scholarly journals Manhattan curves for hyperbolic surfaces with cusps

2018 ◽  
Vol 40 (7) ◽  
pp. 1843-1874 ◽  
Author(s):  
LIEN-YUNG KAO
Keyword(s):  
Riemann Surfaces ◽  
Thermodynamic Formalism ◽  
Compact Surface ◽  
Hyperbolic Surfaces ◽  
Rigidity Results ◽  
Markov Shifts ◽  
Countable State ◽  
Geometric Rigidity ◽  
Convex Cocompact ◽  
Rank 2

In this paper, we study an interesting curve, the so-called Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface; in particular, representations corresponding to Riemann surfaces with cusps. Using thermodynamic formalism (for countable state Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Burger [Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not.1993(7) (1993), 217–225] and Sharp [The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z.228(4) (1998), 745–750] for convex cocompact Fuchsian representations.

scholarly journals GOOD POTENTIALS FOR ALMOST ISOMORPHISM OF COUNTABLE STATE MARKOV SHIFTS

2007 ◽  
Vol 07 (01) ◽  
pp. 1-15 ◽  
Author(s):  
MIKE BOYLE ◽  
JEROME BUZZI ◽  
RICARDO GÓMEZ
Keyword(s):  
Equivalence Relation ◽  
Thermodynamic Formalism ◽  
Strong Version ◽  
Markov Shifts ◽  
Countable State

Almost isomorphism is an equivalence relation on countable state Markov shifts which provides a strong version of Borel conjugacy; still, for mixing SPR shifts, entropy is a complete invariant of almost isomorphism [2]. In this paper, we establish a class of potentials on countable state Markov shifts whose thermodynamic formalism is respected by almost isomorphism.

scholarly journals On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings

2009 ◽  
Vol 61 (4) ◽  
pp. 740-761 ◽  
Author(s):  
Pierre-Emmanuel Caprace ◽  
Frédéric Haglund
Keyword(s):  
Abelian Group ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Free Abelian Group ◽  
Gromov Hyperbolic ◽  
Convex Cocompact ◽  
Rank 2 ◽  
Cocompact Subgroup ◽  
Special Case ◽  
Geometric Action

Abstract.Given a complete CAT(0) space X endowed with a geometric action of a group Ⲅ, it is known that if Ⲅ contains a free abelian group of rank n, then X contains a geometric flat of dimension n. We prove the converse of this statement in the special case where X is a convex subcomplex of the CAT(0) realization of a Coxeter group W, and Ⲅ is a subgroup of W. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

Topological Boundaries for Countable State Markov Shifts

1995 ◽  
Vol s3-70 (3) ◽  
pp. 625-643 ◽  
Author(s):  
Doris Fiebig ◽  
Ulf-Rainer Fiebig
Keyword(s):  
Markov Shifts ◽  
Countable State

scholarly journals Thermodynamic formalism for random countable Markov shifts

2008 ◽  
Vol 22 (1/2, September) ◽  
pp. 131-164 ◽  
Author(s):  
Manuel Stadlbauer ◽  
Yuri Kifer ◽  
Manfred Denker
Keyword(s):  
Thermodynamic Formalism ◽  
Markov Shifts ◽  
Countable Markov Shifts

Zeta functions for certain non-hyperbolic systems and topological Markov approximations

1998 ◽  
Vol 18 (6) ◽  
pp. 1589-1612 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Gibbs Measure ◽  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Thermodynamic Formalism ◽  
Zeta Functions ◽  
Product Formula ◽  
Markov Approximation ◽  
Weighted Functions ◽  
Countable State

We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.

Thermodynamic formalism for countable Markov shifts

1999 ◽  
Vol 19 (6) ◽  
pp. 1565-1593 ◽  
Author(s):  
OMRI M. SARIG
Keyword(s):  
Equilibrium Measure ◽  
Thermodynamic Formalism ◽  
Finite Measure ◽  
Complete Characterization ◽  
Topological Pressure ◽  
Recurrent Functions ◽  
Equilibrium Measures ◽  
Positive Recurrence ◽  
Markov Shifts ◽  
Definition Of

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of $\phi =0$ is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let $L_\phi$ denote the Ruelle operator for $\phi$. We offer a definition of positive recurrence for $\phi$ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a $\sigma$-finite measure $\nu $, a continuous function $h>0$ and $\lambda >0$ such that $L_\phi ^{*}\nu =\lambda \nu$, $L_\phi h=\lambda h $ and $\lambda ^{-n}L_\phi ^nf\rightarrow h\int f\,d\nu$ for suitable functions $f$. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form $$ \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), $$ where $f$ is a suitable function on a suitable shift $X$.

Pressure and equilibrium states for countable state markov shifts

2002 ◽  
Vol 131 (1) ◽  
pp. 221-257 ◽  
Author(s):  
Doris Fiebig ◽  
Ulf-Rainer Fiebig ◽  
Michiko Yuri
Keyword(s):  
Equilibrium States ◽  
Markov Shifts ◽  
Countable State
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