Anosov Diffeomorphisms and γ -Tilings

We consider a toral Anosov automorphism G γ : T γ → T γ given by G γ (x, y) = (ax+ y, x) in the base, where a∈ N\ { 1 } , γ= 1 / (a+ 1 / (a+ 1 / …)) , v= (γ, 1) and w= (- 1 , γ) in the canonical base of R 2 and T γ = R 2 / (vZ× wZ). We introduce the notion of γ-tilings to prove the existence of a one-to-one correspondence between (i) marked smooth conjugacy classes of Anosov diffeomorphisms, with invariant measures absolutely continuous with respect to the Lebesgue measure, that are in the isotopy class of G γ ; (ii) affine classes of γ-tilings; and (iii) γ-solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences.info:eu-repo/semantics/publishedVersio

Tilings and Anosov diffeomorphisms

A. Pinto and D. Sullivan [4] proved a one-to-one correspondence between: (i) C1+ conjugacy classes of expanding circle maps; (ii) solenoid functions and (iii) Pinto-Sullivan’s dyadic tilings on the real line. A. Pinto [1,3] introduced the notion of golden tilings and proved a one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugate to the linear automorphism G(x; y) = (x + y; x), (ii) affine classes of golden tilings and (iii) solenoid functions. Here we extend this last result and we exhibit a natural one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugate to the linear automorphism G(x; y) = (ax+y; ax), where a 2 N, (ii) affine classes of tilings in the real line and (iii) solenoid functions. The solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences

Golden tilings

In this talk we present the definition of a golden sequence {ri}i2N. These golden sequences have the property of being Fibonacci quasi-periodic and determine a tiling in the real line. We prove a one-to-one correspondence between: (i) affine classes of golden tilings; (ii) smooth conjugacy classes of Anosov difeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugate to the Anosov automorphis

Seller-buyer supply chain: a game theory framework

We present a game theoretical framework to study competition and collaboration scenarios in seller-buyer supply chains, representing a manufacturer that wholesales a product to a retailer who, in turn, retails to the final consumer. We study cooperative and non-cooperative scenarios, by considering in the first case a Stackelberg game under both the situations where one of the players, either the buyer or the seller, plays the leading role by taking the initiative and forcing the strategy of the other player, the follower. In the cooperative game, we analyse scenarios where the pro t of both players are increased when applying joint cooperative strategies. We also compute the Nash equilibria. This is undergoing work.info:eu-repo/semantics/publishedVersio

Metals in the shell of Bathymodiolus azoricus from a hydrothermal vent site on the Mid-Atlantic Ridge

Specimens of the mussel Bathymodiolus azoricus were collected from Menez Gwen, a relatively shallow (850 m) hydrothermal vent field on the Mid-Atlantic Ridge. Each bivalve shell (n = 21) was individually cleaned by selective chemical. The residual crystal matrix of each shell was individually analysed for the concentrations of the minor elements magnesium and strontium and the trace elements iron, manganese, copper and zinc. The chemical composition of the crystal matrix is unusual. B. azoricus is identified as a species having one of the most strontium impoverished shells amongst the marine molluscs. For a bimineral species the magnesium concentration is also extraordinary low. Despite originating from a trace metal rich environment; the metal concentrations in the shells were exceptionally low. Mean concentrations of iron, manganese, copper and zinc were 20.6, 3.7, 0.6 and 9.4 microg g(-1) respectively. Minor and trace element concentrations exhibited a marked intra-population variability. Copper concentrations increased and iron and zinc concentrations decreased with increasing shell weight. Due to its insensitivity to the high environmental levels of trace elements and the variability in intra-population concentrations induced by shell weight the crystal matrix of the shell of B. azoricus has little potential for use in environmental trace metal monitoring in areas contiguous to deep-sea hydrothermal vents.info:eu-repo/semantics/publishedVersio

Golden tilings

A. Pinto and D. Sullivan [3] proved a one-to-one correspondence between: (i) Cl+ conjugacy classes of expanding circle maps; (ii) solenoid functions and (iii) Pinto-Sullivan's dyadic tilings on the real line. Here, we prove a one-to-one correspondence between: (i) golden tilings; (ii) smooth conjugacy classes of golden diffeomorphism of the circle that are fixed points of renormalization; (iii) smooth conjugacy classes of Anosov difeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugated to the Anosov automorphism G(x, y) = (x + y, x) and (iv) solenoid functions

Ladrilhamentos dourados da recta real

Apresentaremos a definição de sucessão dourada {r_i}. Estas sucessões possuem a propriedade de serem Fibonacci quasi-periodicas e determinam um ladrilhamento na recta real. Provaremos uma correspondência bijectiva entre: (i) sucessões douradas; (ii) Classes de conjugação diferenciáveis de difeomorfismos de Anosov na classe de conjugação topológica do automorfismo hiperbólico do toro G(x,y) =(x+y,x); (iii) Classes de conjugação diferenciáveis de difeomorfismos da circunferência com número de rotação igual ao inverso do número de ouro e que são pontos fixos do operador renormalização

Anosov diffeomorphisms, renormalization and tilings

In this thesis, we prove a one-to-one correspondence between C^{1+} smooth conjugacy classes of circle diffeomorphisms that are C^{1+} fixed points of renormalization and C^{1+} conjugacy classes of Anosov diffeomorphisms whose Sinai-Ruelle-Bowen measure is absolutely continuous with respect to Lebesgue measure. Furthermore, we use ratio functions to parametrize the infinite dimensional space of C^{1+} smooth conjugacy classes of circle diffeomorphisms that are C^{1+} fixed points of renormalization. We introduce the notion of γ-tilings and we prove a one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, (ii) affine classes of γ-tilings and (iii) solenoid functions. The solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences

Anosov diffeomorphisms and tilings

We consider a toral Anosov automorphism G : T → T given by G(x, y) = (ax + y; x), where a > 1 is a fixed integer, and introduce the notion of γ-tiling to prove the existence of a one-to-one correspondence between (i) smooth conjugacy classes of Anosov diffeomorphisms with invariant measure absolutely continuous with respect to the Lebesgue measure and topologically conjugate to G, (ii) affine classes of - tilings and (iii) solenoid functions. Solenoid functions provide a parametrization of the infinite dimensional space of the mathematical objects described in these equivalences. This talk is based on a joint work with Alberto Pintoinfo:eu-repo/semantics/publishedVersio

Renormalization of circle diffeomorphism sequences and markov sequences

We show a one-to-one correspondence between circle diffeomorphism sequences that are C^{ 1+n}-periodic points of renormalization and smooth Markov sequences.We thank the financial support of LIAAD–INESC TEC through program PEst, USP-UP project, Faculty of Sciences, University of Porto, Calouste Gulbenkian Foundation, FEDER and COMPETE Programmes, PTDC/MAT/121107/2010 and Fundação para a Ciência e a Tecnologia (FCT). J. P.Almeida acknowledges the FCT support given through Grant SFRH/PROTEC/49754/2009