D. Ruelle, Thermodynamic Formalism, (Addison-Wesley, Reading, MA, 1978). Home/Search Document Not in Database Summary Related Articles Check |
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The Manhattan Curve And The Correlation - Of Length Spectra (Correct)
....for valuable discussions on the thermodynamic approach to the Manhattan curve. 1. Thermodynamic Formalism In this section we briefly review the thermodynamic formalism associated to the geodesic flow over a hyperbolic surface. A reference for this material is given by [7] see also [2] [8]) Our motivation for doing this is so that we can write 1 , # 2 ) as the graph of a certain function defined in terms of pressure. Let # denote the surface H # 1 and set M = T 1 #, the unit tangent bundle. The geodesic flow # t : M M is defined by # t (x, v) #(t) #(t) where #(t) ....
D. Ruelle, Thermodynamic formalism, Addison Wesley, Reading, Mass., 1978.
LARGE DEVIATIONS FOR MAPS WITH INDIFFERENT FIXED POINTS.. - Manchester University.. (Correct)
.... (Lemmas 2 4) We let I denote the disjoint union of the two intervals [0, a 0 ] and [a 0 , 1] Given a continuous function f : I R the pressure P (f) is defined by P (f) sup hm (T ) fdm : m is a T invariant probability measure where hm (T ) denotes the entropy of T with respect to m [13], 17] Definition. Given any g (I) we define a transfer operator g : C (I, R) I, R) by Ty=x g(y) h(y) In the special case g = we shall write L . Using the notation g (y) g(y) g(Ty) g(T y) we have that h(y) In particular, h(x) ....
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, New York, 1978.
The Circle Problem on Surfaces of Variable Negative Curvature - Pollicott, Sharp (1997) (Correct)
.... of # [6] Earlier, Lax and Phillips had obtained a more refined result with rather precise error terms under the restriction that # n 2 [3] Subsequently, Lalley [1] recovered Patterson s result in the case n = 1 using methods based on symbolic dynamics [10] and thermodynamic ergodic theory [9]. Since the case where M is compact is covered by the result of Margulis, we shall suppose that M has at least one infinite volume end. This introduces the desirable (but not essential) simplication that then # is a free group. It is worthwhile to note that if the base point x X is chosen so ....
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, New York, 1978.
Orthonormal Expansions of Invariant Densities for Expanding.. - Jenkinson, Pollicott (2002) (Correct)
.... both determinants of trace class transfer operators acting on a suitable Hilbert space of analytic functions (see [Sim] for the general theory of determinants associated to certain Hilbert space operators) For any C function f : I R, the corresponding transfer operator L f is de ned (cf. [Ru2]) by L f v(x) Ty=x v(y)e f(y) As is well known, this operator preserves various function spaces. In particular we will consider such operators as acting on the Hilbert Bergman space A 2 (U) corresponding to appropriate complex domains U . Recall (see for example [HKZ] that A 2 (U) is the ....
.... the only point of accumulation, and all non zero eigenvalues of nite algebraic multiplicity) The leading (i.e. largest in modulus) eigenvalue of any transfer operator L f is well known to be simple, and equal to e P (f) where P (f) P (f; T ) lim n 1 is the pressure of f (cf. [Ru2]) The leading eigenvalue of the Perron Frobenius operator L equals 1, since it is easily checked that P ( log jT j) 0. The corresponding eigenfunction is precisely the invariant density (cf. CE] Our motivation for introducing the functions g (z; t) is the following result (see [JP] ....
D. Ruelle, Thermodynamic Formalism, Wiley, New York, 1978.
Comparison Theorems And Orbit Counting In Hyperbolic Geometry - Pollicott, Sharp (1998) (Correct)
....matrix M n takes the form M n = # # # M 21 P 22 . 0 . M l k 1 M l k 2 . P l k l k # # . The isolated eigenvalues in the spectra of the operators L fn and L f # n ) acting on C and C (XB ) are precisely the eigenvalues of the associated matrices M n and P n [18]. We first show that the eigenvalues of the matrices A n and B n must co incide. Since the eigenvalues can be determined from the asymptotics of the values trace (P n ) trace (P 11 ) trace P l k l k = trace (M n ) as m # we immediately see that the eigenvalues of the ....
D. Ruelle, Thermodynamic Formalism, Addison Wesley, New York, 1978.
Local Limit Theorems For Free Groups - Sharp (2001) (1 citation) (Correct)
....1 e x dx, where denotes the Euclidean norm and where 1 k # 2k #1 2 (0. 1) 18] In fact, this result is similar in spirit to earlier results for subshifts of finite type, hyperbolic di#eomorphisms, and interval maps [1] 4] 5] 10] 12] 17] 19] [20], 23] Here, we shall establish a more precise local limit theorem. First we note a combinatorial restriction. We shall say that # = # 1 , # k ) is even if # 1 # k is even, and odd otherwise. It is clear that if [g] # then # has the same parity as g . Thus, in particular, ....
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, Mass., 1978.
Equilibrium Phase Transitions in Coupled Map - Lattices Pedestrian Approach (Correct)
....the ground state energy. Since the system is one dimensional and has short range interaction no phase transition occurs, i.e. the mean values depend smoothly on the parameters of the system. All the previous considerations are special cases of a much more general mathematical approach (cf. e.g. [14]) The important point which I want to stress here is 5 that these advanced formalisms become simple for piecewise linear maps. The latter are defined to some extent in a geometric way. Fixing partitions fU oe g and fU oe;oe g one defines the map f : U oe;oe U oe in such a way that it acts ....
D. Ruelle, Thermodynamic formalism, Addison--Wesley, Reading, 1978. 12
Exponential Error Terms For Growth Functions On Negatively.. - Pollicott, Sharp (1998) (1 citation) (Correct)
.... in the half plane Re(s) # 1 and thus is a non zero analytic domain for #(s) This allows us to deduce that there exists a constant C 5 0 such that for Re(s) # 2 : h #1 we have # # # # (s) # # C 5 max t , Furthermore, since P ( hr) 0 and # P ( #r) is continuous [17] we can assume without loss of generality that P ( # 1 r) 1 by choosing # 1 su#ciently close to h. 10 To complete the proof we need to show that we can choose the exponent of to be strictly less than 1. To do this we make use of the well known PhragmenLindel of Theorem [22, 5.65] We know ....
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, New York, 1978.
LINEAR ACTIONS OF FREE GROUPS Mark Pollicott and Richard.. - University Of Manchester (Correct)
....is also given by the equivalent variational identity P (f) sup h(#) fd# : # is an invariant probability . If f : R is Holder continuous, then the above supremum is attained at a unique probability measure called the equilibrium state for f . Proposition 4 (Ruelle Operator Theorem [10], 7] R, the spectral radius of the operator L sr is equal to e and this is a simple eigenvalue of strictly maximal modulus. Furthermore, associated to this eigenvalue, there is a strictly positive eigenfunction h s , and an eigenmeasure # s . We adopt the normalization # s (1) 1 and #(h ....
D. Ruelle, Thermodynamic Formalism, Addison Wesley, Redding, Mass., 1978.
Error Terms for Closed Orbits of Hyperbolic Flows - Pollicott, Sharp (Correct)
..... 2.6) Define the curve A(t) h t # and consider those s for which Re(s) A(t) t 1 . The map t P ( tr) is analytic with nowhere vanishing first derivative and, in particular, we can write = 1 P # ( hr) Re(s) h) O( Re(s) h) where P # ( hr) 0 [12], 8] Thus e P ( Re(s)r) if # # and t 2 , for some t 2 t 1 . This causes no loss of generality since increasing # simply corresponds to a narrowing of the region R(#) Hence in this region we have n Z 2 1 2N 1 2N ....
....see that whenever y B n (x) y : d(x, y) where n is chosen such that # # , we have w(y) # L w(x) C 0 1) Im(s) # (C 0 1) Im(s) # (4. 4) for n 0 and w # Im(s) Furthermore, from a standard characterization of equilibrium measures [12], there exists D 0 (independent of n) B n (x) nD . 4.5) Thus we have from (4.4) and (4.5) that Bn (x) c L w d Bn (x) L (B n (x) B n (x) 4.7) su#ciently large. Thus comparing (4.1) and (4.7) we see that ....
D. Ruelle, Thermodynamic Formalism, Addison-Wesley, New York, 1978.
What are SRB measures, and which dynamical systems have them? - Young (1 citation) Self-citation (Ruelle) (Correct)
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D. Ruelle, Thermodynamic Formalism, Addison Wesley, Reading, MA, 1978.
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D. Ruelle, Thermodynamic Formalism, (Addison-Wesley, Reading, MA, 1978).
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Ruelle, D. (1978). Thermodynamic Formalism. Addison Wesley, New York. 14 -2 -1 0 1 2 d=2, |true marginal - belief|
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Ruelle, D. (1978) Thermodynamic Formalism, Addison-Wesley, Reading.
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D. Ruelle, Statistical Mechanics, Thermodynamic Formalism, Addison-Wesley, Reading MA, 1978.
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D. Ruelle, Thermodynamic Formalism, Addison-Wesley, New York, 1978. Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL
INVARIANCE PRINCIPLES FOR INTERVAL MAPS WITH AN.. - University Of Manchester (Correct)
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D. Ruelle, Thermodynamic Formalism, Addison-Wesley, New York, 1978.
Uniqueness of Invariant Measures for Place-Dependent Random.. - Stenflo (Correct)
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D. Ruelle. Thermodynamic formalism. Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota.
Invariant Measures and Their Properties. A Functional Analytic.. - Liverani (2002) (Correct)
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D.Ruelle, Thermodynamics formalism, Addison-Wesley, New York, 1978.
Asymptotic Expansions For Closed Orbits In Homology Classes - Pollicott, Sharp (Correct)
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D. Ruelle, Thermodynamic formalism, Addison-Wesley, New York, 1978.
On Transfer Operators for Continued Fractions with.. - Jenkinson, Gonzalez.. (2002) (Correct)
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D. Ruelle, Thermodynamic Formalism, Wiley, New York, 1978.
Dynamical Sources in Information Theory: A General Analysis of .. - Clement, al. (1999) (1 citation) (Correct)
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Ruelle, D. Thermodynamic formalism, Addison Wesley (1978)
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) D. Ruelle, Statistical Mechanics, Thermodynamic Formalism (Addison-Wesley, Reading MA, 1978)
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