Digital Object Identifier (DOI) 10.1007/s00220-010-1077-9 Commun. Math. Phys. 298, 741–756 (2010) Communications in Mathematical Physics On Dirac Physical Measures for Transitive Flows Radu Saghin 1 , Wenxiang Sun 2 , Edson Vargas 1 1 Departamento de Matematica, IME-USP, Sao Paulo, Brasil. E-mail: rsaghin@crm.cat 2 School of Mathematical Sciences, PKU, Beijing, China Received: 7 September 2009 / Accepted: 16 March 2010 Published online: 23 June 2010 – © Springer-Verlag 2010 Abstract: We discuss some examples of smooth transitive flows with physical measures supported at fixed points. We give some conditions under which stopping a flow at a point will create a Dirac physical measure at that indifferent fixed point. Using the Anosov- Katok method, we construct transitive flows on surfaces with the only ergodic invariant probabilities being Dirac measures at hyperbolic fixed points. When there is only one such point, the corresponding Dirac measure is necessarily the only physical measure with full basin of attraction. Using an example due to Hu and Young, we also construct a transitive flow on a three-dimensional compact manifold without boundary, with the only physical measure the average of two Dirac measures at two hyperbolic fixed points. 1. Introduction Let M be a compact manifold, and f : M → M a continuous map. A probability measure μ on M is invariant under f if for any measurable set A ⊂ M we have μ( f -1 ( A)) = μ( A). The basin of attraction of μ, denoted B(μ), is the set of points x ∈ M such that lim n→∞ 1 n (δ x + δ f (x ) + δ f 2 (x ) + ··· + δ f n-1 (x ) ) = μ, where δ y is the Dirac measure at the point y ∈ M , and the limit is with respect to the weak* topology. This means that for any continuous observable α : M → R, the aver- age of α along the orbit of x will converge to the integral of α with respect to μ, which makes this notion useful in applications. We say that the invariant measure μ is physical if B(μ) has positive Lebesgue measure on M . One has similar definitions for the case of flows. If φ is a continuous flow on the compact manifold M , then a probability measure μ is invariant if μ(φ t ( A)) = μ( A),