scholarly journals On The Minimal Number of Elements of Markov Partitions for Pseudo-Anosov Homeomorpisms

2015 ◽  
Vol 7 (4) ◽  
pp. 157
Author(s):  
Alexey Zhirov
Keyword(s):  
Orientable Surface ◽  
Markov Partition ◽  
Markov Partitions ◽  
Invariant Foliation

In the paper an estimation of the minimal number of elements of Markov partition for generalized pseudo-Anosov homeomorphism of closed non necessary orientable surface is given. It is formulated in terms of characteristic of invariant foliation of generalized pseudo-Anosov homeomorphism.

scholarly journals Symbolic dynamics for non-uniformly hyperbolic systems

2020 ◽  
pp. 1-68
Author(s):  
YURI LIMA
Keyword(s):  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Markov Partition ◽  
Equilibrium Measures ◽  
Markov Partitions ◽  
Recent Advances ◽  
Closed Orbits ◽  
Symbolic Extension ◽  
Uniformly Hyperbolic Systems

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

scholarly journals Generating special Markov partitions for hyperbolic toral automorphisms using fractals

1986 ◽  
Vol 6 (3) ◽  
pp. 325-333 ◽  
Author(s):  
Tim Bedford
Keyword(s):  
Transition Matrix ◽  
Markov Partition ◽  
Natural Conditions ◽  
Markov Partitions ◽  
Toral Automorphisms

AbstractWe show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.

scholarly journals SELF-SIMILAR INVERSE SEMIGROUPS AND SMALE SPACES

2006 ◽  
Vol 16 (05) ◽  
pp. 849-874 ◽  
Author(s):  
VOLODYMYR NEKRASHEVYCH
Keyword(s):  
Markov Partition ◽  
Inverse Semigroups ◽  
Automata Theory ◽  
Markov Partitions ◽  
Self Similar ◽  
Smale Space ◽  
Smale Spaces

Self-similar inverse semigroups are defined using automata theory. Adjacency semigroups of s-resolved Markov partitions of Smale spaces are introduced. It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup. The notions of the limit solenoid and a contracting semigroup is described.

scholarly journals On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces

2021 ◽  
pp. 1-26
Author(s):  
GIOVANNI FORNI
Keyword(s):  
Banach Spaces ◽  
Open Interval ◽  
Orientable Surface ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Invariant Foliation ◽  
Higher Genus ◽  
Compact Orientable Surface ◽  
Compact Surfaces

Abstract We prove that the asymptotics of ergodic integrals along an invariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, is determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [Parabolic dynamics and anisotropic Banach spaces. J. Eur. Math. Soc. (JEMS)21(9) (2019), 2793–2858] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1, e^{h_{\mathrm {top}}})$ .

scholarly journals DOUBLES OF KLEIN SURFACES

2012 ◽  
Vol 54 (3) ◽  
pp. 507-515
Author(s):  
ANTONIO F. COSTA ◽  
WENDY HALL ◽  
DAVID SINGERMAN
Keyword(s):  
Klein Bottle ◽  
Orientable Surface ◽  
Historical Note ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Algebraic Functions ◽  
Genus 2

Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

scholarly journals Cyclic bi-embeddings of Steiner triple systems on 31 points

2001 ◽  
Vol 43 (1) ◽  
pp. 145-151 ◽  
Author(s):  
G. K. Bennett ◽  
M. J. Grannell ◽  
T. S. Griggs
Keyword(s):  
Orientable Surface ◽  
Steiner Triple Systems ◽  
Difference Problem ◽  
Triple Systems ◽  
Relationship Of ◽  
The Relationship

We investigate cyclic bi-embeddings in an orientable surface of Steiner triple systems on 31 points. Up to isomorphism, we show that there are precisely 2408 such embeddings. The relationship of these to solutions of Heffter's first difference problem is discussed and a procedure described which, under certain conditions, transforms one bi-embedding to another.

Construction of Markov Partitions in PL1D Maps

2013 ◽  
Vol 60 (10) ◽  
pp. 702-706 ◽  
Author(s):  
Toni Draganov Stojanovski ◽  
Ljupco Kocarev
Keyword(s):  
Markov Partitions

scholarly journals Genus Ranges of Chord Diagrams

2015 ◽  
Vol 24 (04) ◽  
pp. 1550022 ◽  
Author(s):  
Jonathan Burns ◽  
Nataša Jonoska ◽  
Masahico Saito
Keyword(s):  
Outer Boundary ◽  
Orientable Surface ◽  
Fixed Number ◽  
Chord Diagram ◽  
Boundary Circle ◽  
Line Segments ◽  
Chord Diagrams ◽  
Computer Calculations

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can be, and those that cannot be, realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.

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