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It is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos. The complementary set to the stability intervals is a Cantor set of fractal dimension D = 0.87. This exponent is found to be universal for a large class of functions.PACS numbers: 03.20.+i, 47.20.+m, 74.50.+r The transition to chaos in dynamical systems has been intensively studied by utilizing discrete maps. In particular, Feigenbaum' found that the bifurcation route to chaos, in which the transition is governed by one external parameter, is characterized by universal indices. Recently another route, namely through quasiperiodic behavior, has been studied by several authors. ' ' This transition can be established by varying two frequencies and may be studied by means of the so-called circle map, f(e) = e+n -(It/2~) sin(2~v).The ratio between the frequencies is given by the winding number WPC, n) = lim n-'[f" (0) -&].(2) n In numerical studies Shenker' found that when the ratio W approaches the reciprocal Golden mean the mapping exhibits unusual scaling behavior at the critical point (K =1) where chaos sets in. This transition has been elegantly treated by means of a renormalization-group technique by Feigenbaum, Kadanoff, and Shenker' and Rand eI; al. ' Our objective is quite different. We have studied the global mode-locking phenomenon at the critical point, K=1. A similar mode locking has been observed experimentally in a Josephson junction when an external frequency is applied. ' By numerical iterations of the mapping (1) at different values of 0 we have found evidence for mode locking of the mapping at every single rational value of W. Also, we find that the stability intervals for the different rational values fill up the whole 0 axis. The steps in the 8' vs 0 function thus form a comPlete devil's staircase, of a type which has been found in quite different contexts, such as the one-dimensional Ising model' and the Frenkel-Kontorowa model. ' The complementaryset (on the 0 axis) to a complete devil's staircase is a Cantor set of fractal dimension D smaller than or equal to 1. For the staircase of the circle map we have found D =0.87 but this number seems to be universal for a large class of functions. The dimension D is therefore a bona fide critical index characterizing the transition to chaos.The mapping (1) can be considered as the reduced (due to dissipation} one-dimensional map of a more general map of the plane onto itself. 'The iteration of (1) from a given starting point 8, converges towards a limit cycle or an aperiodic trajectory. A limit cycle is characterized by a rational winding number W =P/Q, where Q is the period of the cylcle and P is the number of sweeps through the unit interval [0; 1] in a cycle when the mapping (1) is considered modulo 1.We denote the interval in 0 for which the iteration converges to a limit cycle P/Q as AQ (P/Q). As shown by Herman' for 0&%&1 the iteration locks in to every single rational number in a finite interval 40. Ho...

Abstract. In stochastic finance, one traditionally considers the return as a competitive measure of an asset, i.e., the profit generated by that asset after some fixed time span ∆t, say one week or one year. This measures how well (or how bad) the asset performs over that given period of time. It has been established that the distribution of returns exhibits "fat tails" indicating that large returns occur more frequently than what is expected from standard Gaussian stochastic processes [1,2,3]. Instead of estimating this "fat tail" distribution of returns, we propose here an alternative approach, which is outlined by addressing the following question: What is the smallest time interval needed for an asset to cross a fixed return level of say 10%? For a particular asset, we refer to this time as the investment horizon and the corresponding distribution as the investment horizon distribution. This latter distribution complements that of returns and provides new and possibly crucial information for portfolio design and risk-management, as well as for pricing of more exotic options. By considering historical financial data, exemplified by the Dow Jones Industrial Average, we obtain a novel set of probability distributions for the investment horizons which can be used to estimate the optimal investment horizon for a stock or a future contract. PACS.Financial data have been recorded for a long time as they represent an invaluable source of information for statistical investigations of financial markets. In the early days of stochastic finance it was argued that the distribution of returns (see definition below) of an asset should follow a normal (Gaussian) distribution [4,5]. However, by analysing large, and often high-frequency, financial data sets, it has been established that these distributions on short time scales -typically less then a month, or so -can posses so-called "fat-tails", i.e. distributions that show strong deviations from that of a Gaussian [1,2,3] with higher probabilities for large events. This is similar to the distributions found for turbulence in air and fluids which have led to comparisons between the statistics of financial markets and that of turbulent fluids [5,6,7,8]. In turbulence, one obtains stretched exponential distributions which find their analogy in finance when considering higher order correlations of the asset price [9,10].In order to get a deeper understanding of the fluctuation of financial markets it is important to supplement this established information of fluctuations in the returns with alternative measures. In the present paper, we therefore ask the following "inverse" question: "What is the typical time span needed to generate a fluctuation or a movement a Email address: ingves@nordita.dk b Email address: mhjensen@nbi.dk c Email address: johansen@nbi.dk Correspondence to: Ingve Simonsen (in the price) of a given size" [11,12,13,14,15,16]. Given a fixed log-return barrier, ρ, of a stock or an index as well as a fixed investment date, the corresponding time span is estimate...