 ◽

1990 ◽

Vol 428 (1875) ◽

pp. 351-377 ◽

Cited By ~ 3

Keyword(s):

Piecewise Linear ◽

Voronoi Tessellation ◽

Tangent Plane ◽

Linear Functions ◽

Piecewise Linear Functions ◽

Dual Transformation ◽

Delaunay Mesh ◽

Definition Of

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

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