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  • 51.
    Edwards, Samuel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Renormalization of integrals of products of Eisenstein series and analytic continuation of representationsManuscript (preprint) (Other academic)
    Abstract [en]

    We combine Zagier’s theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL(2, R) due to Bernstein and Reznikov to study triple products of Eisenstein series of generic (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

  • 52.
    Edwards, Samuel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    The Rate of Mixing for Diagonal Flows on Spaces of Affine Lattices2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 53.
    Edwards, Samuel Charles
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Some applications of representation theory in homogeneous dynamics and automorphic functions2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of an introduction and five papers in the general area of dynamics and functions on homogeneous spaces. A common feature is that representation theory plays a key role in all articles.

    Papers I-IV are concerned with the effective equidistribution of translates of pieces of subgroup orbits in quotient spaces of semisimple Lie groups by discrete subgroups. In Paper I we focus on finite-volume quotients of SL(2,C) and study the speed of equdistribution for expanding translates orbits of horospherical subgroups. Paper II also studies the effective equidistribution of translates of horospherical orbits, though now in the setting of a quotient of a general semisimple Lie group by a lattice subgroup. Like Paper II, Paper III considers effective equidistribution in quotients of general semisimple Lie groups, but now studies translates of orbits of symmetric subgroups. In all these papers we show that the translates equidistribute with the same exponential rate as for the decay of the corresponding matrix coefficients of the translating subgroup. In Paper IV we consider the effective equidistribution of translates of pieces of horospheres in infinite-volume quotients of groups SO(n,1) by geometrically finite subgroups, and improve the dependency on the spectral gap for certain known effective equidistribution results.

    In Paper V we study the Fourier coefficients of Eisenstein series for generic non-cocompact cofinite Fuchsian groups. We use Zagier's renormalization of certain divergent integrals to enable use of the so-called triple product method, and then combine this with the analytic continuation of irreducible representations of SL(2,R) due to Bernstein and Reznikov.

    List of papers
    1. On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)
    Open this publication in new window or tab >>On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)
    2017 (English)In: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 11, p. 155-188Article in journal (Refereed) Published
    Abstract [en]

    Let Gamma be a lattice in G = SL(2, C). We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in Gamma\G. Our method of proof relies on the theory of unitary representations.

    Keywords
    Effective equidistribution, translates, horospheres
    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:uu:diva-315688 (URN)10.3934/jmd.2017008 (DOI)000396538600008 ()
    Funder
    Swedish Research Council, 621-2011-3629Göran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
    Available from: 2017-02-18 Created: 2017-02-18 Last updated: 2018-04-11Bibliographically approved
    2. On the rate of equidistribution of expanding translates of horospheres in Γ\G
    Open this publication in new window or tab >>On the rate of equidistribution of expanding translates of horospheres in Γ\G
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    Let G be a semisimple Lie group and Γ a lattice in G. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space Γ\G.

    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:uu:diva-347856 (URN)
    Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
    3. On the equidistribution of translates of orbits of symmetric subgroups in Γ\G
    Open this publication in new window or tab >>On the equidistribution of translates of orbits of symmetric subgroups in Γ\G
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    We use the method of Burger to study the rate of equidistribution for translates of orbits of symmetric subgroups in homogeneous spaces Γ\G for semisimple Lie groups G and lattices Γ.

    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:uu:diva-347857 (URN)
    Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
    4. Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groups
    Open this publication in new window or tab >>Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groups
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    We use the method of Burger to study the rate of equidistribution for translates of pieces of horospheres in Γ\ SO0(n, 1) for geometrically finite discrete subgroups Γ < SO0(n, 1) with infinite covolume.

    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:uu:diva-347858 (URN)
    Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
    5. Renormalization of integrals of products of Eisenstein series and analytic continuation of representations
    Open this publication in new window or tab >>Renormalization of integrals of products of Eisenstein series and analytic continuation of representations
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    We combine Zagier’s theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL(2, R) due to Bernstein and Reznikov to study triple products of Eisenstein series of generic (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:uu:diva-347859 (URN)
    Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-24
  • 54.
    Ekström, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Selected Problems in Financial Mathematics2004Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.

    In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility.

    In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.

    Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.

    A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.

    In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.

    Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.

    List of papers
    1. Properties of American option prices
    Open this publication in new window or tab >>Properties of American option prices
    2004 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 114, no 2, p. 265-278Article in journal (Refereed) Published
    Abstract [en]

    We investigate some properties of American option prices in the setting of time- and level-dependent volatility. The properties under consideration are convexity in the underlying stock price, monotonicity and continuity in the volatility and time decay. Some properties are direct consequences of the corresponding properties of European option prices that are already known, and some follow by writing solutions of different stochastic differential equations as time changes of the same Brownian motion.

    National Category
    Natural Sciences
    Identifiers
    urn:nbn:se:uu:diva-92190 (URN)10.1016/j.spa.2004.05.002 (DOI)
    Available from: 2004-10-08 Created: 2004-10-08 Last updated: 2017-12-14Bibliographically approved
    2. The perpetual American put option in a level-dependent volatility model
    Open this publication in new window or tab >>The perpetual American put option in a level-dependent volatility model
    2003 In: Journal of Applied Probability, ISSN 0021-9002, Vol. 40, no 3, p. 783-789Article in journal (Refereed) Published
    Identifiers
    urn:nbn:se:uu:diva-92191 (URN)
    Available from: 2004-10-08 Created: 2004-10-08Bibliographically approved
    3. Russian options with a finite time horizon
    Open this publication in new window or tab >>Russian options with a finite time horizon
    2004 In: Journal of Applied Probability, ISSN 0021-9002, Vol. 41, no 2, p. 313-326Article in journal (Refereed) Published
    Identifiers
    urn:nbn:se:uu:diva-92192 (URN)
    Available from: 2004-10-08 Created: 2004-10-08Bibliographically approved
    4. Properties of game options
    Open this publication in new window or tab >>Properties of game options
    Article in journal (Refereed) Submitted
    Identifiers
    urn:nbn:se:uu:diva-92193 (URN)
    Available from: 2004-10-08 Created: 2004-10-08Bibliographically approved
    5. Superreplication of options on several underlying assets
    Open this publication in new window or tab >>Superreplication of options on several underlying assets
    2005 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 42, no 1, p. 27-38Article in journal (Refereed) Published
    National Category
    Natural Sciences
    Identifiers
    urn:nbn:se:uu:diva-92194 (URN)
    Available from: 2004-10-08 Created: 2004-10-08 Last updated: 2017-12-14Bibliographically approved
    6. Convexity of the optimal stopping boundary for the American put option
    Open this publication in new window or tab >>Convexity of the optimal stopping boundary for the American put option
    2004 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 299, no 1, p. 147-156Article in journal (Refereed) Published
    Abstract [en]

    We show that the optimal stopping boundary for the American put option is convex in the standard Black–Scholes model. The methods are adapted from ice-melting problems and rely upon studying the behavior of level curves of solutions to certain parabolic differential equations.

    National Category
    Natural Sciences
    Identifiers
    urn:nbn:se:uu:diva-92195 (URN)10.1016/j.jmaa.2004.06.018 (DOI)
    Available from: 2004-10-08 Created: 2004-10-08 Last updated: 2017-12-14Bibliographically approved
  • 55.
    Ekström, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Tysk, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    The Black-Scholes equation in stochastic volatility models2010In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 368, no 2, p. 498-507Article in journal (Refereed)
    Abstract [en]

    We study the Black-Scholes equation in stochastic volatility models. In particular, we show that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate function spaces. On the other hand, if the boundary is non-attainable, then the boundary behaviour is not needed to guarantee uniqueness, but is nevertheless very useful for instance from a numerical perspective.

  • 56.
    Enstedt, Mattias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    On the existence and non-existence of a solution to the Hartree-Fock equations with an external magnetic field2007Licentiate thesis, monograph (Other academic)
  • 57.
    Enstedt, Mattias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Melgaard, Michael
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Existence of infinitely many distinct solutions to the quasi-relativistic Hartree-Fock equations2009Report (Other academic)
  • 58.
    Eriksson, Jonatan
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    On the pricing equations of some path-dependent options2006Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C1-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.

    List of papers
    1. Monotonicity in the volatility of single-barrier option prices
    Open this publication in new window or tab >>Monotonicity in the volatility of single-barrier option prices
    2006 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 9, no 6, p. 987-996Article in journal (Refereed) Published
    Abstract [en]

    We generalize earlier results on barrier options for puts and calls and log-normal stock processes to general local volatility models and convex contracts. We show that Γ ≥ 0, that Δ has a unique sign and that the option price is increasing with the volatility for convex contracts in the following cases:

    If the risk-free rate of return dominates the dividend rate, then it holds for up-and-out options if the contract function is zero at the barrier and for down-and-in options in general.

    If the risk-free rate of return is dominated by the dividend rate, then it holds for down-and-out options if the contract function is zero at the barrier and for up-and-in options in general.

    We apply our results to show that a hedger who misspecifies the volatility using a time-and-level dependent volatility will super-replicate any claim satisfying the above conditions if the misspecified volatility dominates the true (possibly stochastic) volatility almost surely.

    Keywords
    Barrier option, convexity, volatility, parabolic equation
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-93999 (URN)10.1142/S0219024906003822 (DOI)
    Available from: 2006-02-17 Created: 2006-02-17 Last updated: 2017-12-14Bibliographically approved
    2. When American options are European
    Open this publication in new window or tab >>When American options are European
    (English)In: Decisions in Economics and Finance , ISSN 1593-8883Article in journal (Refereed) Submitted
    Identifiers
    urn:nbn:se:uu:diva-94000 (URN)
    Available from: 2006-02-17 Created: 2006-02-17 Last updated: 2009-02-26Bibliographically approved
    3. On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing
    Open this publication in new window or tab >>On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing
    2007 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 101, no 1, p. 148-160Article in journal (Refereed) Published
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-94001 (URN)
    Available from: 2006-02-17 Created: 2006-03-19 Last updated: 2017-12-14Bibliographically approved
    4. Explicit pricing formulas for turbo warrants
    Open this publication in new window or tab >>Explicit pricing formulas for turbo warrants
    (English)In: Risk magazine, ISSN 1082-6394Article in journal (Refereed) Submitted
    Identifiers
    urn:nbn:se:uu:diva-94002 (URN)
    Available from: 2006-02-17 Created: 2006-02-17 Last updated: 2009-02-26Bibliographically approved
  • 59.
    Eriksson, Olle
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Hodge Decomposition for Manifolds with Boundary and Vector Calculus2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 60.
    Ernst, Thomas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Expansion formulas for Apostol type Q-Appell polynomials, and their special cases2018In: Le Matematiche, ISSN 2037-5298, E-ISSN 0373-3505, Vol. 73, no 1, p. 3-24Article in journal (Refereed)
    Abstract [en]

    We present identities of various kinds for generalized q-ApostolBernoulli and Apostol-Euler polynomials and power sums, which resemble q-analogues of formulas from the 2009 paper by Liu and Wang. These formulas are divided into two types: formulas with only q-ApostolBernoulli, and only q-Apostol-Euler polynomials, or so-called mixed formulas, which contain polynomials of both kinds. This can be seen as a logical consequence of the fact that the q-Appell polynomials form a commutative ring. The functional equations for Ward numbers operating on the q-exponential function, as well as symmetry arguments, are essential for many of the proofs. We conclude by finding multiplication formulas for two q-Appell polynomials of general form. This brings us to the q-H polynomials, which were discussed in a previous paper.

  • 61.
    Eyiah-Donkor, Emmanuel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Horizon-unbiased Utility of Wealth and Consumption2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 62.
    Falck, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Asymptotic Expansions of Integrals and the Method of Steepest Descent2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 63.
    Fernström, Rickard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Farey Fractions2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 64.
    Festuccia, Guido
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
    Hansen, Dennis
    Univ Copenhagen, Niels Bohr Inst, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark..
    Hartong, Jelle
    Univ Libre Bruxelles, Phys Theor & Math Inst, CP 231, B-1050 Brussels, Belgium.;Univ Libre Bruxelles, Int Solvay Inst, CP 231, B-1050 Brussels, Belgium..
    Obers, Niels A.
    Univ Copenhagen, Niels Bohr Inst, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark..
    Torsional Newton-Cartan geometry from the Noether procedure2016In: PHYSICAL REVIEW D, ISSN 2470-0010, Vol. 94, no 10, article id 105023Article in journal (Refereed)
    Abstract [en]

    We apply the Noether procedure for gauging space-time symmetries to theories with Galilean symmetries, analyzing both massless and massive (Bargmann) realizations. It is shown that at the linearized level the Noether procedure gives rise to (linearized) torsional Newton-Cartan geometry. In the case of Bargmann theories the Newton-Cartan form M-mu couples to the conserved mass current. We show that even in the case of theories with massless Galilean symmetries it is necessary to introduce the form M-mu and that it couples to a topological current. Further, we show that the Noether procedure naturally gives rise to a distinguished affine (Christoffel type) connection that is linear in M-mu and torsionful. As an application of these techniques we study the coupling of Galilean electrodynamics to TNC geometry at the linearized level.

  • 65.
    Figueras, Jordi-Lluis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Lilja, Dan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Existence of non-smooth bifurcations of uniformly hyperbolic invariant manifolds in skew product systems2018In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 31, no 12, p. 5573-5588Article in journal (Refereed)
    Abstract [en]

    In this paper we study the anti-integrable limit scenario of skew product systems. We consider a generalization of such systems based on the Frenkel-Kontorova model, and prove that under certain mild regularity conditions on the potential the structure of the orbits is of Cantor type. From our results we deduce the existence of the non-smooth folding bifurcation (conjectured by Figueras and Haro (2015 Chaos 25 123119)). Lastly we present some results which are useful in determining if a potential satisfies the regularity conditions required for the Cantor sets of orbits to exist and are also of independent interest. We also prove the existence of orbits with any fibered rotation number in systems of both one and two degrees of freedom. In particular, our results also apply to two-dimensional maps with degenerate potentials (vanishing second derivative), so extending the results of existence of Cantori to more general twist maps.

  • 66.
    Fjellström, Carmina
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Information Diffusion-Based Modeling of Oil Futures Prices2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 67.
    Freyland, Sara
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The Happy Ending Problem and its connection to Ramsey theory2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 68.
    Gaidashev, Denis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Renormalization for Lorenz maps of monotone combinatorial types2019In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 39, no 1, p. 132-158Article in journal (Refereed)
    Abstract [en]

    Lorenz maps are maps of the unit interval with one critical point of order rho > 1 and a discontinuity at that point. They appear as return maps of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics and a sufficiently flat critical point, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

  • 69.
    Gaidashev, Denis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Johnson, Tomas
    A numerical study of infinitely renormalizable area-preserving maps2012In: Dynamical systems, ISSN 1468-9367, E-ISSN 1468-9375, Vol. 27, no 3, p. 283-301Article in journal (Refereed)
    Abstract [en]

    It has been shown in Gaidashev and Johnson [D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: stable sets, J. Mod. Dyn. 3(4) (2009), pp. 555–587.] and Gaidashev et al. [D. Gaidashev, T. Johnson, and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, in preparation.] that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these Cantor sets for any two infinitely renormalizable maps is conjugated by a transformation that extends to a differentiable function whose derivative is Hölder continuous of exponent α > 0. In this article we investigate numerically the specific value of α. We also present numerical evidence that the normalized derivative cocycle with the base dynamics in the Cantor set is ergodic. Finally, we compute renormalization eigenvalues to a high accuracy to support a conjecture that the renormalization spectrum is real.

  • 70.
    Gaidashev, Denis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Johnson, Tomas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets2009In: Journal of Modern Dynamics, ISSN 1930-5311, Vol. 3, no 4, p. 555-587Article in journal (Refereed)
    Abstract [en]

    It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a ``universal'' area-preserving map $F_*$ ---  a map with orbits of all binary periods $2^k, k \in \fN$.  In this paper, we consider {\it infinitely renormalizable} maps --- maps on the renormalization stable manifold in some neighborhood of  $F_*$ --- and study their dynamics. For all such infinitely renormalizable maps in a neighborhood of the fixed point $F_*$ we prove the existence of a ``stable'' invariant Cantor set  $\cC^\infty_F$ such that the Lyapunov exponents of $F \arrowvert_{\cC^\infty_F}$ are zero, and whose Hausdorff dimension satisfies$${\rm dim}_H(\cC_F^{\infty}) < 0.5324.$$  We also show that there exists a submanifold, $\bW_\omega$, of finite codimension in the renormalization local stable manifold, such that for all $F\in\bW_\omega$ the set $\cC^\infty_F$ is  ``weakly rigid'': the dynamics of any two maps in this submanifold, restricted to the stable set $\cC^\infty_F$, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.

  • 71.
    Galias, Zbigniew
    et al.
    AGH Univ Sci & Technol, Dept Elect Engn, Mickiewicza 30, PL-30059 Krakow, Poland.
    Tucker, Warwick
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor2019In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 266, no 5, p. 2408-2434Article in journal (Refereed)
    Abstract [en]

    In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.

  • 72.
    Ghaderi, Hazhar
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Mountain Pass Theorems with Ekeland’s Variational Principle and an Application tothe Semilinear Dirichlet Problem2011Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 73.
    Ghafari, Paria
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Dimension Reduction and Adaptivity to Price Basket Options2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 74.
    Guckenheimer, John
    et al.
    Cornell University, Department of Mathematics.
    Johnson, Tomas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Meerkamp, Philipp
    Cornell University, Department of Mathematics.
    Rigorous enclosures of a slow manifold2012In: SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, E-ISSN 1536-0040, Vol. 11, no 3, p. 831-863Article in journal (Refereed)
    Abstract [en]

    Slow-fast dynamical systems have two time scales and an explicit parameter representing the ratio of these time scales. Locally invariant slow manifolds along which motion occurs on the slow time scale are a prominent feature of slow-fast systems. This paper introduces a rigorous numerical method to compute enclosures of the slow manifold of a slow-fast system with one fast and two slow variables. A triangulated first order approximation to the two dimensional invariant manifold is computed “algebraically.” Two translations of the computed manifold in the fast direction that are transverse to the vector field are computed as the boundaries of an initial enclosure. The enclosures are refined to bring them closer to each other by moving vertices of the enclosure boundaries one at a time. As an application we use the method to prove the existence of tangencies of invariant manifolds in the problem of singular Hopf bifurcation and to give bounds on the location of one such tangency.

  • 75.
    Guo, Qi
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Minkowski Measure of Asymmetry and Minkowski Distance for Convex Bodies2004Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let C and D be n-dimensional convex bodies. Denote by As(C) and As(D) the Minkowski measures of asymmetry of C and D resp. and by d(C,D) the Minkowski distance between C and D.

    In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that d(C,D) < n(As(C) + As(D))/2 for all convex bodies C,D.

    In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, d(C,D) < (n-1) min(As(C),As(D)) + n.

    In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form conv(C,p) where C is a convex set with known measure of asymmetry and p is a point outside C.

    In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) > n - ε for 0 < ε < 1/δ where δ = 8(n+1), then d(C,S) < 1 + 8(n+1) ε .

  • 76.
    Gustafsson, William
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Evaluating the Longstaff-Schwartz method for pricing of American options2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 77.
    Hallgren, Christoffer
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Temperature effects on ant activity: Analysis of a mathematical model2012Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 78.
    Hansen, Peder
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Pricing exotic power options2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 79.
    Haro, Alex
    et al.
    Univ Barcelona, Dept Matemat & Informat, Gran Via 585, E-08007 Barcelona, Spain.
    Luque, Alejandro
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    A-posteriori KAM theory with optimal estimates for partially integrable systems2019In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 266, no 2-3, p. 1605-1674Article in journal (Refereed)
    Abstract [en]

    In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invariant tori for n-DOF Hamiltonian systems with additional n - d independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.

  • 80.
    Hasan, Harady
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Elliptic Curves: A journey through theory and its applications2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 81.
    Haseeb, Hayat
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    A Comparison of Models for Oil Futures2013Independent thesis Advanced level (degree of Master (Two Years)), 10 credits / 15 HE creditsStudent thesis
  • 82.
    Hofmann, Steve
    et al.
    Univ Missouri, Dept Math, Columbia, MO 65211 USA..
    Le, Phi
    Syracuse Univ, Math Dept, 215 Carnegie Bldg, Syracuse, NY 13244 USA..
    Maria Martell, Jose
    CSIC, CSIC, Inst Ciencias Matemat, UAM,UC3M,UCM, C Nicolas Cabrera 13-15, E-28049 Madrid, Spain..
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The weak-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY2017In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 3, p. 513-558Article in journal (Refereed)
    Abstract [en]

    Let E subset of Rn+1, n >= 2, be an Ahlfors-David regular set of dimension n. We show that the weak- A 1 property of harmonic measure, for the open set Omega: =Rn+1 \ E, implies uniform rectifiability of E. More generally, we establish a similar result for the Riesz measure, p-harmonic measure, associated to the p-Laplace operator, 1 < p < infinity.

  • 83.
    Hu, Jiaqun
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    An Empirical Comparison of Different Approaches in Portfolio Selection2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 84.
    Husin, Axel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    The Hilbert Transform2013Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 85.
    Hynd, Ryan
    et al.
    MIT, Dept Math, Cambridge, MA 02139 USA.
    Lindgren, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Extremal functions for Morrey's inequality in convex domains2019In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 375, no 3-4, p. 1721-1743Article in journal (Refereed)
    Abstract [en]

    For a bounded domain Omega subset of R-n and p > n, Morrey's inequality implies that there is c > 0 such that c parallel to u parallel to(p)(infinity) <= integral(Omega) vertical bar Du vertical bar(p) dx for each u belonging to the Sobolev space W-0(1,p) (Omega). We show that the ratio of any two extremal functions is constant provided that Omega is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the p-Laplacian.

  • 86.
    Hynd, Ryan
    et al.
    Univ Penn, Dept Math, Philadelphia, PA 19104 USA.
    Lindgren, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Uppsala Univ, Dept Math, S-75106 Uppsala, Sweden.
    LIPSCHITZ REGULARITY FOR A HOMOGENEOUS DOUBLY NONLINEAR PDE2019In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 51, no 4, p. 3606-3624Article in journal (Refereed)
    Abstract [en]

    We study the doubly nonlinear PDE vertical bar partial derivative u vertical bar(p-2)partial derivative(t)u - div(vertical bar del u vertical bar(p-2)del u) = 0. This equation arises in the study of extremals of Poincare inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and Holder continuity in time of order (p - 1)/p for viscosity solutions. As an application of our estimates, we obtain pointwise control of the large time behavior of solutions.

  • 87.
    Israelsson, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Basic quantisation rules of semiclassical analysis2016Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Semiclassical analysis is the study of how to connect classical mechanics with quantummechanics in a mathematically rigorous way. What is crucial for quantum mechanics isthat the operators that occur are self-adjoint and map a subset of L2 into L2. This hasbeen proven in this thesis. Also, a technique of asymptotic expansion of functions interms of Planck's constant h is developed, in order to study the so called semiclassicallimit (i.e. the limit as h approaches 0) We also develop different approaches toquantisation, and a calculus of operators (i.e. quantum observables) based on acalculus for their corresponding symbols (or classical observables).

  • 88. Israelsson, Anders
    Mathematical Foundations of Quantum Mechanics2013Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 89.
    Ivarsson, Björn
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Interior regularity of solutions to a compelx Monge--Ampère equation2002In: Arkiv för matematik, Vol. 40, p. 275-300Article in journal (Refereed)
  • 90.
    Ivarsson, Björn
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Regularity and uniqueness of solutions to boundary blow-up problems for the complex Monge--Ampère operator2006In: Bulletin of the Polish Academy of Sciences, Vol. 54, p. 13-25Article in journal (Refereed)
  • 91.
    Ivarsson, Björn
    et al.
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Matero, Jerk
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    The blow-up rate of solutions to boundary blow-up problems for the complex Monge--Ampère operator2006In: Manuscripta Mathematica, Vol. 120, p. 325-345Article in journal (Refereed)
  • 92.
    Jamal, Muhammad
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Nonlinear phenomena and resource exploitation in group living organisms2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 93.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Konstantopoulos, Takis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Yuan, Linglong
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Johannes Gutenberg Univ Mainz, Inst Math, Staudingerweg 9, D-55099 Mainz, Germany..
    On a representation theorem for finitely exchangeable random vectors2016In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 442, no 2, p. 703-714Article in journal (Refereed)
    Abstract [en]

    A random vector X = (X1, ... ,X-n) with the X-i taking values in an arbitrary measurable space (S, Sp) is exchangeable if its law is the same as that of (X-sigma(1), ... ,X-sigma(n)) for any permutation a. We give an alternative and shorter proof of the representation result (Jaynes [6] and Kerns and Szekely [9]) stating that the law of X is a mixture of product probability measures with respect to a signed mixing measure. The result is "finitistic" in nature meaning that it is a matter of linear algebra for finite S. The passing from finite S to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof ("canonical mixing measure") by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on S; but for S =, one can choose a mixing measure on R-n.

  • 94.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Viola, Alfredo
    Univ Republica, Montevideo, Uruguay..
    A Unified Approach to Linear Probing Hashing with Buckets2016In: Algorithmica, ISSN 0178-4617, E-ISSN 1432-0541, Vol. 75, no 4, p. 724-781Article in journal (Refereed)
    Abstract [en]

    We give a unified analysis of linear probing hashing with a general bucket size. We use both a combinatorial approach, giving exact formulas for generating functions, and a probabilistic approach, giving simple derivations of asymptotic results. Both approaches complement nicely, and give a good insight in the relation between linear probing and random walks. A key methodological contribution, at the core of Analytic Combinatorics, is the use of the symbolic method (based on q-calculus) to directly derive the generating functions to analyze.

  • 95.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A piecewise contractive dynamical system and Phragmèn's election method2019In: Bulletin de la Société Mathématique de France, ISSN 0037-9484, E-ISSN 2102-622X, Vol. 147, no 3, p. 395-441Article in journal (Refereed)
    Abstract [en]

    We prove some basic results for a dynamical system given by a piece-wise linear and contractive map on the unit interval that takes two possible values at a point of discontinuity. We prove that there exists a universal limit cycle in the non-exceptional cases, and that the exceptional parameter set is very tiny in terms of gauge functions. The exceptional two-dimensional parameter is shown to have Hausdorff-dimension one. We also study the invariant sets and the limit sets; these are sometimes different and there are several cases to consider. In addition, we prove the existence of a unique invariant measure. We apply some of our results for the dynamical system, involving a study of rational and irrational rotation numbers, to a combinatorial problem involving an election method suggested by Phragmen, and we show that the proportion of elected seats for each party converges to a limit, which is a rational number except for a very small exceptional set of parameters.

  • 96.
    Jawaid, Hassan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Applications of the Heath, Jarrow and Morton (HJM) model to energy markets2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 97.
    Johansson, Anders
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Jordan, Thomas
    Bristol University.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Pollicott, Mark
    Multifractal analysis of non-uniformly hyperbolic systems2010In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 177, no 1, p. 125-144Article in journal (Refereed)
    Abstract [en]

    We prove a multifractal formalismfor Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville-Pomeau map.

  • 98. Johansson, Anders
    et al.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Square Summability of Variations and Convergence of the Transfer Operator2008In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 28, no 4, p. 1145-1151Article in journal (Refereed)
    Abstract [en]

    In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer Math. Soc. 353(l) (2001), 327-347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Oberg. Math. Res. Lett. 10(5-6) (2003), 587-601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys. 11 (1991), 279-307], under square summability of variations.

  • 99. Johansson, Anders
    et al.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Square summability of variations of g-functions and uniqueness of g-measures2003In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 10, no 5-6, p. 587-601Article in journal (Refereed)
  • 100. Johansson, Anders
    et al.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Pollicott, Mark
    Ergodic Theory of Kusuoka Measures2017In: Journal of Fractal Geometry, ISSN 2308-1309, Vol. 4, no 2, p. 185-214Article in journal (Refereed)
    Abstract [en]

    In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role (cf. \cite{kusuoka2}, \cite{kajino}, \cite{str3}). Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the H\"older continuous functions, as well as the highly discontinuous $g$-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpi\'nski gaskets.

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